Carlos Cruz

Francisco Rubio

Diego Venegas-Yazigi

Nathalie Audebrand

Christophe Calers

Evgenia Spodine

Veronica Paredes-Garcia

Verónica Paredes-García

Crystal Growth

Design

Toward a Rational Design of 3d-4f Heterometallic Coordination Polymers based on Mixed Valence Copper Centers

Keywords: . Coordination Polymers, Copper, Gadolinium, Mixed-Valence, 3d-4f, Magnetism 1

établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

researchers. 1,[START_REF] Hao | A 3D Porous Coordination Polymer Transformed from a 1D Nonporous Coordination Polymer for Selectively Sensing of Diiodomethane[END_REF][START_REF] Li | Ni II , Mn II , and Co II Coordination Polymers with 1,4-Naphthalenedicarboxylic Acid Exhibiting Metamagnetic and Antiferromagnetic Behaviors[END_REF][START_REF] Wang | 1-D Helical Chain, 2-D Layered Network and 3-D Porous Lanthanide-Organic Frameworks Based on Multiple Coordination Sites of Benzimidazole-5,6-Dicarboxylic Acid: Synthesis, Crystal Structure, Photoluminescence and Thermal Stability[END_REF] In this sense, the combination of two different metal cations in the same structure, such as first row transition (3d) and lanthanide (4f) cations has been used as a strategy to obtain new CPs with novel structures and potential applications such as magnetic refrigeration, solid state luminescence and catalysis. [START_REF] Bing | Two Lanthanide(III)-Copper(II) Organic Frameworks Based on {OLn 6 } Clusters That Exhibited a Large Magnetocaloric Effect and Slow Relaxation of the Magnetization[END_REF][START_REF] Dong | Copper(I)-Lanthanide(III) Heterometallic Metal-Organic Frameworks Constructed from 3-(3-Pyridyl)Acrylic Acid: Syntheses, Structures, and Properties[END_REF][START_REF] Li | Three-Dimensional Pillared-Layer 3d-4f Heterometallic Coordination Polymers with or without Halides[END_REF][START_REF] He | LnZn(IN) 3 (C 2 H 4 O 2 )] n (Ln = Eu, Sm, and Gd): Chiral Enantiomerically 3D 3d-4f Coordination Polymers Constructed by Interesting Butterfly-like Building Units and -[Ln-O-Zn] n -Helices[END_REF][START_REF] Yang | Six New 3d-4f Heterometallic Coordination Polymers Constructed from Pyrazole-Bridged Cu(II)Ln(III) Dinuclear Units[END_REF][START_REF] Yang | Crystal Structure, and Catalytic Properties of Porous 3d-4f Heterometallic Coordination Polymers Constructed from Pyrazine-2,3-Dicarboxylic Acid[END_REF] Compared with the lanthanide series, first row transition metals have a richer chemistry as a consequence of the various oxidation states (from +1 to +7) and coordination geometries. Specifically, Cu I and Cu II cations can adopt a wide variety of coordination geometries (linear, trigonal, tetrahedral, trigonal bipyramid, square base pyramid, octahedral) as compared to other transition metal ions. The coordination plasticity of the copper cations to accommodate different amounts of donor atoms has been also exploited in the construction of 3d-4f extended structures. [START_REF] Shaowei | Recent Advances in the Construction of Lanthanide-Copper Heterometallic Metal-Organic Frameworks[END_REF] From a magnetic point of view, Cu II CPs have been widely studied due to the simple paramagnetic ground state (S=1/2) of this cation, which allows magneto-structural correlations. [START_REF] Fedin | Electron Paramagnetic Resonance of Switchable Copper-Nitroxide-Based Molecular Magnets: An Indispensable Tool for Intriguing Systems[END_REF][START_REF] Rentschler | Molecule-Based Magnets: Ferro-and Antiferromagnetic Interactions in Copper (II)-Polyorganosiloxanolate Clusters[END_REF] After Bencini et al. [START_REF] Bencini | Crystal and Molecular Structure of and Magnetic Coupling in Two Complexes Containing Gadolinium(III) and Copper(II) Ions[END_REF] found a compound showing ferromagnetic coupling phenomenon between Cu II and Gd III , great efforts have been made to obtain Cu II -Ln III heterometallic compounds with higher dimensionality, seeking for new magnetic materials. 15 Thus, Cu II -Ln III heterometallic networks have been reported in the last decade presenting ferromagnetic behavior based on different N,O-bifunctional organic ligands, such as, iminodiacetic acid, [START_REF] Manna | N. 3D Heterometallic (3d-4f) Coordination Polymers: A Ferromagnetic Interaction in a Gd(III)-Cu(II) Couple[END_REF] glycine, [START_REF] Li | Three Series of 3d-4f Heterometallic Polymers Based on[END_REF] glycinehydroxamic acid, [START_REF] Pavlishchuk | Formation of Coordination Polymers or Discrete Adducts via Reactions of Gadolinium(III)-Copper(II) 15-Metallacrown-5 Complexes with Polycarboxylates: Synthesis, Structures and Magnetic Properties[END_REF] pyrazoledicarboxylic acid [START_REF] Yang | Six New 3d-4f Heterometallic Coordination Polymers Constructed from Pyrazole-Bridged Cu II Ln III dinuclear Units[END_REF] and benzotriazolecarboxylic acid. [START_REF] Xin | Seven 3d-4f Coordination Polymers of Macrocyclic Oxamide with Polycarboxylates: Syntheses, Crystal Structures and Magnetic Properties[END_REF] On the other hand, considering that the full shell of Cu I (3d [START_REF] Yang | Crystal Structure, and Catalytic Properties of Porous 3d-4f Heterometallic Coordination Polymers Constructed from Pyrazine-2,3-Dicarboxylic Acid[END_REF] ) can be used as an antenna to improve lanthanide luminescence, Cu I -Ln III heterometallic coordination polymers (HCPs) have been the focus of research in the solid state light emissive materials. [START_REF] Zeng | 3d-4f Metal-Organic Framework with Dual Luminescent Centers That Efficiently Discriminates the Isomer and Homologues of Small Organic Molecules[END_REF][START_REF] Xu | A Cuprous/Lanthanide-Organic Framework as the Luminescent Sensor of Hypochlorite[END_REF] Literature presents a very rich Cu I -Ln III coordination polymer chemistry in which Cu I is stabilized into extended structures using principally functionalized N-heterocyclic ligands, such as pyridine and pyrazine. [START_REF] Wu | Extended Structures and Magnetic Properties of Lanthanide-Copper Complexes with Picolinic Acids as Bridging Ligands[END_REF]24,[START_REF] Bo | Novel Three-Dimensional Pillared-Layer Ln(III)-Cu(I) Coordination Polymers Featuring Spindle-Shaped Heterometallic Building Units[END_REF][START_REF] Sun | Novel Luminescent Heterobimetallic Ln-Cu(I) 3D Coordination Polymers Based on 5-(4-Pyridyl) Isophthalic Acid as Heteroleptic Ligand. Synthesis and Structural Characterization[END_REF][START_REF] Ay | In Situ Hydrothermal Syntheses, Structures and Photoluminescent Properties of Four Novel Metal-Organic Frameworks Constructed by Lanthanide (Ln=Ce(III), Pr(III), Eu(III)) and Cu(I) Metals with Flexible Dicarboxylate Acids and Piperazine-Based Ligands[END_REF] For the majority of the reported Cu I -Ln III CP the stabilization of Cu I is not always done by the coordination of only a N-organic ligand, but also the presence of coordinated halides anions is found. [START_REF] Zhu | A 3-D Gd(III)-Cu(I) Coordination Polymer Based on Nicotinate and Oxalate Ligands: Synthesis, Crystal Structure, and Magnetic Properties[END_REF][START_REF] Wu | A Novel Three-Dimensional Hetero-Metallic Coordination Polymer: Poly[[diaquadi- 3 -chlorido- 2 -chlorido-tri- 3 -nicotinato-tricopper(I)gadolinium(III)] hemihydrate][END_REF][START_REF] Cao | A Tubular One-Dimensional Polymer Constructed from Alternating Clusters of Europium(III)-Water and Copper(I) Chloride Bridged by 4-(Pyridin-4-yl)Benzoate[END_REF][START_REF] Fang | Three Pillared-Layer 3d-4f Heterometallic Frameworks Based on Tetranuclear Lanthanide Clusters[END_REF][START_REF] Xia | Crystal Structure, and Photoluminescent Properties of a Series of Ln III -Cu I Heterometallic Coordination Polymers Based on Cu 4 I 3 Clusters and Ln-Ina Rod Units[END_REF][START_REF] Fang | Two Series of Sandwich Frameworks Based on Two Different Kinds of Nanosized Lanthanide(III) and Copper(I) Wheel Cluster Units[END_REF][START_REF] Chen | Heterometallic Coordination Polymers Constructed by Linear Ligands[END_REF] To the best of our knowledge, the only example of a 3d-4f HCP containing Cu I /Cu II has been obtained by Luo et al. [Gd 2 Cu II 2 Cu I 5 (IN) 10 (µ 2 -Cl)(µ 2 -OH 2 ) 2 (µ 3 -OH) 2 ](ClO 4 ) 2 (HIN= isonicotinic acid), [START_REF] Luo | The First Self-Penetrating Topology Based on an Unusual -Po Net with Double Edges Constructed from a 12-Connected Gd 2 ( 2 -O carboxylate ) 2 ( 2 -OH 2 ) 2 ( 3 -OH) 2 Cu 2 Core[END_REF] probably due to the coordination requirements of Cu I , Cu II and Ln III cations present in the same structure are not easily satisfied. Furthermore, considering the instability and the poor solubility of the Cu I compounds and the fact that usually compounds based on this cation are obtaining for reduction of Cu II cation, the possibility to have in the same structure Cu I , Cu II and Ln III cations it is a great challenge from a synthesis point of view.

