Hepeng Yao

David Clément

Anna Minguzzi

Patrizia Vignolo

Laurent Sanchez-Palencia

Tan's Contact for Trapped Lieb-Liniger Bosons at Finite Temperature

The universal Tan relations connect a variety of microscopic features of many-body quantum systems with two-body contact interactions to a single quantity, called the contact. The latter has become pivotal in the description of quantum gases. We provide a complete characterization of the Tan contact of the harmonically trapped Lieb-Liniger gas for arbitrary interactions and temperature. Combining thermal Bethe ansatz, localdensity approximation, and exact quantum Monte Carlo calculations, we show that the contact is a universal function of only two scaling parameters, and determine the scaling function. We find that the temperature dependence of the contact, or equivalently the interaction dependence of the entropy, displays a maximum. The presence of this maximum provides an unequivocal signature of the crossover to the fermionized regime and it is accessible in current experiments.

Describing strongly correlated quantum systems from microscopic models and first principles is a central challenge for modern many-body physics. The derivation of universal relations in systems governed by contact interactions is an example of such an approach [START_REF] Tan | Generalized virial theorem and pressure relation for a strongly correlated Fermi gas[END_REF][START_REF] Tan | Energetics of a strongly correlated Fermi gas[END_REF]. Pointlike interactions induce a characteristic singularity of the many-body wavefunction at short interparticle distance and, correspondingly, algebraically decaying momentum tails, n(k) C/k 4 [START_REF] Olshanii | Short-distance correlation properties of the Lieb-Liniger system and momentum distributions of trapped one-dimensional atomic gases[END_REF][START_REF] Tan | Large momentum part of fermions with large scattering length[END_REF]. The 1/k 4 scaling is universal and holds irrespective of the quantum statistics, dimension, temperature, and interaction strength. Furthermore, the weight C of the tails, known as Tan's contact, contains a wealth of information about many quantities characterizing the specific state, e.g. the interaction energy, the pair correlation function, the free-energy dependence on interactions, and the relation between pressure and energy density [START_REF] Tan | Generalized virial theorem and pressure relation for a strongly correlated Fermi gas[END_REF][START_REF] Tan | Energetics of a strongly correlated Fermi gas[END_REF][START_REF] Tan | Large momentum part of fermions with large scattering length[END_REF]. Stemming from the unique possibility to measure it in ultracold gases, the contact has become central to the description of quantum gases. Recent experiments on three-dimensional Fermi and Bose gases have permitted us to validate the universal Tan relations, hence demonstrating that C provides valuable information on a variety of thermodynamic quantities [START_REF] Stewart | Verification of universal relations in a strongly interacting Fermi gas[END_REF][START_REF] Wild | Measurements of Tan's contact in an atomic Bose-Einstein condensate[END_REF][START_REF] Sagi | Measurement of the homogeneous contact of a unitary Fermi gas[END_REF][START_REF] Hoinka | Precise determination of the structure factor and contact in a unitary Fermi gas[END_REF][START_REF] Luciuk | Evidence for universal relations describing a gas with p-wave interactions[END_REF][START_REF] Fletcher | Two-and three-body contacts in the unitary Bose gas[END_REF][START_REF] Laurent | Connecting few-body inelastic decay to quantum correlations in a many-body system: A weakly coupled impurity in a resonant Fermi gas[END_REF].

Interacting one-dimensional (1D) bosons display very different physical regimes at varying interaction strength, from quasicondensates to the emblematic fermionization effect [START_REF] Girardeau | Relationship between systems of impenetrable bosons and fermions in one dimension[END_REF]. So far, the emergence of statistical transmutation in the Tonks-Girardeau regime [START_REF] Paredes | Tonks-Girardeau gas of ultracold atoms in an optical lattice[END_REF][START_REF] Kinoshita | Observation of a onedimensional Tonks-Girardeau gas[END_REF], the suppression of pair correlations [START_REF] Kinoshita | Local pair correlations in one-dimensional Bose gases[END_REF][START_REF] Jacqmin | Sub-Poissonian fluctuations in a 1D Bose gas: From the quantum quasicondensate to the strongly interacting regime[END_REF], and the observation of quantum criticality [START_REF] Yang | Quantum criticality and the Tomonaga-Luttinger liquid in one-dimensional Bose gases[END_REF] have been reported in ultracold atom experiments. However, the experimental characterization of the various quantum degeneracy regimes at finite temperature, identified in Ref. [START_REF] Petrov | Regimes of quantum degeneracy in trapped 1D gases[END_REF], remains challenging. A major difficulty is that most quantities show a smooth monotonic behavior when crossing over different regimes. Understanding whether the contact can provide an efficient probe is one of the motivations of our work.

In most experimental conditions, the gases are confined in longitudinal harmonic traps and thermal effects cannot be neglected. While the homogeneous 1D gas is exactly solvable by Bethe ansatz, the trapped system is not integrable, therefore requiring approximate or ab initio numerical approaches. Previous theoretical studies have investigated the contact for homogeneous bosons at finite temperature [START_REF] Kheruntsyan | Pair correlations in a finite-temperature 1D Bose gas[END_REF][START_REF] Kormos | Expectation values in the Lieb-Liniger Bose gas[END_REF], trapped bosons at zero temperature [START_REF] Olshanii | Short-distance correlation properties of the Lieb-Liniger system and momentum distributions of trapped one-dimensional atomic gases[END_REF][START_REF] Minguzzi | High momentum tail in the Tonks gas under harmonic confinement[END_REF], and at finite temperature in the Tonks-Girardeau limit [START_REF] Vignolo | Universal contact for a Tonks-Girardeau gas at finite temperature[END_REF]. Momentum distributions of strongly interacting, trapped bosons at finite temperature were also computed by quantum Monte Carlo methods [START_REF] Xu | Universal scaling of density and momentum distributions in Lieb-Liniger gases[END_REF].

In this Letter, we provide a complete characterization of Tan's contact of 1D bosons under harmonic confinement for arbitrary interactions, particle number, temperature, and trap frequency, and show that it indeed provides a useful probe of quantum degeneracy regimes. Using a combination of thermal Bethe-ansatz solutions with local-density approximation and exact quantum Monte Carlo calculations, we demonstrate that the contact is a universal function of only two scaling parameters and find the scaling function (see Fig. 1), hence generalizing the results of Ref. [START_REF] Xu | Universal scaling of density and momentum distributions in Lieb-Liniger gases[END_REF]. As a main result, we find that the contact displays a maximum versus the temperature. This behavior is characteristic of the trapped gas with finite, although possibly arbitrarily strong, interactions. We argue that the existence of this maximum is a direct consequence of the dramatic change of correlations and thus provides an unequivocal signature of the crossover to fermionization in the trapped 1D Bose gas. We derive asymptotic limits and discuss a physical picture of the evolution of the contact versus temperature and interaction strength. Finally, we compute the full momentum distributions in various regimes. They show the emergence of the high-momentum tails and assess the experimental observability of our predictions.

Two-parameter scaling.-Consider a 1D Bose gas with repulsive two-body contact interactions, in the presence of the harmonic potential V (x) = mω 2 x 2 /2, with m the particle mass, x the space coordinate, and ω/2π the trap frequency. It is governed by the extended Lieb-Liniger (LL) Hamiltonian

H = j - 2 2m ∂ 2 ∂x 2 j + V (x j ) + g j< δ(x j -x ), (1)
where j and span the set of particles, and g = -2 2 /ma 1D is the coupling constant with a 1D the 1D scattering length. The thermodynamic properties of the interacting gas at the finite temperature T are uniquely determined by the grand potential Ω = -k B T ln Tr e -(H-µN )/kBT , where k B is the Boltzmann constant, N the particle number operator, and µ the chemical potential.

We start from the homogeneous case, V (x) = 0. Using k B T as the unit energy and, correspondingly, the thermal de Broglie wavelength λ T = 2π 2 /mk B T as the unit length, we readily find that the Hamiltonian H/k B T is a function of the unique parameter a 1D /λ T . Since the interactions are short range, Ω is an extensive quantity. It follows that the dimensionless quantity Ω/k B T is a function of the sole intensive, dimensionless parameters µ/k B T and a 1D /λ T , times the length ratio L/λ T . We may thus write

Ω/k B T = (L/λ T ) A h µ/k B T , a 1D /λ T , (2)
with A h a dimensionless function.

For the gas under harmonic confinement, the additional energy scale ω emerges, associated with the length scale a ho = /mω. Within the local-density approximation (LDA), we write the grand potential as the sum of the contributions of slices of homogeneous LL gases with a chemical potential locally shifted by the trap potential energy,

Ω/k B T = ´dx λ T A h [µ -V (x), T, g].
Using Eq. ( 2) and rescaling the position x by the quantity 2 √ πa 2 ho /λ T , we then find

Ω/k B T = a ho /λ T 2 A µ/k B T , a 1D /λ T , (3)
with A a dimensionless function stemming from A h . The scaling forms of the relevant thermodynamic quantities are then readily found from Eq. ( 3). On the one hand, the average particle number N = -∂Ω/∂µ| T,a1D reads

N = (a ho /λ T ) 2 A N µ/k B T , a 1D /λ T . (4)
It follows that the reduced chemical potential µ/k B T is a universal function of only two scaling parameters, namely N λ 2 T /a 2 ho and a 1D /λ T , or, equivalently, ξ γ = -a ho /a 1D √ N , and ξ T = -a 1D /λ T . On the other hand, the contact is expressed using the Tan sweep relation [START_REF] Tan | Large momentum part of fermions with large scattering length[END_REF][START_REF] Barth | Tan relations in one dimension[END_REF][START_REF] Here | we use the normalization condition of the momentum distribution ´dk 2π n(k) = N[END_REF]

, C = (4m/ 2 ) ∂Ω/∂a 1D | T,µ , yielding C = (a 2 ho /a 5 1D )A C µ/k B T , a 1D /λ T ).
Using Eq. ( 4) and writing µ/k B T as a function of ξ γ and ξ T , we find

C = N 5/2 a 3 ho f (ξ γ , ξ T ) , (5)
with f a dimensionless function. In the following, we shall use this two-parameter scaling form. Note that the procedure used to find the scaling form ( 5) is general and can be straightforwardly extended to higher dimensions and Fermi gases.

Scaling function for 1D bosons at finite temperature.-In order to verify the scaling form ( 5) and find the scaling function f for interacting 1D bosons, we use two complementary approaches.

On the one hand, we perform the LDA on the exact solutions of the Yang-Yang (YY) equations [START_REF] Yang | Thermodynamics of a onedimensional system of bosons with repulsive δ-function interaction[END_REF], found by the thermal Bethe ansatz for the grand-potential density

Ω/L = -k B T ˆdq 2π ln 1 + e - (q) k B T , (6)
and the dressed energy,

(k) = 2 k 2 2m -µ-k B T ˆdq 2π 2c c 2 + (k -q) 2 ln 1 + e - (q) k B T , (7)
with c = mg/ 2 = -2/a 1D . We thus find the grand potential in the harmonic trap, and the scaling function A in Eq. ( 3). Applying the procedure presented above, we then find the scaling function f in Eq. ( 5) for the contact.

On the other hand, to assess the accuracy of the LDA, we perform ab initio quantum Monte Carlo (QMC) calculations. We use the same implementation as in Refs. [START_REF] Carleo | Universal superfluid transition and transport properties of twodimensional dirty bosons[END_REF][START_REF] Boéris | Mott transition for strongly interacting one-dimensional bosons in a shallow periodic potential[END_REF], which allows for numerically exact simulation of the Hamiltonian (1) within the grand-canonical ensemble. It yields the total number of particles N , the interaction energy H int , and the contact via the thermodynamic relation C = (2gm 2 / 4 ) H int versus temperature and chemical potential [START_REF]See Supplemental Material for details about the QMC approach, the derivation of the virial expansion, and an analysis of the experimental observability of the contact maximum in 2[END_REF].