In the present work, we report two novel mixed valence Cu I /Cu II -Ln III HCP, [Gd(H 2 O) 4 Cu II Cu I (IDC) 2 ] 1 and [Gd 2 (H 2 O) 2 (C 2 O 4 ) 2 Cu II (IDC) 2 Cu I 2 (4,4'-bipy)]•4.5H 2 O 2, obtained using 1H-imidazole-4,5-dicarboxylic acid (H 3 IDC) as the principal N,O-bifunctional organic linker. H 3 IDC has been already used by our research group, proving to be an excellent ligand to obtain new 3d-4f compounds, giving rise to HCP including a 2D Cu II -Gd III network, [START_REF] Cruz | Cu(II)-Gd(III) 2D-Coordination Polymer Based on Two Different Organic Linkers[END_REF] Co II -Gd III 3D frameworks [START_REF] Cruz | Novel 3d/4f Metal Organic Networks Containing Co(II) Chiral Chains[END_REF] and recently two 3d-Ce III compounds (3d= Cu II , Co II ). 38 In the present work, two new HCP were prepared under solvothermal conditions. 1 presents a 2D structure showing a honeycomb type arrangement, while 2 corresponds to a 3D structure. Structural, thermal stability, spectroscopic and magnetic properties are discussed in detail.

EXPERIMENTAL SECTION

All reagents and solvents used to obtain 1 and 2 were of p.a. quality and were used without any previous purification. Figure 1, summarizes the synthesis of the two compounds. Finally, the reaction vessel was cooled down to room temperature at 3°C/h rate (48 hours); olivegreen needle-like crystals of 1, suitable for X-ray diffraction, were separated by filtration.

Although oxalic acid is not present in the final product, without the addition of this reagent 1 is not obtained. MW: 662.

X-ray Data Collection and Structure Determination

Single-crystal X-ray diffraction data of compound 1 and 2 were collected at 150 K on a D8 VENTURE Bruker AXS diffractometer and processed with the APEX3 program suite, 39 using Mo-Kα as X-ray wavelength. Frame integration and data reduction were carried out with the program SAINT, 40 and SADABS was employed for multiscan-type absorption corrections. 41 Using the Olex2 42 package, the structures were solved with the ShelXT 43 structure solution program using Dual Space Methods and refined with the ShelXL 44 refinement package, using least squares minimization based on F 2 . Crystallographic data, details on data collections and refinement parameters of the crystal structure are summarized in Table 1. Structure drawings have been made with TOPOs software. [START_REF] Blatov | Program Package for Multipurpose Crystallochemical Analysis[END_REF] Additional data concerning the crystals and the refinement parameters are detailed in the supporting information (Table S1).

All hydrogen atoms where properly localized, expect the one belonging to O4W in 1 and O1W, O2W, O3W, O4W and O9 in 2. Nevertheless, all hydrogen atoms were considered in the final formula. Additionally, crystallization H 2 O molecules O1W, O2W, O3W and O4W located into the interstitial spaces of 2 present disorder and were modelled by parts, giving a total amount of four water molecules per unity of complex. The amount of H 2 O of crystallization in the bulk material was obtained experimentally using thermogravimetric analysis.

In situ temperature-dependent X-ray diffraction (TDXD) was performed for 1 in a flux of N 2 with a Malvern Panalytical Empyrean powder diffractometer, using a germanium incident-beam graphite monochromator, selecting the radiation Cu Kα 1 , and a PIXcel 3D detector and equipped with an Anton Paar HTK1200 oven chamber. In situ measurements were carried out at various constant temperatures from ambient temperature until 800°C. Between each measurement step, the sample was heated at a rate of 1.0 °C min -1 to the desired temperature. Diffraction patterns were collected over the angular range 5-60°(2θ) with a counting time of 40 s step -1 and with a step length of 0.013°(2θ).

Magnetic Measurements

Magnetic measurements were carried out using a Quantum Design Dynacool Physical Properties Measurement System (PPMS), equipped with a Vibrating Sample Magnetometer (VSM). The dc data were collected under an external applied field of 1 kOe in the 1.8-300 K temperature range.

Isothermal magnetizations were performed between 0 and 90 kOe at temperatures varying from 1.8 to 8 K. All Magnetic measurement were performed under crystalline power samples in ACS Paragon Plus Environment Crystal Growth & Design plastic sampleholder. Diamagnetic corrections (estimated from Pascal constants) were considered. [START_REF] Bain | Diamagnetic Corrections and Pascal's Constants[END_REF] 2.5. Thermogravimetric analysis Thermogravimetric analysis was performed on a Mettler Toledo TGA/DSC STAR system. The samples were introduced in an alumina holder and heated under a N 2 atmosphere from room temperature to 900 °C with a heating rate of 5 °C/min.

Spectroscopic measurements

UV-Vis-NIR spectra were obtained on a Perkin Elmer Lambda 1050 spectrophotometer UV-Vis/NIR, equipped with a Perkin Elmer 150 mm InGaAs Integrating sphere. The measurements were carried out in the 200-1200 nm range using the solid sample without any support. X-ray diffraction analysis reveals that compound 1 crystallizes in the triclinic space group, 𝑃1 presenting an asymmetrical unit with two different copper (Cu I , Cu II ) and one gadolinium cations, four water molecules and two independent fully deprotonated IDC 3-species. Both IDC 3-anions bind two copper and one gadolinium cations, at the same time presenting a η 3 -κN-κO',O''-κO''',N' coordination mode (Figure 2a). Cu1 cations are bonded to two IDC Gd III cations is completed by four H 2 O molecules (Figure 3c), leading to a bicapped trigonal prism geometry, corroborated by a continuous shape measurement using the SHAPE software. [START_REF] Llunell | SHAPE V2.1[END_REF] Furthermore, 1 presents a honeycomb topology formed by heterometallic [Gd 2 Cu I 2 Cu II 2 ] hexagonal moieties (Figure 4a). Each corner of the [Gd 2 Cu I 2 Cu II 2 ] hexagon is formed by an IDC 3- anion connected to each other by one Gd III and one copper cation (Cu1 or Cu2), forming an alternating Gd III -Cu II -Cu I ordering. The hexagonal honeycomb topology is shown more clearly in a simplified structure in figure 4b, in which it is easy to see how the cations Cu I , Cu II and Gd III are behaving as linear nodes, while the IDC 3-linker can be defined as a tritopic trigonal plane spacer. Moreover, it is interesting to note how of the H-bonding produced by the H 2 O molecules coordinated to Gd cation are the key of the resulting crystal packing of the layers. In this sense if we only consider the arrangement of the four water molecules around of Gd III belonging to one layer, it is possible to observe how this disposition produces two types of interaction between the adjacent layers. Thus, the three water molecules forming a tripod are localized in such way that they are inserted in the hexagons of the cavity of one adjacent layer (Figure 4c), interacting by H-bonding with the carboxylate groups of IDC 3-(Table 2 for H-bonding). This interaction produces a displaced-layer stacking, with each hexagonal cavity filled by a water-tripod avoiding the existence of porosity in the structure, defining a double-layer arrangement. Furthermore, the last H 2 O molecule coordinating the Gd III cations is pointing just opposite to the former watertripod, allowing a second H-bond interaction between the double-layers (Table 2 for H-bonding), giving the final crystal packing (Figure 4d). Since Cu1 and Cu2 cations have an unsaturated coordination sphere (linear and square plane respectively) in the 2D structure of [Gd(H 2 O) 4 Cu II Cu I (IDC) 2 ] 1, this heterometallic compound is a good candidate as starting material to generate a 3D coordination polymer using a linear ligand such as 4,4'-bipyridine as interlayer linker (two step synthesis). However, by adding the 4,4'bipyridine ligand in the same reaction media used for 1 (one pot synthesis) a new compound 2 was formed. This compound is also a Cu I -Cu II -Gd III mixed valence HCP, but now presenting a

RESULTS AND DISCUSSION

3D network [Gd 2 (H 2 O) 2 (C 2 O 4 ) 2 Cu II (IDC) 2 Cu I 2 (4,4'-bipy)]•4.5H 2 O.
Compound 2 crystallizes in the triclinic space group and presents a 3D coordination 𝑃1 framework, with an asymmetrical unit formed by one and a half copper and one gadolinium cations, one water molecule, one independent fully deprotonated IDC 3-species, one oxalate anion (C 2 O 4 2-) and half 4,4'-bipyridine (4,4'-bipy). Additionally, the compound also presents disordered crystallization H 2 O molecules. The coordination mode of the IDC 3-species is the same as the one observed in 1, η 3 -κN-κO',O''-κO''',N' (Figure 2a). 3) Å bond lengths. Cu2 is also weakly interacting with one carboxylate group belonging to IDC 3-with a distance of Cu2-O4=2.627(2) Å (Figure 5b). Furthermore, the N2-Cu2-N3 angle is 178.71( 14)°, while the N-Cu2-O4 is 105.58 (12)° and 74.87 (10)°; therefore the environment around Cu2 can be defined as lineal CuN 2 .

Considering the type of coordination environment, Cu1 can be assigned a 2+ oxidation state with a square planar geometry, while Cu2 presents a 1+ oxidation state with linear geometry.

Meanwhile, Gd1 presents a GdO 9 environment formed by two IDC anions and one H 2 O molecules are bonded to Gd1 centers with Gd-O bond distance in the range of 2.547(2)-2.387(2) Å, leading to a capped square antiprism geometry, corroborated by an continuous shape measurement using the SHAPE software (Figure 5c). [START_REF] Llunell | SHAPE V2.1[END_REF] Moreover, the 3D extended structure of 2 can be disassembled in two different homometallic substructures. The first one corresponding to a cationic 1D substructure formed by fragments. [START_REF] Luo | The First Self-Penetrating Topology Based on an Unusual -Po Net with Double Edges Constructed from a 12-Connected Gd 2 ( 2 -O carboxylate ) 2 ( 2 -OH 2 ) 2 ( 3 -OH) 2 Cu 2 Core[END_REF] In the case of 1 and 2, the inclusion of a Cu I /Cu II mixed-valence compound was also achieved by the partial hydrothermal reduction of a cupric salt precursor. Furthermore, it has been reported by Truong et al. that under hydrothermal condition oxalic acid can act as reducing agent; therefore, this reactive could be the responsible for the reduction of copper salt under autogenerated pressure. This could be related with the fact that if oxalic acid is not present as reagent in the corresponding synthesis, 1 and 2 were not obtained. Moreover, Truong et al. propose that oxalic acid degrades to carbon monoxide or carbon dioxide and that a copper oxalate complex act as intermediate in the Cu I /Cu II reduction process. [START_REF] Truong | Hydrothermal Growth of Cross-Linked Hyperbranched Copper Dendrites Using Copper Oxalate Complex[END_REF] Another interesting feature in the compound reported by Luo et al. is the presence of Cu I centers in linear CuO 2 and CuN 2 environment, similar to that of 2. Linear conformations are common in other d 10 coordination polymers such as Au I 49,50 and Ag I , [START_REF] Chen | Recent Advances in Supramolecular Design and Assembly of Silver (I) Coordination Polymers[END_REF][START_REF] Tong | Pseudo-Polyrotaxane and β-Sheet Layer-Based Three-Dimensional Coordination Polymers Constructed with Silver Salts and Flexible Pyridyl-Type Ligands[END_REF] however Cu I is usually reported presenting trigonal [START_REF] Zhang | A Two-Dimensional Mixed-Valence Cu II /Cu I Coordination Polymer Constructed from 2-(Pyridin-3-yl)-1H-Imidazole-4,5-Dicarboxylate[END_REF][START_REF] Visinescu | Synthesis, crystal structure and magnetic properties of a new cyanide-bridged mixed-valence copper(I)/copper(II) clathrate[END_REF] and tetrahedral [START_REF] Fotouhi | Electrochemical Synthesis and Structural Characterization of a Novel Mixed-Valence Copper (I)-Copper (II) Complex: Page 48 of 58 ACS Paragon Plus Environment Crystal Growth & Design[END_REF][START_REF] De | A Hexadecanuclear Copper(I)-Copper(II) Mixed-Valence Compound: Structure, Magnetic Properties, Intervalence Charge Transfer, EPR, and NMR[END_REF] geometries, while Cu I /Cu II and Cu I CPs containing linear Cu I are more scarce. [START_REF] Zhu | An Unprecedented 1-D Mixed-Valence Cu(II)/Cu(I) Metal-Organic Framework Bearing 2-Phenyl Imidazole Dicarboxylates[END_REF] On the other hand, according to the Robin and Day classification, 1 and 2 can be described as Class I mixed valence compounds, since in both structures the Cu I and Cu II cations can be identified by the difference in their coordination geometries. [START_REF] Day | Mixed Valence: Origins and Developments[END_REF][START_REF] Robin | Mixed Valence Chemistry-A Survey and Classification[END_REF] Additionally, another structural comparison can be made between 1 and two 3D frameworks IDC 3-and anti-anti  2 -O,O'-IDC 3-carboxylate, which are further pillared by µ 2 -N,N'-4,4'-bipy connections, presenting hexagonal cavities into the 3D CP. This fact suggests that the property of generate 2D structures with honey comb topology is an intrinsic property of the IDC 3-ligands, being independent of the metal ions used. 4 Cu II Thermogravimetric analysis between 25 and 900°C under N 2 atmosphere was performed to characterize the thermal stability of 1 and 2 (inset Figure 7). Thermogram of 1 shows two consecutive weight losses, between 95-220° (7.56 wt%) and 220-290°C (2.75 wt%) attributed to the release of the four coordinated water molecules (10.3 wt%). After these two steps, a third weight loss of 27.9% is observed from 350° C, indicating the thermal decomposition of the framework. On the other hand, 2 presents a first weight loss between 40-210 °C with a value of 6.38 wt%, which can be associated to the thermal outflow of the 4.5 crystallization H 2 O molecules, occluded in the interstices of the 3D structure of 2. The percentage associated with this weight loss is a little higher than the obtained from X-ray diffraction analysis (4.0 H 2 O molecules). Later, three consecutive smooth weight losses are observed between 210° and 330°C with a total percentage of 39.17 wt%, associated with the thermal decomposition of the framework. The stability of this type of networks can be associated to the hydrogen bonding interactions between the water molecules and carboxylate oxygen atoms belonging to organic linkers, stabilizing significantly the network packing, [START_REF] Beatty | Open-Framework Coordination Complexes from Hydrogen-Bonded Networks: Toward Host/Guest Complexes[END_REF][START_REF] Zhang | Two Luminescent 2D Layered Copper (I)-Olefin Coordination Polymers with High Thermal Stability[END_REF][START_REF] Hao | A Series of d 10 Metal Coordination Polymers Based on a Flexible Bis (2-Methylbenzimidazole) Ligand and Different Carboxylates: Synthesis, Structures, Photoluminescence and Catalytic Properties[END_REF] Furthermore, in 3D frameworks the guest molecule evacuation from interstitial cavities can be accompanied by collapse of the structure of the extended networks. [START_REF] Abrahams | Assembly of Porphyrin Building Blocks into Network Structures with Large Channels[END_REF][START_REF] Dimos | Microporous Rare Earth Coordination Polymers: Effect of Lanthanide Contraction on Crystal Architecture and Porosity[END_REF] In situ temperature-dependent X-ray diffraction (TDXD) was performed for the 1 framework to establish structural correlations with TG analysis (Figure 7). Comparing the experimental 1 powder diffractogram with the calculated pattern from single crystal (Figure S1), it is clear that the 1 powder phase is pure. Unfortunately, it was no possible to perform the TDXD of 2, due to the low yield in the synthesis of this compound.

[Gd 2 (H 2 O) 2 (C 2 O 4 ) 2 ] 2+ building

Thermal stability of [Gd(H 2 O)

When 1 is heated from room temperature to 290°C under N 2 atmosphere, TDXD experiments show that the diffraction pattern is maintained, proving that the release of the coordination H 2 O molecules does not affect the structural integrity of 1. Until 300C no changes are registered in TDXD, showing that the framework is stable even after the dehydration process. From 300C to 350C the principal diffraction peaks of 1 begin to disappear. Considering that the TG analysis 400) and ( 440) diffraction planes of cubic-Gd 2 O 3 (ICDD PDF2 No. 03-065-3181). [START_REF] Kuzníková | Synthesis and Characterization of Gadolinium Oxide Nanocrystallites[END_REF]

Spectroscopic characterization

UV-Vis-NIR electronic spectra are interesting to complement the characterization of 1 and 2.