The scaling function f , namely the rescaled contact a 3 ho C/N 5/2 , for 1D bosons under harmonic confinement resulting from YY theory and LDA is shown in Fig. 1. Figure 2 shows some sections of the latter (solid lines) along with QMC data (points) for a quantitative comparison. The rescaled contact is plotted as a function of the interaction strength ξ γ for various values of the temperature via the quantity ξ T in Fig. 2(a) and, inversely, as a function of ξ T for various 9), and ( 10) respectively. The QMC data are found from various sets of parameters, corresponding to the various symbols [START_REF]For Fig. 2(a), we used |a1D|/aho = 9.5 (red squares), 0.032 (red diamonds), 1[END_REF].

values of ξ γ in Fig. 2(b). The numerically exact QMC data are computed for a broad set of parameters. When plotted in the rescaled units of Eq. ( 5), they show excellent data collapse among each other and fall onto the LDA curves. Quite remarkably, the agreement holds also in the low-temperature and strongly interacting regime where the particle number is as small as N 5, within less that 3%. Our analysis hence validates the scaling form ( 5) and shows that the LDA is very accurate in computing the contact for the trapped LL model.

Onset of a maximum contact versus temperature.-We now turn to the behavior of the contact. Particularly interesting is the nonmonotonicity of C versus temperature and the onset of a maximum, see Fig. 2(b). This behavior strongly contrasts with that found for the homogeneous gas and the trapped gas in the Tonks-Girardeau limit (a 1D → 0), which are both characterized by a systematic increase of the contact versus temperature [START_REF] Kheruntsyan | Pair correlations in a finite-temperature 1D Bose gas[END_REF][START_REF] Vignolo | Universal contact for a Tonks-Girardeau gas at finite temperature[END_REF]. In the trapped case, the maximum in the contact as a function of ξ T is found irrespective to the strength of interactions but is significantly more pronounced in the strongly interacting regime. From the data of Fig. 1, we extract the temperature T * at which the contact is maximum at fixed ξ γ . In Fig. 3, we plot ξ * T = -a 1D /λ * T as a function of ξ γ . As we discuss now, ξ * T shows significantly different behavior in the strongly and weakly interacting regimes, but, in both cases, it characterizes the onset of the regime dominated by interactions.

We first consider the strongly interacting regime, ρ(0)|a 1D | 1. Using the virial expansion, we obtain the analytical expression for the contact [START_REF]See Supplemental Material for details about the QMC approach, the derivation of the virial expansion, and an analysis of the experimental observability of the contact maximum in 2[END_REF] ξ * T 0.490 ± 0.005 extracted from the data (dashed red line in Fig. 3).

C = 2N 5/2 πa 3 ho ξ γ ξ T √ 2 - e 1/2πξ 2 T ξ T Erfc(1/ √ 2πξ T ) , (8)
The existence of a maximum of the contact in the strongly interacting regime can be inferred from the competition of two different behaviors. On the one hand, at low temperatures, |a 1D | λ T (ξ T 1), both quantum and thermal fluctuations are dominated by repulsive interactions and the gas is fermionized. The contact is then found from the Bose-Fermi mapping, i.e. C = (2 2 /gm) ´dx ρ(x)e K (x), where e K is the kinetic-energy density [START_REF] Cazalilla | One-dimensional optical lattices and impenetrable bosons[END_REF]. Since the gas is weakly degenerate, the latter follows from the equipartition theorem, i.e. e K (x) = ρ(x)k B T /2, and the density profile can be taken as noninteracting, i.e. ρ(x) = (N/ √ 2πL th ) exp(-x 2 /2L 2 th ), with L th = k B T /mω 2 . It yields

C = 2 √ 2N 5/2 ξ γ ξ T /a 3 ho , ξ -1 γ ξ T 1, (9)
thus recovering the results of Ref. [START_REF] Vignolo | Universal contact for a Tonks-Girardeau gas at finite temperature[END_REF] by a different approach.

On the other hand, at high temperature, λ T |a 1D | (ξ T 1), the weakly degenerate Bose gas is dominated by thermal fluctuations. In this case, the contact can be estimated by the mean-field expression H int = g ´dx [ρ(x)] 2 . Using the thermal density profile, we then find

C 2 √ 2N 5/2 ξ γ /πξ T a 3 ho , ξ -1 γ , 1 ξ T . (10)
Both Eqs. ( 9) and ( 10) are in good agreement with the numerical calculations, see red-dotted and dash-dotted lines in Fig. 2(b). These expressions show that the contact increases with temperature in the fermionized regime but decreases when thermal fluctuations dominate over interactions. The maximum of the contact thus provides a nonambiguous signature of the crossover to fermionization. The situation is completely different in the weakly interacting regime, ρ(0)|a 1D | 1. In this case, the gas is never fermionized. At low temperature, (|a 1D |/ρ(0) 3 ) 1/4 λ T , the gas forms a quasicondensate characterized by suppressed density fluctuations [START_REF] Kheruntsyan | Pair correlations in a finite-temperature 1D Bose gas[END_REF]. The contact is then found from the mean-field expression for H int , using the Thomas-Fermi (TF) density profile ρ(x) = (µ/g)(1 -x 2 /L 2 TF ) with L TF = 2µ/mω 2 . In this regime one has [START_REF] Olshanii | Short-distance correlation properties of the Lieb-Liniger system and momentum distributions of trapped one-dimensional atomic gases[END_REF]

C = ηN 5/2 ξ 5/3 γ /a 3 ho , 1, ξ T ξ -1 γ ( 11
)
with η = 4 × 3 2/3 /5.

In the high-temperature and weakly interacting regime, λ T (|a 1D |/ρ(0) 3 ) 1/4 , the interactions are negligible and the bosons form a nearly ideal degenerate gas. Using the corresponding density profile ρ

(x) = λ -1 T Li 1/2 exp α -x 2 /2L 2 th with α(ξ γ , ξ T ) = ln[1 - exp(-1/(2πξ 2 γ ξ 2 T ))], we find C = 16 √ πN 5/2 ξ 5 γ ξ 3 T /a 3 ho G(α), ξ -1 γ ξ T ξ -2 γ (12) with G(α) = ´dx Li 2 1/2 [exp(α -x 2 )].
The function G(α) decays at least as λ 4

T and thus C decreases with the temperature. Therefore, in the weakly-interacting regime, the maximum contact signals the crossover from the quasicondensate regime to the ideal Bose gas regime. The position of the maximum of the contact may be estimated by equating Eqs. ( 11) and ( 12). The calculation is significantly simplified by neglecting quantum degeneracy effects in Eq. ( 12). Then, G(α) π/2 exp(2α) and we find

ξ * T ∼ ξ -ν γ , ν = 2/3. (13)
The numerical data are well fitted by Eq. ( 13) with ν as an adjustable parameter (see dotted blue line in Fig. 3), yielding ν fit = 0.6 ± 0.06, in good agreement with the theoretical estimate ν th = 2/3 [START_REF]Quantum degeneracy effects tend to increase the value of ν for[END_REF].

Maximum entropy versus interaction strength.-To further interpret the onset of a maximum contact versus temperature, we note that it is equivalent to the onset of a maximum entropy S versus interaction strength. For fixed number of particles, it is a direct consequence of the Maxwell identity [36]

∂C/∂T | a1D,N = (4m/ 2 ) ∂S/∂a 1D | T,N . (14)
In the homogeneous LL gas, the entropy at fixed temperature and number of particles decreases monotonically versus the interaction strength, since repulsive interactions inhibit the overlap between the particle wavefunctions, hence diminishing the number of available configurations. In the trapped gas, however, this effect competes with the interaction dependence of the available volume. More precisely, starting from the noninteracting regime, the system size increases sharply with interaction strength, while the particle overlap varies smoothly. In this regime, the number of available configurations and the entropy thus increase with the interaction strength. At the onset of fermionization, interaction-induced spatial exclusion becomes dramatic and the particles strongly avoid each other. In turn, as opposed to the noninteracting regime, the volume increases very slightly. In this regime, the number of available configurations thus decreases when the interactions increase. This picture confirms that the maximum of the entropy as a function of the interaction strength, or equivalently the maximum of the contact as a function of the temperature, signals the fermionization crossover. Experimental observability.-Our predictions can be investigated with quantum gases where the Tan contact is extracted from radio-frequency spectra or momentum distributions [START_REF] Stewart | Verification of universal relations in a strongly interacting Fermi gas[END_REF][START_REF] Wild | Measurements of Tan's contact in an atomic Bose-Einstein condensate[END_REF][START_REF] Luciuk | Evidence for universal relations describing a gas with p-wave interactions[END_REF]37]. Figure 4 shows momentum distributions found from QMC calculations in the strongly interacting regime close to zero temperature [ξ T 1, Fig. 4(a)] and at the contact maximum [ξ T ξ * T , Fig. 4(b)]. In both cases, an algebraic decay at large momenta is observed, with an amplitude matching our estimate for the contact [38].

In ultracold atom experiments, 1D systems are produced in the strongly interacting regime when loaded in 2D arrays of tubes [START_REF] Paredes | Tonks-Girardeau gas of ultracold atoms in an optical lattice[END_REF][START_REF] Kinoshita | Observation of a onedimensional Tonks-Girardeau gas[END_REF][START_REF] Meinert | Probing the excitations of a Lieb-Liniger gas from weak to strong coupling[END_REF], which raises the question of the effect of averaging the momentum distributions over the tubes. However, in the strongly interacting regime, the relative amplitude of the maximum with respect to the zero-temperature value, C * /C 0 , is lower bounded by its value in the central tube [START_REF]See Supplemental Material for details about the QMC approach, the derivation of the virial expansion, and an analysis of the experimental observability of the contact maximum in 2[END_REF]. For the parameters of Ref. [START_REF] Meinert | Probing the excitations of a Lieb-Liniger gas from weak to strong coupling[END_REF], we find C * /C 0 5.1, which should be sufficient for a clear identification of the maximum. Moreover, the condition for the tubes to be in the quasi-1D regime, k B T ω ⊥ , where ω ⊥ is the transverse trap frequency, can be fulfilled using a strong-enough transverse confinement [START_REF]See Supplemental Material for details about the QMC approach, the derivation of the virial expansion, and an analysis of the experimental observability of the contact maximum in 2[END_REF].

Conclusion.-Summarizing, we have provided a complete characterization of the Tan contact for the trapped Lieb-Liniger gas with arbitrary interaction strength, number of particles, temperature, and trap frequency. We have derived a universal scaling function of only two parameters and we have shown that it is in excellent agreement with the numerically exact QMC results over a wide range of parameters. As a pivotal result, we found that the contact exhibits a maximum versus the temperature for any interaction strength. This behavior is mostly marked in the gas with large interactions and provides an unequivocal signature of the crossover to fermion-ization. In outlook, the analysis of the Tan contact can be used to identify critical behaviors [START_REF] Chen | Critical behaviours of contact near phase transitions[END_REF]. It can further be extended to the excited states and multicomponent quantum systems [START_REF] Decamp | High-momentum tails as magnetic-structure probes for strongly correlated SU(κ) fermionic mixtures in onedimensional traps[END_REF][START_REF] Decamp | Exact density profiles and symmetry classification for strongly interacting multi-component Fermi gases in tight waveguides[END_REF][START_REF] Kormos | Exact three-body local correlations for excited states of the 1D Bose gas[END_REF][START_REF] Johnson | Long-lived nonthermal states realized by atom losses in onedimensional quasicondensates[END_REF][START_REF] Pâţu | Universal Tan relations for quantum gases in one dimension[END_REF][START_REF] Pâţu | Universality and quantum criticality of the one-dimensional spinor Bose gas[END_REF]. P.V. acknowledges Mathias Albert for useful discussions. This research was supported by the European Commission FET-Proactive QUIC (H2020 Grant No. 641122) and Paris region DIM-SIRTEQ. It was performed using HPC resources from GENCI-CCRT/CINES (Grant No. c2017056853). Numerical calculations make use of the ALPS scheduler library and statistical analysis tools [START_REF] Troyer | Parallel object oriented Monte Carlo simulations[END_REF][START_REF] Albuquerque | The ALPS project release 1.3: Open-source software for strongly correlated systems[END_REF][START_REF] Bauer | The ALPS project release 2.0: Open source software for strongly correlated systems[END_REF]. D.C. acknowledges support from the Institut Universitaire de France.

small values of ξγ. In the asymptotic limit ξγ → 0, they become dominant. Then G(α) π 2 / |α| and ν 1. However, in this regime, the maximum is extremely weak and hardly visible in practice.