Cu II complexes show d-d absorption bands in the Vis-NIR range, while the complete 3d shell in Cu I compounds prevents metal-centered electronic transitions. At the same time, Cu I compounds may present metal to ligand charge transfer (MLCT), which are generally more intense and at higher energies as compared to the d-d transitions. [START_REF] Armaroli | Photochemistry and Photophysics of Coordination Compounds: Copper[END_REF] Figure S2 shows the solid state UV-Vis-NIR spectra of both compounds in the region between 200 and 850 nm. A broad band covering almost all the studied region is observed in both spectra. At lower wavelengths, a band centered near 260 nm can be associated to a ligand-centered absorption, belonging to the organic species [START_REF] Xiong | Three Transition-Metal Polymers from Imidazole Dicarboxylates-Bearing Methoxyphenyl Groups: Syntheses, Crystal Structures and Properties[END_REF] At higher wavelengths, a band centered at 582 and 535 nm for 1 and 2, respectively can be related with the overlapping of the three spin allowed d-d transitions of Cu II centers in a square planar geometry, , and . [START_REF] Manch | The Structure and Spectra of Nickel(II) and Copper(II) Complexes[END_REF] An

𝑑 𝑥𝑧 /𝑑 𝑦𝑧 →𝑑 𝑥 2 -𝑦 2 𝑑 𝑧 2 →𝑑 𝑥 2 -𝑦 2 𝑑 𝑥𝑦 →𝑑 𝑥 2 -𝑦 2
additional band can be observed at 376 and 395 nm for 1 and 2 associated to metal to ligand charge transfer between Cu I centers and the heterocyclic organic ligands. [START_REF] Barbieri | Luminescent Complexes beyond the Platinum Group: The d 10 Avenue[END_REF] These results also permit to corroborate the existence of a mixed-valence Cu I /Cu II system in 1 and 2.

In compound 1 and 2, two separate emission process could be expected, one of them in the UV region corresponding to a f-f transition belonging of Gd III cation, [START_REF] Strasser | Phosphorescence of gadolinium(III) chelates under ambient conditions[END_REF] and the other one an emission of the Cu I -ligand fragment which should appear in the visible region. [START_REF] Zeng | 3d-4f Metal-Organic Framework with Dual Luminescent Centers That Efficiently Discriminates the Isomer and Homologues of Small Organic Molecules[END_REF][START_REF] Xu | A Cuprous/Lanthanide-Organic Framework as the Luminescent Sensor of Hypochlorite[END_REF] To explore the luminescence properties of 1 and 2, solid state emission spectra were recorded at room temperature using as excitation the energy maxima observed in the absorption spectra.

Nevertheless, no emission was observed at the studied conditions. These results could be related with the fact that in 1 and 2 the f-excited states of Gd III are above of the excited state of the organic ligand, inhibiting a possible ligand→Gd antenna effect. [START_REF] Bünzli | Taking Advantage of Luminescent Lanthanide Ions[END_REF][START_REF] Cotton | Lanthanide and Actinide Chemistry[END_REF] On the other hand, the Cu II cations in 1 and 2 could be acting as a quencher of the Cu I -ligand emission, since the Nevertheless, below 50 K an abrupt increase of the  m T values was observed for both compounds, reaching a value of 13.06 and 19.46 emu K mol -1 for 1 and 2, respectively at 1.8 K, indicating that a ferromagnetic behavior is displayed.

Additionally, field dependence of magnetization at 1.8, 3, 5 and 8 K represented as N B vs H is shown in Figure 9 for 1 and 2. In both cases, the magnetization increases rapidly reaching a Since both 1 and 2 are extended structures, the connectivity of the spin carriers is the key for the magnetic behavior analysis, i.e. to define the proper magnetic network. In the case of 1, the structure can be reduced to Cu II -Gd III chains linked through an anti-anti carboxylate bridge, where the Cu-Gd-chains are interconnected by Cu I diamagnetic centers. However, the lack of an analytical model to fit the experimental magnetic data for a regular Cu-Gd ferromagnetic chain, forces to use a simpler model to quantify the magnetic interaction between both cations.

In this sense, considering the isotropic character of the Cu II and Gd III cations the magnetic data were analyzed using the spin-only Hamiltonian , which describes the super-𝐻 = -2𝐽𝑆 𝐶𝑢 𝑆 𝐺𝑑 exchange interaction between one Cu II and one Gd III cation. Assuming that Cu II and Gd III have equal g values, the experimental data can be fitted with the following analytical expression [START_REF] Costes | A Genuine Example of a Discrete Bimetallic (Cu, Gd) Complex: Structural Determination and Magnetic Properties[END_REF][START_REF] Koner | Two New Diphenoxo-Bridged Discrete Dinuclear Cu II Gd III Compounds with Cyclic Diimino Moieties: Syntheses, Structures, and Magnetic Properties[END_REF]

𝑚 𝑇 = 4𝑁𝑔 2 𝛽 2 𝑇 𝑘𝑇 ( 7 + 15𝑒 ( 8𝐽 𝑘𝑇 ) 7 + 9𝑒 ( 8𝐽 𝑘𝑇 ) ) Page 22 of 58 ACS Paragon Plus Environment Crystal Growth & Design
Where N is the Avogadro number, g is the Lande factor,  is the Bohr magneton, k is the Boltzmann constant and J is the coupling constant. The best fit of the parameters was obtained using a J=0.60 cm -1 and g= 2.02, with an agreement factor R=4×10 -3 (R= m T obs )- m T exp ) [START_REF] Hao | A 3D Porous Coordination Polymer Transformed from a 1D Nonporous Coordination Polymer for Selectively Sensing of Diiodomethane[END_REF] / m T obs ) [START_REF] Hao | A 3D Porous Coordination Polymer Transformed from a 1D Nonporous Coordination Polymer for Selectively Sensing of Diiodomethane[END_REF] .

In the case of 2, the compound presents a higher structural dimensionality, where the Cu II and Gd III spin carriers are bridged by a carboxylate anion and both Gd-Gd interactions can be produced by carboxylate or oxalate anions. However, it has been previously reported that due to the internal nature of the 4f electrons, the Gd III -Gd III magnetic interaction are sufficient small to be neglected. [START_REF] Cañadillas-Delgado | Intramolecular Ferro-and Antiferromagnetic Interactions in Oxo-Carboxylate Bridged Digadolinium(Iii) Complexes[END_REF][START_REF] Cañadillas-Delgado | Two-and Three-Dimensional Networks of Gadolinium(III) with Dicarboxylate Ligands: Synthesis, Crystal Structure, and Magnetic Properties[END_REF] Therefore, the magnetic network can be reduced to a more simple model considering only the interaction Cu II -Gd III hetero-spins carries. 15 From a magnetic point of view, the extended 3D crystal network of 2 can be described as a linear trinuclear Gd III -Cu II -Gd III core, with Gd III -Cu II centers connected through an anti-anti carboxylate bridge. This model leads to a spin-only Hamiltonian of , which gives rise to the following analytical A nonlinear least-squares fit between 300-1.8 K leads to a value of J = 0.22 cm -1 , g=2.0 (R=8×10 -6 , R= m T obs )- m T exp ) 2 / m T obs ) 2 ). For 1 and 2 the positive J value corroborates the ferromagnetic nature of the interaction between the Cu II -Gd III spin carriers as has been reported

𝐻 = -2𝐽𝑆 𝐺𝑑1 𝑆 𝐶𝑢 -2𝐽𝑆 𝐶𝑢 𝑆 𝐺𝑑2 expression 82 𝑚 𝑇 = 𝑁𝑔 2 𝛽 2 𝑇 2𝑘𝑇 𝐴𝐴 𝐵𝐵
for analogous systems, [START_REF] Manna | N. 3D Heterometallic (3d-4f) Coordination Polymers: A Ferromagnetic Interaction in a Gd(III)-Cu(II) Couple[END_REF][START_REF] Winpenny | The Structures and Magnetic Properties of Complexes Containing 3d-and 4f-Metals[END_REF][START_REF] Hamamatsu | Chiral Ferromagnetic Chain of Copper(II)-Gadolinium(III) Complex[END_REF] however the quality of the fit are different. In 1, the dinuclear model cannot reproduce the experimental data in the low temperature range, due to the  m T value at 1.8 K (13.06 emu K mol -1 ) is higher than the expected for a ferromagnetic coupled Cu-Gd isolated binuclear system (10 emu K mol -1 ). This fact indicates that a higher magnetic dimensionality should be consider for 1, i.e. a genuine 1D magnetic network is observed.

Meanwhile in 2, the experimental data are well reproduced in the whole temperature range by the use the trinuclear Gd-Cu-Gd model, indicating that in the case of 2 the Gd-Gd interactions are small enough to be hidden by the stronger Gd-Cu-Gd interactions, thus behaving as isolated trinuclear cores in the crystal lattice.

Spin polarization has been postulated by Gatteschi et al. to explain this magnetic interaction between Cu II -Gd III cations. [START_REF] Benelli | Crystal Structure, and Magnetic Properties of Tetranuclear Complexes Containing Exchange-Coupled Ln 2 Cu 2 (Ln = Gd, Dy) Species[END_REF][START_REF] Andruh | Crystal Structure and Magnetic Properties of [Ln 2 Cu 4 ] Hexanuclear Clusters Where Ln Mechanism of the Gd(III)-Cu(II) Magnetic Interaction[END_REF] This mechanism is based on the partial delocalization of the 3d unpaired electron of the Cu II toward the empty 6s or 5d orbitals of Gd III with the same spin of the seven f electrons, in order to obey Hund's rule. Thus, a ferromagnetic pathway can be established when a magnetic orbital of one paramagnetic site has a non-zero overlap with an empty orbital of the other site. [START_REF] Benelli | Magnetism of Lanthanides in Molecular Materials with Transition-Metal Ions and Organic Radicals[END_REF] Most of the studied Cu II -Gd III systems are 0D complexes formed by dioxo bridges (alcoxo or phenoxo). The magneto-structural correlations show that planar conformation in Cu-dioxo-Gd moieties and shorter Cu … Gd distances lead to stronger ferromagnetic interactions. 88 On the other hand, carboxylate bridges have been less studied since this group mediates weaker magnetic interactions than the oxo bridge due to the longer Cu … Gd distances; as far as we know no systematic studies have been made up to date.

Regarding the different types of carboxylate bridges, the syn-anti conformation has been reported to mediate poor [START_REF] Chen | Structures, and Magnetic Properties of Carboxylate-Bridged Tetranuclear Copper(II)-Lanthanoid(III) Complexes [Cu 2 Ln 2 (Betaine) 10 (H 2 O) 8 ](ClO 4 )[END_REF] or null magnetic interaction between Cu II and Gd III , [START_REF] Wu | Structure and Magnetic Study of a New Linear 1-D Cu(II)-Gd(III) Coordination Polymer[END_REF][START_REF] Li | A Unique 3-D 3d-4f Heterometallic Coordination Polymer with Double Betaine Ligand: Crystal Structure and Magnetic Properties[END_REF] while the antianti conformation produces weak ferromagnetic interactions. [START_REF] Manna | N. 3D Heterometallic (3d-4f) Coordination Polymers: A Ferromagnetic Interaction in a Gd(III)-Cu(II) Couple[END_REF][START_REF] Cruz | Cu(II)-Gd(III) 2D-Coordination Polymer Based on Two Different Organic Linkers[END_REF] For example, Liu et al reported Figure 8.  m T vs T plot at 1 kOe between 300 and 1.8 K for 1 and 2. In the inset, scheme of the Cu II -Gd III and Cu II -Gd III -Cu II moieties used in 1 and 2 for magnetic data analysis.

2. 1 . 6 A

 16 Synthesis of [Gd(H 2 O)4 CuII Cu I (IDC) 2 ] 1 mixture of Cu(NO 3 ) 2 •3H 2 O (0.5 mmol), Gd 2 O 3 (0.13 mmol), H 3 IDC (0.5 mmol) and H 2 C 2 O 4 •2H 2 O (0.5 mmol) in 14 mL of H 2 O and 1 mL of ethanol (EtOH) was treated with 80 μL (0.6 mmol) of triethylamine (TEA). The resultant mixture was placed in a 23 mL Teflon-lined stainless-steel autoclave vessel and heated at 165 °C under self-generated pressure for 4 days.

  connected in a monodentate mode through N atoms with a trans conformation, with bond distances of Cu1-N2=1.872(2) Å and Cu1-N3=1.880(2) Å. Cu1 is also weakly interacting with the two carboxylates groups belonging to each IDC 3-ligand with Cu-O bond distances of Cu1-O=2.576(2) Å and 2.781(2) Å (Figure3a). Meanwhile, Cu2 cations present a coordination environment formed by two IDC 3anions (Figure 3b) bonded by N-imidazole and O-carboxylate, forming two independent NCCO five member rings, with bond distances of Cu2-N1=1.933(2) Å, Cu2-N4=1.922(2) Å, Cu2-O1=1.973(2) Å, Cu2-O5=1.976(2) Å and N-Cu-O angles of 98.99(9)° (N1-Cu2-O5) and 82.44(9)° (N4-Cu2-O5). Taking into account the bond distance for both copper cations, Cu1 should be better described with a CuN 2 environment, having a linear geometry with a N-Cu-N angle of 177.28(1)º. In the case of Cu2, the cation can be well described with a CuN 2 O 2 environment presenting a distorted square plane geometry, since the Cu-O and Cu-N bond distances are comparable in length, and N-Cu-O angles are close to 90. Furthermore, comparing the bond distances and coordination geometries of the two types of copper cations, it is possible to infer that Cu1 should present a 1+ oxidation state, while Cu2 must present a 2+ oxidation state, which is in perfect agreement with the charge balance of the framework. On the other hand, Gd1 presents a GdO 8 coordination sphere, with Gd-O bond distances in the range of 2.558(2)-2.260(2) Å, formed by two κO',O''-IDC 3species, bonded by two Ocarboxylate atoms generating two independent seven member rings. The coordination around

  The 4,4'-bipy molecule serves as a ditopic ligand, connecting two copper centers in a µ 2 -κN-κN' coordination fashion, while the C 2 O 4 2-anions coordinate Gd III in µ 2 -O,O'-κO'',O''' mode (Figure 2b-c). Similarly, as in 1, to achieve the electron neutrality of the network and balance the negative charge belonging to IDC 3-and C 2 O 4 2-species 2 must present copper cations in two different oxidation states. Cu1 presents a CuN 2 O 2 environment formed by two IDC 3anions (Figure 5a) coordinated by Nimidazole and O-carboxylate, forming two NCCO five-member rings, with Cu1-O1=1.962(3) Å and Cu1-N=Å, Cu1-N1=1.915(3) Å bond length distances. Moreover, since the N-Cu-O angles are near to 90° (N1-Cu1-O1=96.01(12)°, N1-Cu1-O1=83.99(12)°), the geometry around Cu1 can be well defined as square planar. On the other hand, Cu2 is bonded to one IDC 3anion and one 4,4'-bipy, connected by means of nitrogen atoms, presenting Cu2-N2=1.846(3) Å, Cu2-N3=1.852(

  one (κO,O'-IDC 3-) and two (κO',O''-IDC 3-) carboxylate groups. Also, two κO,O'-κO'',O'''-C 2 O 4 2-

  units, in which the GdIII centers are bridged by two different oxalate Page 14 of 58 ACS Paragon Plus Environment Crystal Growth & Design anions (Figure6a) forming a zigzag chain along c axis. Meanwhile, the second substructure corresponds to anionic mixed valence Cu I /Cu II linear chains, formed by[Cu II (IDC) 2 Cu I 2 (4,4'bipy)] 2-as building units (Figure6b).

  Figure 6c depicts the cross-linked connectivity between both homometallic substructures, showing clearly the role of the IDC 3-anion in the assembly of both homometallic chains. The IDC 3-anion of the Cu I /Cu II chains anchors Gd III centers of two adjacent [Gd 2 (H 2 O) 2 (C 2 O 4 ) 2 ] 2+ chains, with a κO,O'-IDC 3-coordination fashion, generating the 3D framework. As stated in the introduction section, the unique example of a 3d-4f mixed valence compound is [Gd 2 Cu II 2 Cu I 5 (IN) 10 (µ 2 -Cl)(µ 2 -OH 2 ) 2 (µ 3 -OH) 2 ](ClO 4 ) 2 , which was synthesized by hydrothermal reaction of isonicotinic acid (HIN) as bifunctional ligand, Gd 2 O 3 and CuCl 2 as metal precursors in the presence of HClO 4 and (NH 4 )HCOO, giving a 3D structure constructed by the self-assembly of 12-connected Gd III 2 Cu II 2 subunits interconnected by Cu I -IN -

{[Zn 3 (

 3 IDC) 2 (4,4'-bipy) 3 ]•(4,4′-bipy)•8H 2 O} n obtained by Lu et al. 60 and {[Co 3 (IDC) 2 (4,4'bipy) 3 ]•6H 2 O•DMF} n obtained by Wang et al. 61 It is quite surprising that although these both frameworks are constructed only by M II cations (M II = Zn II and Co II ), presents [M 3 (IDC) 2 ] n 2Dsub-structures with exactly the same honeycomb topology as the one observed in 1. In both cases, the layers are formed by M II cations, connected through alternating imidazolate  2 -N,N'-

Page 18 of 58 ACS

 58 Paragon Plus Environment Crystal Growth & Design shows a non-significant weight loss in this temperature range, the loss of the crystallinity process can be suggested. Moreover, above 450C new peaks start to appear at higher  values, indicating that new crystalline phases are formed. The peaks position at 43.3 and 50.4 correlate well with (111) and (200) diffraction planes of metallic Cu 0 phase (ICDD PDF2 No. 01-089-2838), 67,68 indicating that the reduction of Cu I /Cu II cations to fcc-Cu 0 is taking place. Starting from 650°C, new peaks at 28.6, 33.1 and 47.5 can be associated to the (222), (

Page 21 of 58 ACS

 58 Paragon Plus Environment Crystal Growth & Design plateau of 8  B at 40 kOe for 1 and 15  B at 50 kOe for 2, being consistent with the saturation values expected for the eight unpaired electrons of a core of GdCu II Cu I and fifteen unpaired electrons of a core of Gd 2 Cu II Cu I 2 .