[36] Equation ( 14 In some cases, however, the linear fit is not sufficient for extrapolating correctly the QMC results. This occurs in the stronglyinteracting regime for low to intermediate temperatures. In such cases, we use a third-order polynomial, a 3 ho C/N 5/2 = a + b( /β) + c( /β) 2 + d( /β) 3 , to extrapolate the finite-numerical data. An example is shown on the right panel of Fig. S1. In this case, the extrapolation corrects the QMC result for the smallest value of ( /β = 0.0005) by roughly 25%.

For all the QMC results reported on the Fig. 2 of the main paper, we have performed a systematic third-order polynomial extrapolation, even when a linear extrapolation was sufficient.

S2. DERIVATION OF THE TAN CONTACT IN THE LARGE-TEMPERATURE AND LARGE-INTERACTION LIMIT

We derive here Eq. ( 8) of the main paper for the contact at large, finite temperature (k B T N ω) and large interactions |a 1D |/a ho 1. As a first step, the Tan contact at large temperature and arbitrary interactions can be estimated using the first term of the virial expansion [START_REF] Vignolo | Universal contact for a Tonks-Girardeau gas at finite temperature[END_REF]

C = 4mω λ T N 2 c 2 (S2)
where 2) . The ν's are the solutions of the transcendental equation

c 2 = λ T ∂b 2 ∂|a 1D | and b 2 = ν e -β ω(ν+1/
f (ν) = Γ(-ν/2) Γ(-ν/2 + 1/2) = √ 2 a 1D a ho . ( S3
)
By exploiting the Euler reflection formula

Γ(z)Γ(1 -z) = π sin(πz) , (S4)
one can re-write Eq. (S3) under the form

f (ν) = -cot(πν/2) Γ(ν/2 + 1/2) Γ(ν/2 + 1) . ( S5
)
By using the asympotic expansions

Γ(z) √ 2π z z-1/2 e -z 1 + 1 12 z + O(1/z 2 ) (S6)
and

Γ(z + 1/2) √ 2π z z e -z 1 - 1 24 z + O(1/z 2 ) , (S7)
we obtain the following asymptotic expression for f (ν)

f (ν) -cot(πν/2) 1 ν/2 + 1/2 - 2 ν cot(πν/2). (S8)
In the Tonks-Girardeau regime, corresponding to a 1D = 0, one has ν = 2n + 1, with n ∈ N. Thus, in the regime |a 1D |/a ho 1, we obtain an explicit expression for ν, by writing This yields the following explicit expression for c 2 : (S11)

2 2n + 1 cot(πν/2) √ 2 |a 1D | a ho (S9) namely ν n = 2 π acot( √ 2n + 1|a 1D |/a ho ) + 2n, n ∈ N. ( S10
c 2 = λ T ν (-β ω) ∂ν ∂|a 1D | e -β ω(νn+1/2) = λ T n (-β ω) 2 π √ 2n + 1 a ho -1 1 + (2n + 1)
In order to evaluate analytically the sum in Eq. (S11), we replace ν with 2n + 1 in the exponential. Indeed, the first-order correction in |a 1D | gives a negligible contribution in the limit β → 0 and a 1D → 0. We finally get

c 2 = √ 2 1 2πξ 2 T - e 1/2πξ 2 T 2 3/2 πξ 3 T Erfc(1/ √ 2πξ T ) . (S12)
Thus, the contact at large temperatures and large interactions can be approximated by

C = 4 √ 2N 2 ξ T |a 1D |a 2 ho 1 2πξ 2 T - 1 2 3/2 πξ 3 T e 1/2πξ

(S13)

We have checked that this expression is in excellent agreement with the calculation of Eq. (S2) using the numerical solution of Eq. (S3), see Fig. S3.

S3. EXPERIMENTAL OBSERVABILITY OF THE MAXIMUM CONTACT IN A 2D ARRAY OF 1D TUBES

We address here a more detailed explanation for the question of observability of the contact maximum in ultracold-atom experiments. In subsection S3 A, we discuss the effect of averaging the momentum distribution over the 2D array of tubes. In it reads as

ξ T ξ γ 1 √ 2πN × ω ⊥ ω . ( S20
)
In experiments, the typical value of ω ⊥ /ω varies from a few hundreds to a few thousands. In Fig. S3 we reproduce the Fig. 3 of the main paper, together with the condition (S20) for two values of the atom number N and the parameters of Ref. [START_REF] Meinert | Probing the excitations of a Lieb-Liniger gas from weak to strong coupling[END_REF], ω/2π = 15.8Hz and ω ⊥ /2π = 14.5kHz. The regions where the validity condition is not fulfilled is shown in black for N = 2 and gray for N = 10. We conclude that the value ξ * T corresponding to the maximum of the contact is well inside the validity regime deep enough in the strongly-interacting regime, ξ γ 1. It is thus possible to observe the maximum contact in this regime. Moreover, one can further extend the validity region by increasing the value of the ratio ω ⊥ /ω, i.e. by increasing the transverse confinement.

Figure 1 .

 1 Figure 1. Reduced Tan contact a 3 ho C/N 5/2 for 1D Bose gases in a harmonic trap, versus the reduced temperature ξT = -a1D/λT and the reduced interaction strength ξγ = -aho/a1D √ N . The results are found using thermal Bethe ansatz solutions combined with localdensity approximation (see main text).

Figure 2 .

 2 Figure 2.Reduced Tan contact a 3 ho C/N 5/2 versus the scaling parameters, as found from LDA (solid lines) and QMC calculations (points). (a) Reduced contact versus the interaction, ξγ = -aho/a1D √ N , at the fixed temperatures ξT = -a1D/λT = 0.0085 (blue), 0.28 (green), and 18.8 (red). (b) Reduced contact versus the temperature via the quantity ξT at the fixed interaction strengths ξγ = 10 -2 (blue), 1.58 × 10 -1 (green), and 15.0 (red). The black dashed, red dotted, and red dash-dotted lines correspond to Eqs. (S13),[START_REF] Luciuk | Evidence for universal relations describing a gas with p-wave interactions[END_REF], and (10) respectively. The QMC data are found from various sets of parameters, corresponding to the various symbols[START_REF]For Fig. 2(a), we used |a1D|/aho = 9.5 (red squares), 0.032 (red diamonds), 1[END_REF].

  see black dashed line in Fig.2(b). It has a maximum at ξ * T = 0.485, in very good agreement with the asymptotic scaling

Figure 3 .

 3 Figure 3. Behavior of the temperature at which the contact is maximum versus the interaction strength. Shown is the value of ξ * T (solid black line with shaded gray error bars) as found from the data of Fig. 1, together with the asymptotic behaviors ξ * T 0.49 for the strongly interacting regime (dashed red line) and ξ * T ∝ ξ ν fit T , with νfit 0.6 for the weakly interacting regime (dotted blue line).

Figure 4 .

 4 Figure 4. Log-log plots of momentum distributions found by QMC calculations in the strongly interacting regime. (a) Low temperature: ξγ = 4.47 and ξT = 0.0085. (b) Temperature at the maximum contact: ξγ = 1.26 and ξT = 0.49. The solid blue lines with shaded statistical error bars are the QMC results, the dashed red lines are algebraic fits to the large-k tails, and the dotted green lines are the momentum distributions of the nondegenerate ideal gas. The insets show the same data in lin-lin scale.

  ) follows from the sweep relation C = (4m/ 2 ) ∂F /∂a1D| T,N and the thermodynamic definition of the entropy S = -∂F/∂T | a 1D ,N , where F (N, T, a1D) = Ω(µ, T, a1D) + µN is the Helmoltz free energy. [37] R. Chang, Q. Bouton, H. Cayla, C. Qu, A. Aspect, C. I. Westbrook, and D. Clément, Momentum-resolved observation of thermal and quantum depletion in a Bose gas, Phys. Rev. Lett. 117, 235303 (2016). [38] Fitting the tails of the momentum distributions found from the QMC calculations by n(k) Cfit/k p fit , we obtain pfit and Cfit = a 3 ho Cfit/N 5/2 . For Fig. 4(a), pfit = 3.80 ± 0.20 and Cfit = (1.01 ± 0.14) × 10 -3 while C 0.97 × 10 -3 . For Fig. 4(b), pfit = 3.72 ± 0.10 and C fit = 0.23 ± 0.04 while C 0.22. The agreement with the expected exponent p = 4 is better than 7% and with the contact C better than 5%.

)Figure

  Figure S2. Ca 3ho /N 2 calculated using the numerical solution of Eq. (S3) (blue continuous line) and using the analytical expression (S13) (magenta dashed line). We have considered |a1D|/aho = 0.4.

Figure S3 .

 S3 FigureS3. Reproduction of Fig.3of the main paper together with the validity condition of the quasi-1D regime, Eq. (S20), for ω ⊥ /ω = 30, relevant for the experiments of Ref.[START_REF] Meinert | Probing the excitations of a Lieb-Liniger gas from weak to strong coupling[END_REF]. The dark regions show the excluded regions for N = 2 atoms (black) and N = 10 atoms (gray).

Supplemental Material for

Tan's Contact for Trapped Lieb-Liniger Bosons at Finite Temperature In this supplemental material, we provide details about the QMC approach (Sec. S1), the derivation of the virial-expansion formula (Sec. S2), and a detailed analysis of experimental observability of the contact maximum in experiments using a 2D array of 1D tubes in strongly-interacting regime (Sec. S3).

S1. QUANTUM MONTE CARLO SIMULATIONS A. Path-integral Monte-Carlo approach

The quantum Monte Carlo (QMC) calculations exploit the same implementation as detailed in Refs. [START_REF] Carleo | Universal superfluid transition and transport properties of twodimensional dirty bosons[END_REF][START_REF] Boéris | Mott transition for strongly interacting one-dimensional bosons in a shallow periodic potential[END_REF]. The continuousspace path integral formulation allows us to simulate the exact Hamiltonian, Eq. ( 1) of the main paper, for an arbitrary trap V (x), within the grand-canonical ensemble. The statistical average of the number of world lines yields the total number of particles N , and the interaction energy H int is computed from the zero-range two-body correlator. The contact is then found using the thermodynamic relation

The world lines are discretized into an adjustable number M of slices of elementary imaginary propagation time = 1/M k B T each, and sampled using the worm algorithm [START_REF] Boninsegni | Worm algorithm for continuous-space path integral Monte Carlo simulations[END_REF][START_REF] Boninsegni | Worm algorithm and diagrammatic Monte Carlo: A new approach to continuous-space path integral Monte Carlo simulations[END_REF]. Each calculation is run for various values of and polynomial extrapolation is used to eliminate systematic finite-time discretization errors (see below).

B. Finite-scaling

The QMC results are exact in the → 0 limit. In order to find the final results reported on the Fig. 2 of the main paper, we proceed as follows. For each set of physical parameters (interaction strength, chemical potential, temperature, and trap frequency), we perform a series of QMC calculations for different values of and extrapolate the result to the limit → 0.

For most of the calculations, we are able to use a sufficiently small value of and a linear extrapolation is sufficient. We fit the QMC data with a 3 ho C/N 5/2 = a + b( /β), with a and b as fitting parameters. We then use the quantity a as the final result for a 3 ho C/N 5/2 . An example is shown on the left panel of Fig. S1 below. In this case, the linear extrapolation only corrects the QMC result for the smallest value of ( /β = 0.01) by less than 4%. subsection S3 B, we discuss the validity of the quasi-1D gas condition.