Figure 2 .

 2 Figure 2. Coordination modes of (a) IDC 3-, (b) 4,4'-bipiridine and C 2 O 4 2-in 1 and 2.

Figure 3 .

 3 Figure 3. Coordination environment of (a) Cu1, (b) Cu2 and (c) Gd1 cations present in 1.

Figure 4 .Figure 5 .

 45 Figure 4. (a) Hexagonal honeycomb 2D extended structure, (b) simplified structure of 1, (c)

Figure 6 .

 6 Figure 6. View of the assembly of the extended structure of 2 through a axis, (a) in pink, Gd III -

Figure 7 .

 7 Figure 7. Thermogram between 25 and 900C under N 2 atmosphere for 1 and 2 (inset) and in

Figure 9 . 2 .

 92 Figure 9. Isothermal magnetization represented by reduced magnetization (N  ) at 1.8, 3, 5 and

Figure 3 Figure 4 Page 34 Figure 5 Figure 6 Page

 343456 Figure 3

Figure 7 Figure 8 Page 35 of 58 ACS

 7858 Figure 7

  59 g/mol. Yield of 62%, based on the lanthanide salt. Calculated analysis

	for Cu 2 GdC 10 H 10 N 4 O 12 (%): C, 18.13%; H,1.52%; N,8.46%. (%) Found: C, 18.76%; H, 1.56%;
	N, 7.72%.
	2.2. Synthesis of [Gd 2 (H 2 O) 2 (C 2 O 4 ) 2 Cu II (IDC) 2 Cu I 2 (4,4'-bipy)]•4.5H 2 O 2
	2 was obtained using the same procedure as that of synthesis of 1 but adding 4,4'-bipyridine
	(0.5 mmol) to the mixture; olive-green prismatic crystals of 2, suitable for X-ray diffraction were
	mechanically separated from the reaction medium. MW: 1260.68 g/mol. Yield: 13%, based on
	the lanthanide salt. Calculated analysis for Cu 3 Gd 2 C 24 H 23 N 6 O 22.5 (%): C, 22.86; H, 1.84; N, 6.67.

(%) Found: C, 22.52; H, 1.76; N, 7.06.

  Cu I (IDC) 2 ] 1 and [Gd 2 (H 2 O) 2 (C 2 O 4 ) 2 Cu II (IDC) 2 Cu I 2 (4,4'-

	bipy)]•4.5H 2 O 2

Table 1 .

 1 GdCu(L 1 ) 2 (L 2 )(H 2 O) 2 ]•mH 2 O} n (H 2 L 1 = quinolinic acid, HL 2 = nicotinic acid), in which the anti-anti carboxylate bridges of quinolinate and nicotinate ligands generate a Cu … Gd distance of 6.010 Å, with a ferromagnetic coupling of J= 0.426 cm -1 .[START_REF] Liu | Three New Cu(II)-Ln(III) Heterometallic Coordination Polymers Constructed from Quinolinic Acid and Nicotinic Acid: Synthesis, Structures, and Magnetic Properties[END_REF] On the other hand, Zou et al. found a slightly stronger magnetic interaction of J = 1.30(3) cm -1 between Cu II -Gd III for a Both 1 and 2 was obtained by the partial reduction of Cu II under autogenerated pressure, being the stabilization of Cu I achieved only thanks to the coordination of N-donor ligands. Interestingly, in the case of 1, the simplicity of the chemical composition it is highlighted offering a path of synthesis where only one ligand is necessary to coordinate and stabilize three different cations. The proposed modification of the synthesis of 1 result successfully in the obtention of a second new mixed valence 3d-4f HCP 2, with a superior structural complexity. Compound 1 presents a honeycomb topology 2D network in which the crystal packing is achieved by hydrogen bonding between coordinated water molecules and the carboxylate groups. Meanwhile, in the case of 2 the addition of the extra organic linker permits Crystal data and structure refinement for 1 and 2

	Page 25 of 58

93

These facts are in agreement with that observed in 1 and 2, in which the shorter Cu … Gd= 6.1378 Å distance, present in 1, produces a slightly higher magnetic coupling (J = 0.60 cm -1 ) compared with the coupling constant in 2, (J = 0.22 cm -1 for a Cu … Gd= 6.226 Å). Hence, beyond of the structural complexity that present 1 and

Table 2 .

 2 Table 2 Hydrogen Bonds for 1 Table 1 Crystal data and structure refinement for 1 and 2

	Compound code	1	2
	Compound formula	[Gd(H 2 O) 4 Cu II Cu I (IDC) 2 ]	[Gd 2 (H 2 O) 2 (C 2 O 4 ) 2 Cu II (IDC) 2 Cu I 2 (4,4'-bipy)]•4.5H 2 O
	Empirical formula	C 10 H 10 Cu 2 GdN 4 O 12	Cu 1.5 GdC 12 H 11.5 N 3 O 11.25
	Formula weight	662.53	630.34
	Temperature/K	150	150
	Crystal system	Triclinic	Triclinic
	Space group	𝑃1	𝑃1
	a/Å	6.6701(5)	8.5247(7)
	b/Å	11.3064(9)	9.6931(9)
	c/Å	12.0027(11)	12.0429(11)
	α/°	62.710(3)	102.413(3)

ACS Paragon Plus EnvironmentCrystal Growth & Design

ASSOCIATED CONTENT Supporting Information.

The following files are available free of charge, experimental and calculated Powder X-Ray diffraction pattern of 1 (Figure S1), UV-Vis spectra for 1 and 2 (Figure S2) and complete crystallographic refinament data for 1 and 2 (Table S1 andS2


Andrea Diaz
email: andrea.diaz@ifpen.frstef.proost@econ.kuleuven.be

Stef Proost
email: stef.proost@econ.kuleuven.be

Second-best urban tolling with distributive concerns

Keywords: tax reform, congestion pricing, urban tolls

La collection "Les cahiers de l'économie" a pour objectif de présenter des travaux réalisés à IFP Energies nouvelles et à IFP School, travaux de recherche ou notes de synthèse en économie, finance et gestion. La forme peut être encore provisoire, afin de susciter des échanges de points de vue sur les sujets abordés. Les opinions émises dans les textes publiés dans cette collection doivent être considérées comme propres à leurs auteurs et ne reflètent pas nécessairement le point de vue d' IFP Energies nouvelles ou d' IFP School.

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Introduction

Academic interest in curbing urban congestion has increased considerably in the last few decades as this problem has moved up the political agenda. Economists advocate road pricing as an efficient instrument to use infrastructures. Indeed, imposing a road toll that reflects marginal external congestion costs makes consumers use the road up to the point where marginal social costs equalize marginal social benefits. Optimal road pricing therefore ensures that the only trips made are those that bring the highest benefits for society. This is only true, however, as long as tolling is analyzed in a first-best framework. Additional conditions, e.g. not being able to toll all roads in a network and pre-existing distortions on the labor market, complicate the optimal design of urban congestion tolls.

The related literature is mainly focused on the interaction of road taxes with taxes on labor income (see: Mayeres and Proost, 1997; Parry and Oates, 2000; Parry and Bento, 2001; Van Dender, 2003; Parry and Small, 2005; De Borger, 2009). Road taxes have a positive welfare impact by reducing congestion externalities. But at the same time they have a negative welfare impact, since an increase in commuting costs discourages agents to supply labor.

Which effect (externality reduction or reduced labor supply) prevails has become a central question in transport economics. Parry and Bento (2001) showed that the welfare impact of a road tax differs according to the use of the tax revenues. According to them, using revenues raised from road taxes to reduce tax rates on labor increases social welfare. This is because reduced congestion and reduced labor taxes compensate workers for the congestion toll. Other revenue uses, such as the classic case of making poll transfers, do not compensate the negative labor supply impact and reduce welfare. On the other hand, Mayeres and Proost (1997) demonstrated that as long as equity objectives are relevant, obtaining significant welfare gains from recycling tax revenues requires a careful balance of several options. They show that imposing a tax on congestion externalities may need a reconfiguration of all taxes, and that a reduction of labor taxes is not necessarily the best option. Thus, although urban congestion tolls have gained importance during the last decade, they require further investigation before being broadly recommended. Differences across road users and the composition of the economy (proportion of low-income groups) might change the effect of this instrument on social welfare. Similarly, the use of toll revenues and the distributive objectives of the transport policy could benefit certain consumer groups at the expense of others. This paper contributes to this line of research by analyzing the optimal design of urban tolls to address work-related traffic congestion in a second-best setting that considers tolling a single road in a network and the use of distortive taxes in the labor and transport markets. Our approach is close to that of Parry and Bento (2001) but we add two dimensions to their model.