A. Averaging over the tubes

In most cases, strong transverse confinement is realized by applying a 2D optical lattice in the directions y and z, orthogonal to the 1D direction x. For sufficiently strong lattices, it creates an array of independent 1D tubes, indexed by the labels (j, ) ∈ Z 2 . The total contact then reads as

where C(j, ) is the contact in the corresponding tube. Each tube is populated with a number N (j, ) of atoms, which depends on the loading procedure of the atoms in the 2D lattice. Since the number of atoms is maximum in the central tube (j = = 0), we have ξ γ (j, ) ≥ ξ γ (0, 0), and the condition for having all tubes in the strongly-interacting regime reduces to ξ γ (0, 0) 1. In that regime, the temperature dependence of the contact around the maximum is independent of ξ γ , and thus independent of the tube. Indeed, as shown by Eq. ( 8) of the main paper, the parameter ξ γ just appears as a prefactor. In particular, the maximum contact is located at the universal value ξ * T 0.485, which is identical for the tubes. Using Eq. ( 8) of the main paper, we then find

At zero temperature, the contat may be found using the mapping between the strongly-interacting Bose gas and the stronglydegenerate ideal Fermi gas. It yields the value [3]

We then find that the relative amplitude of the maximum contact with respect to its zero-temperature value fullfils the inequality

Therefore, the relative ampitude of the maximum contact is larger than a fraction of the interaction parameter ξ γ (0, 0) 1 and should be observable. For instance, for the parameters of Ref. [START_REF] Meinert | Probing the excitations of a Lieb-Liniger gas from weak to strong coupling[END_REF], we find ξ γ (0, 0) 7.5 and C * /C 0 5.1.

Note that the lower bound in Eq. (S17) is universal in the sense that it does not depend on the distribution of atoms in the various lattice tubes. Note also that it is immune to shot-to-shot fluctuations of the atom numbers in the tubes.

Finally, a more precise value of the relative amplitude of the maximum contact is found by compting the sums in Eqs. (S15) and (S16) for realistic distributions of the atom numbers among the tubes. Using the estimate

relevant to the experiments of Refs. [START_REF] Paredes | Tonks-Girardeau gas of ultracold atoms in an optical lattice[END_REF][START_REF] Meinert | Probing the excitations of a Lieb-Liniger gas from weak to strong coupling[END_REF], we find C * /C 0 0.8 × ξ γ (0, 0), which is only about 20% larger than the atom distribution-independent lower bound, Eq. (S17). For the parameters of Ref. [START_REF] Meinert | Probing the excitations of a Lieb-Liniger gas from weak to strong coupling[END_REF], it yields C * /C 0 6.1. It may be further increased by lowering the total number of atoms, although at the expense of atom detectability. In this respect, mestable He atoms appear particularly suited for they allow for atom-resolved detection and measurement of momentum distributions over up to six decades [37].

B. Validity condition of the quasi-1D regime

The condition for generating truly quasi-1D tubes in the experiment reads as k B T ω ⊥ , where ω ⊥ is in the angular frequency of the transverse confinement induced by the 2D lattice, see for instance Ref. [START_REF] Olshanii | Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons[END_REF]. This condition may be written using the scaling parameters ξ γ = -a ho /a 1D

√

N and ξ T = -a 1D /λ T . Using the relations a ho = /mω, λ T = 2π 2 /mk B T , and


CERMSEM Pascal Gourdel
email: gourdel@univ-paris1.fr

CERMSEM Leila Triki
email: leila.triki@malix.univ-paris1.fr

Incomplete markets and monetary policy

Keywords: Cash-in-advance constraints, incomplete markets, nominal assets, monetary equilibrium, money, nominal interest rate JEL Classification: C62, D52, E40, E50, G10

We consider an extension of a general equilibrium model with incomplete markets that considers cash-in-advance constraints. The total amount of money is supplied by an authority, which produces at no cost and lends money to agents at short term nominal rates of interest, meeting the demand. Agents have initial nominal claims, which in the aggregate, are the counterpart of an initial public debt. The authority covers its expenditures, including initial debt, through public revenues which consists of taxes and seignorage, and distributes its eventual budget surpluses through transfers to individuals, while no further instruments are available to correct eventual budget deficits. We define a concept of equilibrium in this extended model, and prove that there exists a monetary equilibrium with no transfers. Moreover, we show that if the price level is high enough, a monetary equilibrium with transfers exists.

Introduction

In the canonical general equilibrium model, all trade takes place in a barter economy, precluding the role of money as a medium of exchange. In 1965, Frank Hahn [START_REF] Hahn | On some problems proving the existence of an equilibrium in a monetary economy[END_REF] has argued that it was difficult to justify a positive price for fiat money (i.e. paper money) (this is known as the "Hahn Paradox") which stipulates by a backward induction reasoning in a finite-period economy that money cannot have positive value. As discussed by Dubey-Geanakoplos [START_REF] Dubey | The value of money in a finite horizon: a role for banks[END_REF], there are several ways to overcome this paradox. Among them, one can consider an infinite-horizon model (Samuelson [15], Grandmont-Younes [START_REF] Grandmont | On the efficiency of a monetary equilibrium[END_REF]), and in these cases, money has value because it is a store of value. Another way to overcome Hahn's Paradox is to introduce an external agent, who stands ready to trade commodities for money (Lucas [13], Magill-Quinzii [START_REF] Magill | real effects of money in general equilibrium[END_REF]). Alternatively, following Lerner [START_REF] Lerner | Money as a creature of the state[END_REF], one could postulate the existence of a government that is owed in taxes. In these two latter cases, money has value because an external agent gives something in exchange for it.

The present paper considers the presence of an external agent, an authority, and the theoretical work underlying it, Drèze-Polemarchakis [START_REF] Drèze | Intertemporal general equilibrium monetary and theory[END_REF], [START_REF] Drèze | Monetary Equilibria[END_REF], consists in formulating an intertemporal general equilibrium model with money, introducing reasonable assumptions that guarantee the existence of equilibria in this extended model. In order to define a general competitive equilibria in a monetary economy, we modify the canonical Walrasian model by introducing an incomplete financial market and money balances that facilitate transactions. Fiat money produced at no cost by banks serves as medium of exchange. An authority lends money to agents against promise of reimbursement with interest rate, or equivalently, in exchange for interest bearing bonds. All initial holdings of money are the counterparts of debts to banks. In the monetary vocabulary of monetary economies, this is a model of "inside money" 1 . It is a model appropriate for economies where an authority issues money in exchange for offsetting claims. There is no default, and the authority raises revenue from taxes and seignorage. It distributes its eventual surpluses as lump-sum transfers to agents. The demand for money at given commodity prices and interest rates results from the preference maximizing choice of individuals. As store of value, non-interest-bearing fiat money is dominated by interest-bearing nominal assets. Balances, prices and rates of interests do not enter as arguments of preferences of the agents.

Over a finite horizon with no public debt, and no taxes, Drèze and Polemarchakis [START_REF] Drèze | Monetary Equilibria[END_REF] proved the existence of equilibria for arbitrarily set nominal rates of interest and price levels at all terminal nodes. In a recent joint paper with Bloize [START_REF] Bloize | Monetary economy over infinite horizon[END_REF], they proved the existence over an infinite horizon economy under uncertainty and complete asset markets. The primitive of the model include nominal claims held by individuals (that in the aggregate are the counterpart of initial public debt). Their work extends for Woodford [START_REF] Woodford | Monetary policy and price level determinacy in a cash-in-advance economy[END_REF] in the case of heterogeneous agents, which is in term similar to cash-in-advance economies with a representative agent as in Lucas-Stockey [START_REF] Lucas | Money and interest in a cash-in-advance economy[END_REF]. Woodford [START_REF] Woodford | Monetary policy and price level determinacy in a cash-in-advance economy[END_REF] asserts that the price level is determinate so as to balance the initial public debt and public revenu from taxes and seignorage. Similarly, Dubey-Geanakoplos [START_REF] Dubey | Monetary equilibria with missing markets[END_REF] obtain deterministic equilibria considering the case of a given initial shock of outside money. On the other hand, Bloize-Drèze-Polemarchakis [START_REF] Bloize | Monetary economy over infinite horizon[END_REF] obtain indeterminacy of equilibria since they assume that the public authority can redistribute its eventual surpluses.

In this paper, we propose an extension of Bloize-Drèze-Polemarchakis [START_REF] Bloize | Monetary economy over infinite horizon[END_REF], in an incomplete market setting, and over a finite, two-period horizon. The main results are:

• The existence of a monetary equilibrium with no transfers, under reasonnable assumptions.

• The existence of a monetary equilibrium above a lower bound of the overall price level, when the authority faces a budget surplus. Two alternative assumptions on the public portfolio (the portfolio that the authority supplies) are proposed and the results are compared.

The paper is organized as follows: we begin by introducing the primitive of the model, as well as the time and uncertainty setup (Section 2). We also define an appropriate notion of monetary equilibria, and state the assumptions under which existence will be proved. Section 3 proves the existence of equilibria with no transfers. Finally, Section 4 proves existence when the authority faces a budget surpluses.

A 2-Period Monetary Economy

We consider a finite set I of agents, two periods t = 0 and t = 1 with a finite set S of states of the world at the second period. We denote Σ = {0} ∪ S, where 0 is the state of the world known with certainty at t = 0. The state of the world σ ∈ Σ is called a date-event state. There is a finite set of goods L available for trades at both periods, a finite set J of 1-period maturity nominal assets that agents can buy at t = 0 and which yield monetary returns at t = 1. We denote by y • the family:

y • = (y i , i ∈ I) • The commodity market E c is described by E c = (X • , u • , e • , g • , ξ • )
where, for each agent i ∈ I, X i ⊂ R LΣ + is the consumption set of agent i. A vector x i ∈ X i is a consumption plan. The utility function u i : X i -→ R describes the preferences of agent i ∈ I. The initial endowments are given by e i ∈ R LΣ + and every agent i pays taxes g i ∈ R LΣ + to the authority. Notice that in particular, it can be assumed that g i = η i e i , for some 0 η i < 1 with η • 0 being some tax rates across individuals. Our commodity taxes then reduce to a wealth tax. The public authority also issues transfers t which are elements of R Σ . These transfers are distributed to individuals according to given shares ξ i ∈]0, 1[ such that i∈I ξ i = 1 and each agent receives the amount ξ i t.

• The financial market E f is described by E f = (R, Θ • , θ) where R ∈ R S×J is the return matrix, J S, and for every (s, j) ∈ S × J, R(s, j) ∈ R is denominated in units of account. For each agent i ∈ I, Θ i ⊂ R J is the portfolio set of agent i. Given an agent i ∈ I and a portfolio

θ i ∈ Θ i , (R(σ)θ i , σ ∈ Σ) ∈ R Σ denotes the image of θ i by R.
Finally, the portfolio θ ∈ R J is the total amount of each asset available for trade, fixed by the authority.

• The money market is described by E m = (w • , r) where w • ∈ R ΣI and for each i ∈ I, w i (0) ∈ R + are initial individual nominal claims against the authority (corresponding to the public debt). For convenience, we introduce the following notation: for every agent i ∈ I, w i = (w i (σ), σ ∈ Σ) where w i (s) = 0, and for all states s ∈ S. We setw = i∈I w i . Short term nominal rates of interest r are positive element of R Σ exogenously given.

Finally, a monetary economy is the triplet

E = (E c , E f , E m ).
2.1. The transactions demand for money. Let us begin by introducing these notations: Let p ∈ R LΣ + the commodity price vector. We define the payoff matrix V ∈ R Σ×J by V = (-q R). This operator summarizes the financial structure of the economy, given that q ∈ R J is the asset price vector. Let an agent i ∈ I. We denote net trades by z i = (x i -e i ), where x i ∈ X i and e i initial endowments, z i + = (x i -e i ) + net purchases, z i -= (x i -e i ) -net sales 2 . We will denote by m i ∈ R Σ + initial money balance, and by m i ∈ R Σ + terminal money balance.

An important modeling choice concerns the treatment of time. There are two periods. Formally, a date is a point of time. For purpose of interpretation, the length of time period is thought as non-trivial. Precise timing of transactions does not affect preferences while it does affect money balances and accounting. Taking this into account, we follow the convention that budget constraints will be written at beginning-of-period, given a path of interest rate r ∈ R Σ 2 For a scalar, z + = max {z, 0} and z -= max {-z, 0}; for a vector, z + = (. . . , z k+ , . . . ) and z -= (. . . , z k-, . . . ). Notice that z = z + -z -. Moreover, recall that the functions z -→ z + and z -→ z -are convex.

and a vector h ∈ R Σ , we introduce, for every state σ ∈ Σ:

h(σ) = 1 1 + r(σ) h(σ) (2.1)
Consider a date-event σ ∈ Σ. The transaction demand for money follows the scheme of cashin-advance constraints introduced by Clower [START_REF] Clower | A reconsideration of the micro-foundations of monetary theory[END_REF]. An agent i ∈ I acquires cash balances m i (σ) by borrowing initially from the authority in exchange for bonds at the rate of interest r(σ) ∈ R + , according to the constraint b i (σ) + m i (σ) = 0. Subsequently, he purchases commodities according to the constraint:

p(σ) • z i + (σ) m i (σ), (2.2)
He accumulates end-of-period balances from the sale of commodities according to the constraint

p(σ) • z i -(σ) = m i (σ), (2.3)
At the end of the period, or at the beginning of a subsequent, fictitious period that serves for accounting purposes, the agent settles his debt according to the constraint:

(1 + r(σ)) m i (σ) + p(σ) • g i (σ) -(V θ i )(σ) -ξ i t(σ) m i (σ) + (1 + r(σ))w i (σ)
where g i (σ) ∈ R L + are commodity taxes payed to the authority, w i (σ) ∈ R + are nominal initial claims against the authority that agent receive at beginning-of-period, θ i is the portfolio he chooses to acquire, and ξ i t(σ) ∈ R is the amount of transfers that his share ξ i allows him to obtain. According to equations 2.2 and 2.3, the budget equation of agent i at date-event σ ∈ Σ is summarized by:

(1 + r(σ))[p(σ) • z i + (σ)] + p(σ) • g i (σ) p(σ) • z i -(σ) + (V θ i )(σ) + ξ i t(σ) + (1 + r(σ))w i (σ) p(σ) • [z i + (σ) + g i (σ)] 1 1 + r(σ) p(σ) • z i -(σ) + ( V θ i )(σ) + w i (σ) + ξ i t(σ) p(σ) • (z i (σ) + g i (σ)) + p(σ) • z i -(σ) 1 1 + r(σ) p(σ) • z i -(σ) ( V θ i )(σ) + w i (σ) + ξ i t(σ) p(σ) • (z i (σ) + g i (σ)) + r(σ) 1 + r(σ) p(σ) • z i -(σ) ( V θ i )(σ) + w i (σ) + ξ i t(σ)
For each commodity price p ∈ R ΣL + and each consumption plan x ∈ R ΣL + , we define the vector p x ∈ R Σ by

p x = (p(σ) • x(σ), σ ∈ Σ) ∈ R Σ
where the operator • is the scalar product in R ΣL . For each interest rate r ∈ R Σ + and each money balance m ∈ R Σ + , we define the vector r

• m ∈ R Σ by r • m = (r(σ)m(σ), σ ∈ Σ) ∈ R Σ
We get to the overall budget constraints:

p (x i -e i + g i ) + r • (p (x i -e i ) -) V θ i + w i + ξ i t.
(2.4) 2.2. Authority. The authority enters a date-event 0 with a given public liability w(0) and covers this beginning-of-period expenditure and end-of-period supply of security θ ∈ R J by collecting commodity taxes g(0) ∈ R L + , given that money balances m(0) ∈ R + are supplied so as to accommodate the market demand, where

m(0) = r(0)p(0) • i∈I x i (0) -e i (0)
-At the end-of-period, the authority distributes its eventual budget surpluses as transfers to individuals t(0) ∈ R determined by the beginning-of-period constraint:

t(0) = r(0)p(0) • i∈I x i (0) -e i (0) -+ p(0) • g(0) + q • θ -w(0) (2.5)
These transfers are distributed among agents according to their exogenous shares ξ i ∈ [0, 1], and vary accordingly to different consumption allocation

x • ∈ R ΣLI + .
At date-event s ∈ S, given end-of-period returns of assets, and collected taxes g(σ) ∈ R L + the eventual budget surpluses distributed among agents amount to:

t(s) = r(s)p(s) • i∈I x i (s) -e i (s) -+ p(s) • g(s) -( Rθ)(s) (2.6)
The overall constraint faced by the authority sums up to:

t = r • p i∈I (x i -e i ) -+ p g -V θ -w (2.7)
We can now go through the definition of an equilibrium and state the main result of the paper.

Definitions and notations. Given a commodity price vector p ∈ R LΣ

+ and an asset price q ∈ R J , we introduce the budget set of an agent i ∈ I by:

B i (p, q, t) := (x i , θ i ) ∈ X i × Θ i : p (x i -e i + g i ) + r • [(p (x i -e i ) -] V θ i + w i + ξ i t
A consumption plan x i ∈ X i and a composition of portfolio θ i ∈ Θ i are budget feasible for agent i ∈ I if these actions belong to budget set B i (p, q, t).

Given a commodity price vector p ∈ R LΣ

+ , agent i's behavior in this economy is summarized by the demand correspondence d i (p, q, t) defined by:

d i (p, q, t) := (x i , θ i ) ∈ B i (p, q, t), B i (p, q, t) ∩ [P i (x i ) × Θ i ] = ∅ where P i (x i ) := {y ∈ X i : u i (y) > u i (x i )}. Definition 2.1. A collection (x • , θ • , p, q, t) ∈ R LΣI + × R JI × R LΣ + × R J + × R Σ is a monetary equilibrium of a monetary economy E = (E c , E f , E m ) if (i) For each agent i ∈ I, (x i , θ i ) ∈ d i (p, q, t), (ii)
The public plan t satisfies the authority's budget constraints:

t = r • p i∈I (x i -e i ) -+ p g -V θ -w. (iii)
Commodity and asset markets clear: i∈I x i = i∈I e i and i∈I θ i = θ.

A monetary equilibrium is said to be with no-transfers if t = 0.

Assumptions. Before stating the assumptions, let us introduce the following notation:

A vector v = (v(σ), σ ∈ Σ)) in R Σ is said to be positive, denoted by v > 0, if ∀σ ∈ Σ, v(σ) 0, v = 0,
and it is said to be strictly positive, denoted by v 0 if, ∀σ ∈ Σ, v(σ) > 0.

The commodity market

E c = (X • , u • , e • , g • , ξ •
) is subject to the following assumptions: for each agent i ∈ I, C1 The consumption set X i is a closed, convex subset of the positive orthant of R LΣ , and e i 0. C2 The utility function is continuous, strictly monotone and strictly quasi-concave. 3 C3 There exists a consumption plan x i ∈ X i such that x i -e i + g i 0. This is a strong survival assumption in this extended model. After paying his taxes, agent i can still consume.

The financial market E f = (R, Θ • , θ) is subject to the following assumptions: F1 For each agent i ∈ I, the portfolio set Θ i is equal to R J . F2 The return matrix R has full rank. For convenience, we assume R > 0 4 . F3 The public portfolio is non-negative, i.e. θ 0.

Non-risky asset NRA The public portfolio is a non-risky portfolio:

Rθ 0.
Assumptions (C1) to (F2) are the standard assumptions considering an incomplete market framework. We provide hereafter specific assumptions due to the extension of the incomplete market framework that we consider.

Transfers T1 Transfers t are distributed among agents through given shares ξ • , i.e. each agent receives the amount ξ i t.

Public Revenue PR Aggregate taxes g = i∈I g i are strictly positive.

Definition 2.2. A monetary economy

E = (E c , E f , E m )

is said to be standard if it satisfies the above assumptions

Initial public debt M1 The total amount of initial liabilities is positive: w(0) > 0.

Finally, we propose in the following two additional assumptions on the financial and money market E f = (R, Θ • , θ): 3 The utility function u i is strictly quasi-concave if: 4 Assuming that the public portfolio is a non-risky portfolio (refer to Assumption (NRA) defined later), there is no loss of generality in considering R > 0. One may refer to Lemma 4.3 for the proof of this result.

∀x i , ∀ y i ∈ X i , and ∀λ ∈ [0, 1], u i (x i ) > u i (y i ) implies u i (λx i + (1 -λ)y i ) > u i (y i )
Public Portfolio PP The public portfolio consists only in safe bonds5 , i.e. θ = I1 J6 .

Neither (PP) implies (NRA), nor is the converse true. Assumption (PP) is a restrictive assumption, but allows us to precise properties on first period price levels that is lost when one considers only (NRA) (refer to Theorem 2.2, or Theorem 2.3). The results to be proved are the following:

Theorem 2.1. Let E be a standard monetary economy. Under assumptions (M1), for every path of rate of interest r 0 fixed by the authority, there exists a monetary equilibrium with no transfers (x • , θ • , p, q) of E.

Remark 2.1. In the previous theorem, we may consider a weaker version of Assumption (PR), namely, requiring g(σ) > 0 in all states σ ∈ Σ.

Before stating the existence of a monetary equilibrium with transfers, let us introduce the following notations: We endow the dimensional space R n with norm 1: for any vector

h ∈ R n , h = n d=1 |h d |.
And we denote by B(n, k) the closed ball on R n of radius k > 0, with center 0.

Let d ∈ R Σ ++ .
We call d(σ) the overall price level at date-event σ ∈ Σ when d is defined by: d(0) = p(0) + q and d(s) = p(s) , s ∈ S.

Theorem 2.2. Let E be a standard monetary economy. Under assumption (PP), for every path of rate of interest r 0 fixed by the authority, there is d ∈ R Σ + , such that, for every d d , d 0, there exists a monetary equilibrium with transfers (x • , θ • , p, q, t) of E with p(0) + q = d(0) and p(s) = d(s) for every date-event s ∈ S of the second period.

We also prove that by choosing a higher price level (c d ), we show that, at equilibrium, transfers are positive.

Theorem 2.3. Let E be a standard monetary economy. Under assumption (PP), for every path of rate of interest r 0 fixed by the authority, there is c ∈ R Σ , c 0 such that, for every c c there exists a monetary equilibrium with positive transfers (x • , θ • , p, q, t) of E with p(0) + q = c(0) and p(s) = c(s) for every date-event s ∈ S of the second period.

If we drop assumption (PP), we get the following corollaries of Theorem 2.2 and Theorem 2.3: Corollary 2.1. Let E be a standard monetary economy. For every path of rate of interest r 0 fixed by the authority, there is e ∈ R S , e 0 such that, for every e e there exists a monetary equilibrium with transfers (x • , θ • , p, q, t) of E with p(s) = e(s) for every date-event s ∈ S of the second period.

Corollary 2.2. Let E be a standard monetary economy. For every path of rate of interest r 0 fixed by the authority, there is e ∈ R S , e 0 such that, for every e e there exists a monetary equilibrium with positive transfers (x • , θ • , p, q, t) of E with p(s) = e(s) for every date-event s ∈ S of the second period.

It is important to notice here that Corollary 2.1 is a consequence of Theorem 2.2, while Corollary 2.2 is a consequence of Theorem 2.3. A more reasonable assumption on the public portfolio (in particular (NRA)) prevents us to get precise information on first period price levels.

The next section is devoted for the proof of the case of no transfers. Section 4 will study the case of transfers.

Existence of Monetary Equilibrium with No Transfers

The proof follows the usual scheme considering an incomplete market setting, The general method of proof is the usual incomplete market arguments as in Duffie [START_REF] Duffie | Stochastic equilibria with incomplete financial markets[END_REF], Florenzano [START_REF] Florenzano | General equilibrium of financial markets: An introduction[END_REF], Werner [START_REF] Werner | Equilibrium in economies with incomplete financial markets[END_REF], among others.

We begin by identifying compact, convex sets for consumption sets and portfolio sets. Adapting the work of Bloize-Drèze-Polemarchakis [START_REF] Bloize | Monetary economy over infinite horizon[END_REF] in an incomplete market framework, we modify budget sets by introducing an index µ ∈ R Σ + of the reciprocal of the overall price level leading to well-behaved correspondences. Applying Kakutani's fixed point theorem in (p, q, x • , θ • , µ) leads to the existence of an abstract equilibrium, an equilibrium concept which is defined below (Definition 3.2). The last step of the proof consists in showing that under (PR)-(NRA)-(M1), the introduced index is strictly positive, and the abstract equilibrium is achieved as an monetary equilibrium with no transfers.