First, instead of a congested road and uncongested public transit, we model two congested transport options. They can be both roads or one of them can be public transit. Considering the tax interaction problem in a two-road framework is particularly interesting since Parry and Bento (2001) find that the optimal congestion toll should equal marginal external congestion costs, whereas the two-route literature indicates that when just one of the roads can be taxed, the optimal toll should be set below marginal external costs (see e.g. Rouwendal and Verhoef, 2004). Second, Parry and Bento consider homogeneous consumers without paying attention to income distribution issues. However, we know that at the origin of labor taxes there is often the income distribution objective. We consider heterogeneity in labor productivity, in order to take into account differences across commuters. Differences in productivity imply differences in values of time. This in turn determines the sorting of commuters over the tolled and the untolled route. Tolling the faster route will tend to attract the most productive commuters.

Therefore, the tax can act as an instrument imposed on high-income consumers and used either to redistribute resources to low-income consumers or to obtain additional efficiency gains by lowering labor taxes for all commuters.

Our analysis shows that the optimal toll differs from the Pigouvian tax. The toll can be lower or higher than the marginal external cost of the tolled road. The magnitude of the deviation depends on several aspects: the government distributive concerns, who uses the tolled road, who benefits from redistribution, how easily consumers switch to other alternatives. The marginal external cost takes into account not only congestion on the tolled road but also the increase of congestion on non-tolled alternatives caused by the tolling policy. It also takes into account the values of time of the different commuters that use the road.

A numerical exercise shows that the impact of an urban toll differs among drivers. Those who cannot afford to pay for the toll are obligated to seek other alternatives that can be welfare reducing (if limited and expensive). On the other hand, consumers that can pay the toll benefit from the reduction of congestion generated by those who leave the tolled road.

The social impact of this instrument depends, therefore, not only on the use of the toll revenues but also on the distributive concern of the transport policy. Low-productivity households benefit from poll transfers but take no advantage of labor tax reductions. On the other hand, high-productivity consumers benefit from both policies, but they can gain more from labor tax reductions. If income distributional concerns seek to favor low-productivity workers, the policymaker would prefer to recycle toll revenues through poll transfers.

Increasing the toll and reducing labor taxes is not necessarily welfare optimal because poor can gain more from poll transfers. This paper is organized as follows. In Section 2, we develop an analytical model and analyze the problem faced by homogeneous households. In Section 3, we introduce heterogeneity in labor productivity and establish four different equilibriums of road use given the types of consumers. In Section 4, we analyze the social planner's problem and derive the optimal toll rules for the different equilibriums and two ways of recycling the toll revenues: poll transfers and labor tax cuts. In section 5, we present a numerical illustration. In the last section, we present some conclusions.

Analytical model: the household's problem

We start with a simple model of a representative household whose utility function depends on aggregate consumption of market goods (X, whose price is normalized to one), leisure time (t L ), and the number of days devoted to work (D).

(1)

The representative household owns a car and uses it to commute to work by taking either one of the two parallel congested roads (of given capacities) that connect residential areas to workplaces, as illustrated in Figure 1. A congestion toll (τ) related to distance ( is applied on one of the two roads (route T), while the other (route U) remains untolled. The right-hand side of (2) corresponds to total household's income composed of work income and lump-sum government transfers ( ). Work income in a period is the product of the daily net wage and the number of days worked in the period, where is labor productivity, W is the gross daily wage and is a tax levied on wage. We assume that households are homogeneous in all respects except that they exhibit different exogenous levels of labor productivity. Thus, for the same level of labor supplied, high-productivity households get a higher work income than low-productivity households.

The left-hand side of (2) corresponds to household expenditures on aggregate consumption and commuting. Each day of work requires a commuting round trip which requires time and monetary costs. When commuting by the untolled road, only fuel costs are relevant3 , represents fuel price (such that , where is the resource fuel cost and the fuel tax), and car (or fuel) efficiency. Commuting by the tolled road implies paying for the fuel consumption plus the toll. However, this road allows faster trips, while the untolled road requires more time and higher fuel consumption due to a longer distance: with .

Households also face a time constraint:

(3)

This indicates that the household's time endowment in a period ( ) equates the sum of labor supplied (D U +D T ), commuting time (by the untolled or by the tolled road) and leisure time. t U is the time per unit of distance when using the untolled road (the inverse of the speed -h/km) and t T the time per unit of distance required when using the tolled road.

Households choose how many days to work in a period (hours of work per day are fixed), and how to commute to work in those days (by the tolled or the untolled road). Note that by choosing the optimal number of workdays (D T and D U ) in a period, households indirectly set total income and total leisure time in the period.

The first-order conditions of maximizing utility (1) subject to (2) and ( 3) are (see Appendix A for detailed derivations):

(4)

(5)

These expressions equate the private benefit from an extra day of work (net wage) with the private cost of working (monetary and time costs) plus the marginal disutility from commuting. The monetary cost of transport consists of the fuel consumption charge in the case of commuting by the untolled road (4), whereas it consists of the fuel consumption charge plus the road toll charge when commuting by the tolled road (5). The total time cost includes both daily hours spent at work and daily time lost on commuting.

As a result of considering time as a resource, we get the monetary value of time for each household . This is the ratio between the Lagrange multiplier of the time constraint and the Lagrange multiplier of the income constraint (see Appendix A). The value of spending time in transport 4 (Value of Transport Time -VTT) is represented in ( 4) and ( 5) by the value of time foregone by commuting minus the marginal disutility of commuting ( where R=U,T). We could also express ( 4) and ( 5) as:

(4') (5')
That is, the daily net wage minus the value of daily leisure time foregone by working equals the daily total cost of commuting. The right-hand side of (4') and (5') represents the generalized private cost of commuting by the untolled and tolled road respectively. Equating 4 For a detailed explanation of travel time valuation, see Small and Verhoef (2007) and Jara-Diaz (2000). 5 Wardrop principle: "For a given origin-destination pair of substitute roads, all used routes should have equal average cost and there should be no unused routes with lower costs" (Small and Verhoef, 2007). 6 We assume we can exclude corner solutions where only one of the two roads is used. 7 If the marginal disutility of commuting were the same by the two roads such as if the household has no particular preference for one of the roads, condition (6) would be reduced to:

, so that the household would take into consideration only the extra-gasoline and extra-time costs.

Household's individual decision depends on its own value of time

, which also determines its willingness to pay for a trip. The opportunity cost of time indirectly depends on labor productivity. As high-productivity households will normally get higher wages, they should exhibit higher values of leisure time, whereas low-productivity households exhibit lower values . Thus, a sufficiently high toll should make high-productivity households stay on the tolled road and therefore save high-valued time. In contrast, as lowproductivity households have lower budgets, they should be more sensitive to monetary cost and should prefer taking the untolled road in order to save money.

We finally define the differentiable demand functions for each road , . Assuming that they exist allows us to get the household's indirect utility function as a function of exogenous parameters (see Appendix A).

Use of the congested roads by heterogeneous households

Increases in road use create congestion by reducing average travel speed and raising commuting time for all road users. When deciding which road to use, households do not take into account their impact on raising the commuting time of all other users, thus congestion externalities appear. The average number of cars using each road every day depends on the number of days in a period the representative household decides to commute. Thus, travel time on both routes is endogenous and , such that time required to commute depends on the number of trips households decide to make by each road in a period, with , and .

In the presence of congestion, households take into account the average congestion and weight it at their own value of time. However, they do not take into account the value of time of their co-travelers, on whom they impose a congestion externality. First-best pricing calls for tolling both roads. The tolls should reflect the marginal external costs of each road, such that households face the marginal social cost of their trips. However, we are interested in analyzing the second-best configuration where a single road can be tolled.

We assume that the economy is populated by high-productivity and low-productivity households in a known share (such that ). Both kinds of households independently choose the number of trips they make in a period ( and ).

As households differ in their willingness to pay for commuting, differentiating them according to the road used may be useful.

As a start, we may expect consumers with higher values of time taking road T while consumers with lower values of time taking road U8 . From the right-hand side of equations From ( 7) and ( 8) we can establish four different equilibriums of use of the roads by the households, similar to those established in Small and Yan (2001). Let us assume for a moment that there is not specific preference for a road. That is, and . Thus, conditions ( 7) and (8) become:

Optimal toll: the social planner's problem

The government raises revenues, to finance a fixed set of public goods , and a head subsidy G, from three tax instruments: fuel taxes (τ g ), tolls (τ) and wage taxes (τ w ). The government maximizes social welfare10 , subject to the following budget constraint:

(9)

Note that total budget collected by the government depends on household decisions: D U and D T .