3.1. Truncating the economy. Given assumptions (C2) and (F2), we may restrict ourselves to positive commodity and asset prices. We consider the following compact, convex set for commodity and asset price vectors: Π = (p, q) ∈ R LΣ + × R J + : p(0) + q = 1 and p(s) = 1, ∀s ∈ S

We provide hereafter the definition of a truncated monetary economy. The following lemma establishes that in order to prove Theorem 2.1, we can suppose without any loss of generality that commodity and financial sets are compact.

Definition 3.1. If E c = (X • , u • , e • , g • , ξ • ) is a commodity market, and E f = (R, Θ • , θ) is a financial market, then for any k > 0, we let E c k and E f k defined by E c k = (X • k , u • k , e • , g • , ξ • ) E f k = (R, Θ • k , θ) where, for each i ∈ I, X i k = X i ∩ B(Σ × L, k). We set u i k as the restriction of u i to X i k , Θ i k = Θ i ∩ B(J, k).
Let X the set of attainable commodity allocations, i.e.

X := x ∈ i∈I X i : i∈I (x i -e i ) = 0 (3.1)
For every i ∈ I, X i is the projection of X on X i . The sets ( Θ i , i ∈ I) of attainable portfolios are defined as follows:

Θ i := {θ i ∈ Θ i : ∃(p, q) ∈ Π, ∃x i ∈ X i , p (x i -e i + g i ) + r • (p (x i -e i ) -) = V θ i + w i }. We set Θ = i∈I Θ i . Lemma 3.1. Let E c commodity market and E f a financial market. Then (a) There exists k > 0 such that ∀i ∈ I, X i ⊂ int B(Σ × L, k), Θ i ⊂ int B(J, k) (3.2) (b) If k > 0 is
sufficiently large such that 3.2 is satisfied, then for each money market E m , any monetary equilibrium of the truncated economy

(E c k , E f k , E m ) is a monetary equilibrium of the initial economy (E c , E f , E m ).
The proof of Lemma 3.1 is referred to the appendix (A1). We can now fix k > 0. Following Lemma 3.1, we can suppose without any loss of generality that for each i ∈ I, the sets X i and Θ i are compact. Let us introduce the following notation: Consider a set V ⊂ R n . We recall that the convex hull of V, denoted by co (V ) ⊂ R n is the smallest convex set containing V .

For convenience of notation, we set v(x) ∈ R Σ + such that:

v(x, σ) = i∈I x i (σ) -e i (σ) -, ∀σ ∈ Σ.
Since µ ∈ R Σ is also a variable involved in our fixed-point argument, we propose the following convex set:

M := co      µ ∈ R Σ + ∃x ∈ i∈I X i , ∃(p, q) ∈ Π, µ(0)w(0) = r(0)p(0) • v(x, 0) + p(0) • g(0) + qθ µ(s)( R(s)θ) = r(s)p(s) • v(x, s) + p(s) • g(s)      Claim 3.1. The set M ⊂ R Σ is compact.
Proof. The compactness of M follows from assumptions (M1)-(F2)-(NRA) and from the compactness of X and Π. Let (x i ν , p ν , q ν ) be a sequence in X i × Π and (µ ν ) a sequence in M . Then, for each ν ∈ N, we have

µ ν (0)w(0) = r(0) p(0) • i∈I x i ν (0) -e i (0) -+ p ν (0) • g(0) + q ν θ i µ ν (s)( R(s)θ) = r(s) p ν (s) • i∈I x i ν (s) -e i (s) -+ p ν (s) • g(s)
According to assumption (M1), w(0) > 0, and according to (NRA), (R(s)θ) > 0, ∀s ∈ S, thus we can suppose that µ ν converges µ. Note that µ ∈ M .

3.2.

Modifying budget sets. For each (p, q, µ) ∈ Π × M , we define β i (p, q, µ) the following modified budget set of agent i ∈ I defined by the set of actions (x i , θ i ) ∈ X i × Θ i such that:

p(0) • (x i (0) -e i (0) + g i (0)) + r(0) p(0) • (x i (0) -e i (0)) - -qθ i + µ(0)w i (0) p(s) • (x i (s) -e i (s) + g i (s)) + r(s) p(s) • (x i (s) -e i (s)) - µ(s)( Rθ i )(s), ∀s ∈ S
The associated demand correspondence is defined by:

δ i (p, q, µ) := (x i , θ i ) ∈ β i (p, q, µ), β i (p, q, µ) ∩ [P i (x i ) × Θ i ] = ∅
Let us begin by introducing the notion of abstract equilibrium Definition 3.2. An abstract equilibrium consists in a collection (p, q, x • , θ • ) and an index of the reciprocal overall price levels µ ∈ R Σ + , such that: (i) For every agent i ∈ I, (x i , θ i ) ∈ δ i (p, q, µ)

(ii) The authority's constraints are satisfied

r(0)p(0) • i∈I (x i (0) -e i (0)) -+ p(0) • g(0) = µ(0)w(0) -q • θ r(s)p(s) • i∈I (x i (s) -e i (s)) -+ p(s) • g(s) = µ(s)( Rθ)(s), ∀s ∈ S
(iii) Markets clear: i∈I x i = i∈I e i and i∈I θ i = θ, Remark 3.1. If µ 0 , and (p, q, x • , θ • , µ) is an abstract equilibrium of E, then (p , q , x • , θ • ) is a monetary equilibrium given that

p (σ) = p(σ) µ(σ)
, ∀σ ∈ Σ and q = q µ(0) .

For each i ∈ I, for each (p, q, µ) ∈ Π × M , we denote by β i the interior of the set β i on Π × M .

Lemma 3.2. For every agent i ∈ I, the correspondence β i has non-empty values on Π × M .

The proof of this lemma is referred in appendix (A2). We have the following properties for the modified correspondences: Claim 3.2. For each agent i ∈ I, (i) the correspondence β i is u.s.c. on Π × M with compact convex values. (ii) the correspondence β i is l.s.c. on Π × M . (iii) the demand correspondence δ i is u.s.c. on Π×M with non-empty compact, convex values.

The proof of this claim is given in appendix (A3).

Applying Kakutani's fixed point theorem. Let us define the correspondence

F : Π × i∈I X i × i∈I Θ i × M -→ Π × i∈I X i × i∈I Θ i × M such that: F (p, q, x • , θ • , µ) = Φ(x • , θ • ) × i∈I δ i (p, q, µ) × Γ(x • , p, q) where Φ(x • , θ • ) = (p, q) ∈ Π : ∀(p , q ) ∈ Π, (p -p ) • i∈I x i -e i + ( q -q ) • i∈I θ i -θ 0) and Γ(x • , p, q) = µ ∈ M : µ(0)w(0) = r(0)p(0) • i∈I x i (0) -e i (0) -+ p(0) • g(0) + qθ µ(s)( Rθ)(s) = r(s)p(s) • i∈I x i (s) -e i (s) -+ p(s) • g(s)
Correspondence F is u.s.c. with non-empty convex compact values. Applying Kakutani's fixed point theorem, there exists (p, q, x • , θ

• , µ) ∈ Π × i∈I X i × i∈I Θ i × M such that: ∀i ∈ I, (x i , θ i ) ∈ δ i (p, q, µ), (3.3) ∀(p, q) ∈ Π, (p -p) • i∈I (x i -e i ) + ( q -q) • ( i∈I θ i -θ) 0, (3.4) µ(0)w(0) = r(0) p(0) • i∈I x i (0) -e i (0) -+ p(0) • g(0) + qθ (3.5) µ(s)( Rθ)(s) = r(s) p(s) • i∈I x i (s) -e i (s) -+ p(s) • g(s) (3.6)
We will now show that the obtained fixed point is an abstract equilibrium. In order to do this, we need only to prove that commodity and asset markets clear. This follows from Claims 3.3 to 3.8.

For convenience, we introduce the following sets ∆(σ), ∀σ ∈ Σ: ∆(0) := {(p(0), q) ∈ R L + ×R J + : p(0) + q = 1} ∆(s) := {(p(s), s ∈ S) ∈ R SL + : p(s) = 1} Note that ∆(σ) for all date-event σ ∈ Σ are simply projections of Π. For a given set U ∈ R n , we denote by U • the negative polar cone of U , i.e. the cone of vectors η ∈ R n such that η • u 0, for every u ∈ U . Proof. Taking p(s) = p(s) for every date-event s ∈ S in fixed point property (3.4), one has, for all prices (p(0), q) ∈ ∆(0):

p(0) • i∈I (x i (0) -e i (0)) + q • ( i∈I θi -θ) p(0) • i∈I (x i (0) -e i (0)) + q • ( i∈I θi -θ) (3.7)
Moreover, fixed point property (3.3) states that (x i , θ i ) ∈ β i (p, q, µ). Summing first period constraints among all agents and recalling fixed point property (3.5), we get:

p(0) • i∈I x i (0) -e i (0) + q • ( i∈I θi -θ) 0 (3.8)
According to inequalities (3.7) and (3.8), one has: Proof. Consider a date-event s ∈ S. According to fixed point property (3.4), by choosing p(σ) = p(σ) for all date-event σ ∈ Σ \ {s} and q = q, one has:

p(0) • i∈I x i (0) -e i (0) + q • ( i∈I θi -θ) 0, ∀(p(0), q) ∈ ∆(0). Thus i∈I x i (0) -e i (0) ; ( i∈I θi -θ) ∈ [∆(0)] • = R LJ -, i.e.
p(s) • i∈I x i (s) -e i (s) p(s) • i∈I x i (s) -e i (s) , ∀ p(s) ∈ ∆(s).
Moreover, fixed point property (3.3) states that (x i , θ i ) ∈ β i (p, q, µ). Summing second period constraints among all agents and recalling fixed point property (3.6), we get:

p(s) • i∈I x i (s) -e i (s) µ(s)R(s)( i∈I θi -θ) (3.9)
Since µ 0, R > 0 (assumption (F2)), and ( i∈I θi -θ) 0 (Claim 3.3), focusing on date-event s, one has:

p(s) • i∈I x i (s) -e i (s) 0, ∀p(s) ∈ ∆(s) (3.10) This means that x i (s) -e i (s) ∈ [∆(s)] • = R L -, i.e. i∈I x i (s)
i∈I e i (s). Claim 3.5. Budget constraints of all agents i ∈ I are saturated:

p(0) • (x i (0) -e i (0) + g i (0)) + r(0) p(0) • (x i (0) -e i (0)) -+ qθ i = µ(0)w i (0) (3.11) p(s) • (x i (s) -e i (s) + g i (s)) + r(s) p(s) • (x i (s) -e i (s)) -= µ(s)( Rθ i )(s) (3.12)
Proof. We will only consider the case where s ∈ S (for s = 0, the proof is similar). Assume on the contrary that (3.12) does not hold, i.e. there exists i ∈ I, and a date-event s ∈ S where the equality is not true, i.e. p(s) • (x i (s) -e i (s) + g i (s)) + r(s) p(s) • (x i (s) -e i (s)) -< µ(s)( Rθ i )(s).

In view of Claims (3.3)-(3.4) and of our choice of k, there exists some consumption plan x i ∈ B(ΣL, k) satisfying x i > x i and

p(s) • (x i (s) -e i (s) + g i (s)) + r(s)(p(s) • (x i (s) -e i (s)) -) ( V θ i )(s) + w i (s).
Thus, (x i , θ i ) ∈ β i (p, q, µ). Following assumption (C1), u i (x i ) > u i (x i ), which yields a contradiction to the fixed point property 3.3.

Claim 3.6. Commodity prices are strictly positive, i.e. p 0.

Proof. Indeed, if not, there exists a date-event σ ∈ Σ and a good ∈ L such that p(σ, ) = 0. Let us consider an agent i ∈ I. In view of our choice of k and Claims (3.3)-(3.4), we can find some consumption plan x i ∈ B(ΣL, k) such that x i (σ) > x i (σ) and (x i , θ i ) ∈ β i (p, q, µ).

Following assumption (C1), u i (x i ) > u i (x i ), which yields a contradiction to the fixed point property (3.3).

Claim 3.7. At first period, we have the following property:

i∈I x i (0) = i∈I e i (0) and q • i∈I θ i -θ = 0
Proof. Indeed, according to Claim (3.5), summing among all agents equalities (3.11), one has

p(0) • i∈I x i (0) -e i (0) + q • i∈I θ i -θ = 0
Since (p(0), q) ∈ R LJ + , x i (0) -e i (0) 0, and i∈I θ i -θ 0, one has:

p(0) • i∈I x i (0) -e i (0) = 0 and q • i∈I θ i -θ = 0 (3.13)
Since p(0) 0, we get i∈I x i (0) = i∈I e i (0).

Claim 3.8. Asset markets clear and second period commodity markets clear:

i∈I θ i = θ and i∈I x i (s) = i∈I e i (s)
Proof. Indeed, according to the previous claim, one has q• i∈I θ i -θ = 0. Moreover, referring to claim (3.4), inequality (3.8) tells us that R(s) i∈I θ i -θ 0. Thus, by setting θ = -i∈I θ i -θ one has V θ 0. Assume that there exists a date-event σ ∈ Σ such that

V (σ)θ > 0. (3.14)
Let an agent i ∈ I. According to Claim 3.5, budget constraints of agents are saturated at fixed point. Consider an agent i ∈ I, One has,

p (x i -e i + g i ) + r • (p (x i -e i ) -) = V θ i
Hence, for λ > 0, one has (x i , θ i + λθ) ∈ β i (p, q, µ). Moreover, recalling inequality (3.14), one has at date-event σ ∈ Σ,

p(σ) • (x i (σ) -e i (σ) + g i (σ)) + r(σ)(p(σ) • (x i (σ) -e i (σ)) -) < V (σ)(θ i + λθ)
One can find an allocation x i ∈ B(ΣL, k) such that (x i , (θ i + λθ)) ∈ δ i (p, q, µ) which contradicts fixed point property (3.3). As for the clearance of second period commodity markets, it is straightforward by summing among all agents second period's saturated budget constraints (3.12) and the fact that p(s) 0 for all date-event s ∈ S (Claim 3.6).

We have proved that there exists an abstract equilibrium. In order for the abstract equilibrium to be achieved as an equilibrium, we need to show that at every abstract equilibrium, µ 0. The following lemma shows under what condition this is satisfied. Lemma 3.3. Under (M1)-(NRA)-(PR), at every abstract equilibrium µ 0, and the abstract equilibrium is achieved as an equilibrium.

Proof. Assume that (PR) is satisfied. Referring to 3.5, one has:

µ(0)w(0) p(0) g(0) + q • θ.
Since p(0) 0, the fact that g(0) 0 and recalling assumption (M1), w(0) > 0, one has µ(0) > 0. Moreover, referring to 3.6, one has at a state s ∈ S of the second period:

µ(s)(Rθ)(s) p(s) g(s) > 0.
Since θ is a non-risky portfolio, one has µ(s) > 0, ∀s ∈ S.

Hence, we have proved the existence of an abstract equilibrium with strictly positive index of the reciprocal of the overall price level. Following remark 3.1, we get that there exists a monetary equilibrium (p , q , x • , θ

• ) with no transfers, where

p (σ) = p(σ) µ(σ)
, ∀σ ∈ Σ and q = q µ(0) .

Existence of Monetary Equilibrium with Transfers

Let E be a standard economy satisfying assumption (PP) and (NRA). Let d ∈ R Σ ++ . Given assumptions (C1) and (F2), we may restrict ourselves to positive commodity and asset prices. We consider the following convex, compact set for commodity and asset price vectors:

Π d = (p, q) ∈ R LΣ + × R J + : p(0) + q = d(0) and p(s) = d(s), ∀s ∈ S 4.1.
Truncating the economy. We provide hereafter the definition of a truncated monetary economy. The following lemma establishes that in order to prove Theorem 2.2 and Theorem 2.3, we can suppose without any loss of generality that commodity and financial sets are compact.

Transfers are also considered to belong to an adequate compact, convex set.

Definition 4.1. If E c = (X • , u • , e • , g • , ξ •
) is a commodity market, and E f = (R, Θ • , θ) is a financial market, then for any h > 0, we let E c h and E f h defined by

E c h = (X • h , u • h , e • , g • , ξ • ) E f h = (R, Θ • h , θ) where, for each i ∈ I, X i h = X i ∩ B(Σ × L, h). We set u i h as the restriction of u i to X i k , Θ i h = Θ i ∩ B(J, h), and T h = B(Σ, h).
Let X is the set of attainable commodity allocations defined in (3.1). We set T the set of attainable transfers, i.e.

T = t ∈ R Σ : ∃x • ∈ X, ∃(p, q) ∈ Π d , t = r • p i∈I x i -e i -+ p g -V θ -w
Let the set of attainable portfolios be defined by:

Θ i = {θ i ∈ Θ i : ∃(p, q) ∈ Π d , ∃x i ∈ X i , ∃t ∈ T , p (x i -e i + g i )+ r • (p (x i -e i ) -) = V θ i +w i +ξ i t} We set Θ = i∈I Θ i .
Notice here that the attainable portfolio sets depend now on transfers t ∈ T , thus compactness of Θ will crucially depend on the compactness of T . Lemma 4.1. Let E c commodity market and E f a financial market. Then (a) There exists h > 0 such that ∀i ∈ I, X i ⊂ int B(Σ × L, h), Θ i ⊂ int B(J, h) and T ⊂ B(Σ, h) (4.1) (b) If h > 0 is sufficiently large such that 4.1 is satisfied, then for each money market E m , any monetary equilibrium of the truncated economy

(E c h , E f h , E m ) is a monetary equilibrium of the initial economy (E c , E f , E m ).
The proof of this lemma is referred in appendix (A4). We can now fix h > 0. Following Lemma 4.1, we can suppose without any loss of generality that for each i ∈ I, the sets X i and Θ i are compact. For convenience, we set T = T h . 4.2. Modifying budget sets. We will begin by defining the adequate price level that one should consider. Let an agent i ∈ I. According to assumption (C3), for every agent i ∈ I, there exists a consumption plan x i ∈ X i such that x i -e i + g i 0. Thus, for every date-event σ ∈ Σ, there exists χ i (σ) > 0 such that x i -e i + g i -χ i I1 Σ .

We will need the following notation: g(σ) = Inf {g(σ, l), l ∈ L}. Let For each i ∈ I, for each (p, q, t) ∈ Π d × T , we define the following correspondances:

d (0) := Max        0 : w(0) -min i∈I { w i (0) ξ i } min{1, g(0)} (1 + r(0))        (4.2) d (s) := R(s)θ g(s) + min i∈I { χ i (s) ξ i } , ∀s ∈ S (4.
B i d (p, q, γ(t)) := {(x i , θ i ) ∈ X i × Θ i : p (x i -e i + g i ) + r • p (x i -e i ) - V θ i + w i + ξ i γ(t)}. β i d (p, q, γ(t)) := {(x i , θ i ) ∈ X i × Θ i : p (x i -e i + g i ) + r • p (x i -e i ) - V θ i + w i + ξ i γ(t)}. d i d (p, q, γ(t)) := {(x i , θ i ) ∈ X i ×Θ i : (x i , θ i ) ∈ B i d (p, q, γ(t)
) and [P i (x i )×Θ i ]∩B i d (p, q, γ(t)) := ∅} As we shall see in the following lemma, the constructed price level d ∈ R Σ + leads to the non-emptyness of β i d on Π d × T . Lemma 4.2. For every agent i ∈ I, the correspondence β i d has non-empty values on Π d × T . For the proof of Lemma 4.2, refer to appendix (A5). We have the following properties for the modified correspondences: 7 Recall that T = T h , and given the definition of K, one can always choose h > 0 big enough in order for γ to be defined from T into T .

(ii) the correspondence B i d is l.s.c. on Π d × T . (iii) the demand correspondence is u.s.c. on Π d × T with non-empty compact, convex values.

The constructed correspondences are well behaved. We may now apply a fixed point theorem that will lead us to the existence of a monetary equilibrium.

F d : Π d × i∈I X i × i∈I Θ i × T -→ Π d × i∈I X i × i∈I Θ i × T such that: F d (p, q, x • , θ • , t) = Φ d (x • , θ • ) × i∈I d i d (p, q, γ(t)) × Γ d (p, q, x • , θ • )
where

Φ d (x • , θ • ) = (p, q) ∈ Π d : ∀(p , q ) ∈ Π d , (p -p ) • i∈I (x i -e i ) + ( q -q ) • ( i∈I θ i -θ) 0) and Γ d (p, q, x • ) = t ∈ T : t = r • p i∈I x i -e i -+ p g -V θ -w
The correspondence F d is u.s.c. with non-empty convex compact values. Applying Kakutani's fixed point theorem, there exists (p, q, x • , θ

• , t ) ∈ Π d × i∈I X i × i∈I Θ i × T such that: ∀i ∈ I, (x i , θ i ) ∈ d i d (p, q, γ(t)), (4.6) ∀(p, q) ∈ Π d , (p -p) • i∈I (x i -e i ) + ( q -q) • ( i∈I θ i -θ) 0, (4.7) t = r • p i∈I x i -e i -+ p g -V θ -w. (4.8)
Notice here that the fixed point obtained satisfies (x i , θ i ) ∈ d i (p, q, γ(t)). According to the authority's constraints (4.8) and the definition of γ (refer to 4.4), one has γ(t) = t. Thus, (x i , θ i ) ∈ d i d (p, q, t). Finally, in order to prove that the obtained fixed point is achieved as a monetary equilibrium, we need only to show that commodity and asset markets clear. These proofs are very similar to the case of no transfers, one needs only to replace conditions (3.5) and (3.6) by the new authority's constraint (4.8).

Application:

The case of positive transfers. This section is devoted to the proof of Theorem 2.3: In choosing a higher price level, we get to positive transfers. (1 + r(0)) and c (s) R(s)θ g(s) , ∀s ∈ S (4.9) If p(0) = 0, similarly, one has:

p(0) • (x i (0) -e i (0) + g i (0)) + r(0)p(0) • (x i (0) -e i (0)) -= (1 -r(0))p(0) • (x i (0) -e i (0)) + p(0) • g i (0) = p(0) 1 + r(0) • x i (0) -e i (0) + g i (0) < 0.
and (x i , 0) belongs to β i (p, q, µ).

If p(0) = 0, one has q = 0. Since 0 ∈ int Θ i and for all σ ∈ S,

p(σ) • (x i (σ) -e i (σ) + g i (σ)) + r(σ) p(σ) • (x i (σ) -e i (σ)) -< 0.
By a continuity argument, one can choose a portfolio θ i ∈ Θ i such that

q • θ i < 0 p(s) • (x i (s) -e i (s) + g i (s)) + r(s) p(s) • (x i (s) -e i (s)) -< µ(s)R(s)θ i , ∀s ∈ S which means that (x i , θ i ) ∈ β i (p, q, µ).
A3. Proof of Claim 3.2:

Proof. : Let us begin by showing property (i): Let i ∈ I and (x n , θ n , p n , q n , µ n ) be a sequence in X i × Θ i × Π × M . Following standard compactness argument, one can assume that the sequence (x n , θ n , p n , q n , µ n ) converges to (x, θ, p, q, µ) and such that (

x n , θ n ) ∈ β i (p n , q n , µ n ). For each n ∈ N, p n (0) • (x n (0) -e i (0) + g i (0)) + r(0) p n (0) • (x n (0) -e i (0)) - -q • θ n + µ n (0)w i (0) p n (s) • (x n (s) -e i (s) + g i (s)) + r(s) p n (s) • (x n (s) -e i (s)) - µ n (s)( Rθ n )(s), ∀s ∈ S
Passing to the limit, we get (x, θ) belongs to β i (p, q, µ).

Let us now show that β i is l.s.c. on Π × M : Let (p, q, µ) ∈ Π × M . Since β i (p, q, µ) has non-empty, convex values (refer to lemma 3.1), we have β i (p, q, µ) = cl β i (p, q, µ). Finally, the claim follows from the fact that β i (p, q, µ) has an open graph.

Finally, the u.s.c. follows from the continuity of the utility functions. Indeed, δ i (p, q, µ) is the argmax of u i on β i (p, q, µ). Since u i is continuous and β i is continuous on Π × M , it follows from Berge's Maximum theorem that δ i is u.s.c. on Π × M with non-empty values. The convexity of δ i (p, q, µ) follows from the quasi-concavity of u i . A4. Proof of Lemma 4.1:

Proof of Part (a) :

This part follows from the compactness of the sets X, T and Θ. Indeed, the compactness of X follows from Assumptions (C1). Following Assumptions (F1) and (F2), for each i ∈ I, the set Θ i is bounded. Indeed, for this end, let us begin by showing that T is a closed and bounded subset of R Σ . Let (x • ν , p ν , q ν ) be a sequence in T × X × Π d . Following standard compactness argument, we may assume that the sequence (x • ν , p ν , q ν ) converges to (x • , p, q). Let (t ν ) a sequence in T . For each ν ∈ N, we thus have t ν = r • p ν i∈I x i -e i -+ p ν g -V θ -w (4.11) Note that p(s) • (x i (s) -e i (s) + g i (s)) < -χ i (s)d(s). But remark that by construction of d(s) (refer to 4.3), one has -χ i (s)d(s) ξ i K(s). Indeed, Recalling the definition of K(s), one has -χ i (s)d(s) ξ i K(s), thus p(s) 1+r(s) •(x i (s)-e i (s)+g i (s)) < ξ i γ(t, s). Moreover, if p(0) = 0, similarly to the case s ∈ S , one has: p(0) • (x i (0) -e i (0) + g i (0)) + r(0)p(0) • (x i (0) -e i (0)) -= p(0) 1 + r(0)

• x i (0) -e i (0) + g i (0) < 0 At date-event σ = 0, one has w i (0) + ξ i γ(t, 0) 0. Indeed, by construction (refer to 4. (1 + r(0))w i (0) + ξ i Min {1, g(0)}d(0) -(1 + r(0))w(0) 0

(1 + r(0))w i (0) + ξ i K(0) 0.

Since γ(t, 0) = Max { t(0), K(0)}, one has w i (0) + ξ i γ(t, 0) 0. Hence, (x i , 0) belongs to β i (p, q, t).

If p(0) = 0, one has q = 0. Since 0 ∈ int Θ i and for all s ∈ S, p(s) • (x i (s) -e i (s) + g i (s)) + r(s) p(s) • (x i (s) -e i (s)) -< ξ i γ(t, s), by a continuity argument, one can choose a portfolio θ i ∈ Θ i such that q•θ i < 0 ≤ w i (0)+ξ i γ(t, 0) and p(s) • (x i (s) -e i (s) + g i (s)) + r(s) p(s) • (x i (s) -e i (s)) -< R(s)θ i + ξ i γ(t, s), ∀s ∈ S and (x i , θ i ) belongs to β i (p, q, γ(t)).

A6. Proof of Claim 4.1:

Proof. : Let us begin by showing property (i): Let i ∈ I and (x n , θ n , p n , q n , γ(t n )) be a sequence in X i ×Θ i ×Π d ×T . Following classical compactness argument, one can assume that the sequence Notice that, for > 0 small enough, referring to (4.14), we have R(k) = Rθ 0, one has R 0. Let us check that Span R = Span R. By construction, Span R ⊂ Span R. Reciprocally, since J j=1 R(j) = R(k), one has R(k) ⊂ Span R. Finally, for all j ∈ J, j = k, one has

R(j) = 1 R(j) - 1 J R(k) = 1   R(j) - 1 J J j=1 R(j)   ⊂ Span R
The proof of Corollaries 2.1 and 2.2 follow:

Proof. Consider a standard economy E := (E c , E f , E m ) satisfying assumption (NRA), i.e. Rθ 0. According to Lemma 4.3, there exists ( R, θ) such that R 0, Span R = Span R, R θ = Rθ and θ = I1 R J .

If we set E := (E c , E f , E m ) where we modify the financial market E f = (R, Θ • , θ) by E f = ( R, Θ • , θ). The auxiliary economy E f satisfies assumption (PP), (F1) and (F2). Thus, according to Theorem 2.3, there exists a monetary equilibrium (p, q, x • , θ • ) of E.

Since Span R ⊂ Span R, we have the following property: ∀k ∈ J, ∃γ ∈ R J : R(k) = j∈J γ(j) R(j)

For every asset k ∈ J, define q(k) = j∈J γ(j) q(j). If (p, q, x • , θ • ) is a monetary equilibrium of E, then (p, q, x • , θ • ) is a monetary equilibrium of E, with ∀i ∈ I, ∀k ∈ J, θ i (k) = j∈J γ(j) θ i (j).

Recalling the price levels properties for (p, q), we have p(s) = c(s), whereas we do not know anything on first period price levels p(0) + q .

Claim 3 . 3 .

 33 At the first period, we have the following property: i∈I x i (0) i∈I e i (0), and i∈I θi θ

Claim 3 . 4 .

 34 i∈I x i (0) i∈I e i (0) and i∈I θi θ. At the second period, commodity markets satisfy: i∈I x i (s) i∈I e i (s), ∀s ∈ S

3 )

 3 Consider price levels d 0, d d , and let the mapping γ, from T into T , 7 be defined by: γ(t) := γ((t, σ), σ ∈ Σ) where γ(t, σ) = max{ t(σ), K(σ)},(4.4) given that K(0) := d(0)Min {1, g(0)} -w(0)(1 + r(0)), K(s) := d(s)g(s) -R(s)θ, ∀s ∈ S.(4.5)

Claim 4 . 1 .

 41 For each agent i ∈ I, (i) the correspondence B i d is u.s.c. on Π d × T with compact convex values.

4. 3 .

 3 Applying Kakutani's fixed point theorem. Let us define the correspondence

Claim 4 . 2 .

 42 Under assumptions (M1)-(PR) and (PP), there exists a price level c ∈ R Σ + above which transfers are positive elements of R Σ , for any consumption allocation x • ∈ R ΣLI . Proof. Let c ∈ R Σ ++ , c = (c (σ), σ ∈ Σ). Consider a state s ∈ S and let c (0) w(0) Min 1, g(0)

  )g(s) + d(s) min i∈I χ i (s) ξ i R(s)θ d(s)g(s) + d(s) χ i (s) ξ i R(s)θ χ i (s) ξ i d(s) d(s)g(s) -R(s)θ -χ i (s)d(s) ξ i d(s)g(s) -R(s)θ

{

  w i (0) ξ i } (1 + r(0)) Min {1, g(0)} d(0)Min {1, g(0)} + (1 + r(0)) min i∈I w i (0) ξ i (1 + r(0))w(0)

Given a dimensional space R n , we denote I 1n the vector in R n with all components equal to one.

Notice that it is equivalent to consider any public portfolio θ 0 in assumption (PP), given an adequate corresponding choice of the return matrix R.

Appendix

A1. Proof of Lemma 3.1:

Proof of Part (a) : This part follows from the compactness of the sets X, and Θ. Indeed, the compactness of X follows from Assumptions (C1). Following Assumptions (F1) and (F2), for each i ∈ I, the set Θ i is closed and bounded: Indeed, let us consider (x i ν , p ν , q ν ) be a sequence in X i × Π and (θ i ν ) a sequence in Θ i . Then, for each ν ∈ N, we have

By a classical compactness argument, we may suppose that the sequence (x i ν , p ν , q ν ) converges to (x i , p, q). If the sequence (θ i ν ) is not bounded, then, passing to a subsequence if necessary, we can suppose that lim n θ i ν = +∞. Multiplying (4.12) by 1/ θ i ν and passing to the limit, there exists κ ∈ R J with V κ = 0 where κ = 1. Assumption (F2) implies that if V κ = 0 then κ = 0: a contradiction. It follows that the sequence (θ i ν ) is bounded, and passing to a subsequence if necessary, we can suppose that there exists θ i ∈ R J such that (θ i ν ) converges to θ i and θ i ∈ Θ i .

Proof of Part (b) :

Let (x • , θ • , p, q) be a monetary equilibrium with no-transfers of E = (E c k , E f k , E m ). Suppose that it is not a monetary equilibrium of E. Then for some i, there exists (x i , θ i ) ∈ X i × Θ i such that u i (x i ) > u i (x i ) and (x i , θ i ) is budget feasible. Since (x i , θ i ) belongs to int B(Σ × L, k) then, it is easy to find 0 < λ 1 such that

) is budget feasible. Indeed, we see in the following that budget sets are convex: for this, we need only to recall that,

Finally, from Assumption C.2, we also have

which yields a contradiction.

A2. Proof of Lemma 3.2:

Proof. Let (p, q, µ) ∈ Π×M . Let an agent i ∈ I. According to (C3), we can choose a consumption plan x i ∈ X i such that x i -e i +g i 0. Since g i 0 and x i -e i -g i 0, we get (x i -e i ) + = 0 and (x i -e i ) -= -(x i -e i ) 0. Thus, for every state s ∈ S, one has:

Passing to a subsequence if necessary, we can suppose that there exists t ∈ R Σ such that (t ν ) converges to t and t ∈ T . Let us consider an agent i ∈ I. We now show that Θ i is a closed and bounded subset of R J . Let us consider (x i ν , p ν , q ν , t ν ) be a sequence in X i × Π d × T . By a classical compactness argument, we may assume that the sequence (x i ν , p ν , q ν , t ν ) converges to (x i , p, q, t). Let (θ i ν ) be a sequence in Θ i . Then, for each ν ∈ N, we thus have

If the sequence (θ i ν ) is not bounded, then, passing to a subsequence if necessary, we can suppose that lim n θ i ν = +∞. Multiplying (4.12) by 1/ θ i ν and passing to the limit, there exists κ ∈ R J with κ = 0 where κ = 1: a contradiction. It follows that the sequence (θ i ν ) is bounded, and passing to a subsequence if necessary, we can suppose that there exists θ i ∈ R J such that (θ i ν ) converges to θ i and θ i ∈ Θ i .

Proof of Part (b) :

Let (x • , θ • , p, q, t) be a monetary equilibrium of

). Suppose that it is not a monetary equilibrium of E. Then for some i, there exists (

it is easy to find 0 < γ 1 such that

) is budget feasible. Indeed, we see in the following that budget sets are convex: for this, we need only to recall that,

Finally, from Assumption C.2, we also have

which yields a contradiction.

A.5 Proof of Lemma 4.2

Proof. Let an agent i ∈ I. According to (C3), we can choose a consumption plan x i ∈ X i such that x i -e i + g i -χ i I1 Σ . Since g i 0 and x i -e i -g i 0, we get (x i -e i ) + = 0 and (x i -e i ) -= -(x i -e i ) 0. Thus, for every state s ∈ S, one has:

Passing to the limit, we get (x, θ) belongs to B i d (p, q, γ(t)).

Let us now show that B i is l.s.c. on Π d : Let (p, q) ∈ Π d . Since β i d (p, q, γ(t)) has non-empty, convex values (refer to lemma 4.1), we have B i d (p, q, γ(t)) = cl β i d (p, q, γ(t)). Finally, the claim follows from the fact that β i d (p, q, γ(t) has an open graph.

Finally, the u.s.c. follows from the continuity of the utility functions. Indeed, d i (p, q, γ(t)) is the argmax of u i on B i d (p, q, γ(t)). Since u i is continuous and

Proof. The proof will be one in two steps. We begin by showing that there exists ( R, θ) such that R θ = Rθ, Span R = Span R and θ = e k . 8

Indeed, according to assumption (NRA), Rθ 0. This implies that θ = 0, i.e. there exits an asset k ∈ J such that θ(k) = 0. We posit

It is evident that R θ = Rθ. We now show that Span R = Span R. By construction, Span R ⊂ Span R. Reciprocally, in order to show that Span R ⊂ Span R, one only needs to show that R(k) ⊂ Span R. This follows from the fact that

We now show that there exists ( R, θ) such that R θ = R θ, Span R = Span R and θ = I1 R J Let > 0 and define:

R(j) 8 The vector e k designates the vector in R J with all its components equal to 0 except for the k th one.