The social welfare function for our two groups of consumers becomes

.

In what follows, we analyze the welfare improving changes of the four possible equilibriums studied in the previous section. For each case, toll revenues are returned to the individuals either through poll transfers or recycled through labor-tax cuts.

High-income households commuting by T, and low-income households commuting by U

Assume that the economy is at the equilibrium where high-income households take only road

T and low-income households take only road U. Each household chooses the optimal number of commuting trips that maximizes its individual utility. We assume that an increase of the toll reduces the number of trips high-income commuters make on road T. As road U is not priced, users on this road keep the number of trips fixed. Assuming equal labor tax rates for both types of households, the government's budget constraint becomes:

(10)

Toll revenues used to finance poll transfers

The welfare impact of an increase in the congestion tax is obtained by maximizing the social welfare function subject to the budget constraint (10) (see Appendix B for detail). The optimal toll per unit of distance, when revenues are returned to households as poll transfer is given by:

(11) where , , , , , and is the elasticity of demand of high-income households for the tolled road.

The optimal congestion toll has two main components. The marginal external congestion cost (mecc) and other taxes levied per trip. The mecc measures the increase in traffic-time cost to all road users caused by congestion arising from an extra trip per period. In equation ( 11), it is represented by the product of the increase in commuting time per distance from an additional trip , the value of time of the commuter and the number of trips made per period . Other taxes per trip appear in (11) as the complementary relationship between work-related trips and the labor market makes that all the taxes (per kilometer) levied per day of work serve to tackle the externality caused by each day of work, namely congestion. Thus, for example, if the sum of the fuel and the labor tax exceeds the mecc, rather than taxing road T commuters, the government should subsidize them. Equation ( 11) therefore suggests an optimal combination of the toll, the fuel tax and the labor tax, rather than a unique optimal toll level.

Each term in ( 11) is multiplied by a factor that depends on the government distributive concerns ( and ). They depend on the social weight given to the groups ( ) and on the ratio of the marginal utility of income of both types of consumers . This component captures the value that the government attaches to a unit of income of low-productivity households with respect to a unit of income of high-productivity households. This involves trading off the utility of one group against that of the other group. Normally, when the decision maker attaches a higher weight to a unit of income of a low-productivity consumer, thus . Conversely, if the government attached the same weight to both groups ( ), we would have , and . Thus the government would not be concerned with redistribution.

If we have that, if the demand for the tolled road is inelastic , and are higher than one, so that the optimal toll level could be higher than the mecc if the government cannot raise enough revenues from other taxes. In this case, the toll becomes an instrument to distribute income among consumers. On the contrary, if the demand for the tolled road is elastic, and are lower than one and the toll have to be set lower than the mecc. In this case, the toll cannot longer be used as an instrument to distribute income, as every euro of tax revenues then has a high efficiency cost. When the government gives the same weight to both kinds of consumers, , so that the optimal toll equal the difference between the mecc and the sum of the other taxes.

Toll revenues used to cut labor taxes

Following the same procedure, we obtain the optimal toll when the incremental toll revenues are used to cut labor tax rates (see Appendix B.1):

where , . This expression differs from (11) in that and take into account the proportion of labor supplied by lowproductivity households. Therefore, labor productivity enters in the distributive concern term, so that redistributing income through the labor tax implies that what drives the toll level is the proportion of labor supplied by low-productivity consumers rather than their proportion in the economy .

High-income households commuting by T and U, and low-income households commuting by U

In this subsection, we assume that the economy is at the second type of equilibrium, where high-income households take road T and road U, and low-income households take only road U. As the roads are substitutes, we assume that if the toll increases, high-income consumers reduce the number of trips they make by road T and increase the number of trips they make by road U. To keep things simple we assume that low-income users do not change their number of trips by road U as the toll increases11 . Thus, the government's budget constraint becomes:

(13)

Toll revenues used to finance poll transfers

Here, we consider the case where the incremental toll revenues are used to finance lump-sum transfers (see Appendix B.2):

(14) 14

Because in this case high-income commuters have the possibility to exchange trips on road T for trips on road U as the toll increases, we get the term , which

gives the number of trips added to U per trip removed from T12 . Although equation ( 14) has the same structure as ( 11), it incorporates the marginal external congestion cost caused on road U by the fraction of trips removed from T and added to U.

As before, ( 14) implies that the optimal toll should be set as the difference between a fraction of the mecc and other taxes per trip. The externality-correction term refers to the difference between the marginal cost of congestion on road T and the marginal cost of congestion imposed on road U by commuters that change roads to avoid the toll. This is a typical second best result: mitigate the distortion on one market only to the extent that it does not aggravate the distortion on the other market (Small and Verhoelf, 2007 p. 140).

Toll revenues used to cut labor taxes

The optimal toll when the incremental toll revenues are used to cut labor taxes is given by (see Appendix B.2):

(15)

This equation has the same structure as ( 14) and contains the externality-correction term.

Again, the only difference between ( 14) and ( 15) is the redistributive term, which takes into account the proportion of labor supplied by low-productivity households as in (12).

Equations ( 14) and ( 15) imply therefore a toll level lower than that implied by ( 11) and ( 12), respectively, as the former includes a mecc that is reduced by the effect of traffic diversion.

High-income households commuting by T, and low-income households commuting by T and U

Let us assume that the economy is at the third type of equilibrium, where high-income households take road T, and low-income households take both road T and road U. As before, we assume that if the toll increases, low-productivity consumers reduce the number of trips they make by road T and increase the number of trips they make by road U. In addition, we assume that high-productivity consumers reduce their number of trips by road T only as result of the toll increase. However, they do not move to road U. Thus, the government's budget constraint becomes:

(16)

The optimal toll when revenues are used to make poll transfers is as follows (for detail see

Appendix B.3): (17)
where , and

. Although ( 17) is more complex than previous equations, we can identify the same structure. The optimal toll should be set as the difference between a fraction of the externality-correction term and the level of other taxes per trip. The externalitycorrection term here consists of three terms: the mecc imposed on road T by both high-income and low-income households and the mecc imposed on road U by low-income households. In this case, the value of time of both types of consumers appears in the equation as they both take the tolled road. As indicated by Small and Verhoef (2007 p. 145), when tolls cannot be differentiated among user groups, the second-best toll depends on a weighted average (by the price sensitivity of demand) of the marginal external costs for the different groups.

The price elasticity of demand of both groups appears in (17) as both kinds of consumers take road T. Each term in this expression is weighted by the price elasticity of each type of household, as the response of consumers to toll increases depends on their price elasticity.

In this section we only derive the expression for the optimal toll when revenues are used to finance poll transfers. The complexity of the expression makes that deriving the second case does not add any further insight other than the already found in (17). That is, when both kinds of individuals use the tolled road, the price elasticity of both groups should be taken into account.

High-income households commuting by T and U, and low-income households commuting by T and U

Finally, assume that the economy is at the fourth type of equilibrium, where both kinds of households take both roads. Next, assume that if the toll increases, both groups reduce the number of trips they make by road T and increase the number of trips they make by road U.

Moreover, assume that road users reduce the number of trips they make on this road only as a result of the toll increase. The government's budget constraint becomes:

The optimal toll when the incremental toll revenues are used make poll transfers is as follows (see Appendix B.4):

. Equation ( 19) is the more general equation derived in this analysis. It takes into account the use of the roads by both kinds of consumers. Interpretation is as explained before.

Optimal tolling rule: summary

The optimal tolling rule can be formulated as an optimal deviation from the marginal external congestion costs as follows:

where and stand for distributional concerns (a combination of the social weights given to each group, the marginal utility of income of each group, the proportion of low-income households in the economy, the price elasticity of demand), stands for the marginal external congestion cost and for the preexisting taxes in the economy.

The urban toll on work-related traffic differs from the Pigouvian tax given the second-best conditions considered in the analysis. These can be summarized as: (i) When only a single road in a network can be tolled, the marginal external cost induced on substitute roads by traffic diversion should be subtracted from the marginal external cost of congestion on the tolled road. (ii) When the toll cannot be differentiated among heterogeneous drivers, the marginal external cost should be an elasticity-weighted average of the marginal external cost for all the drivers using the tolled road. (iii) When patterns of traveling interfere with other markets' equilibriums, the taxes levied (per trip) on the other markets should be considered in the tolling rule.

Numerical illustration

This section presents the results of a numerical simulation 13 of a road network such as illustrated in Figure 1. The average distance of a daily (round) commuting trip is 40 Km if using the tolled road and 60 Km if using the untolled road. The maximum commuting demand in a period (a year) is 200 000 trips. The slope of the congestion function is such that the freeflow speed is 60 Km/h, which is reduced to 30 Km/h when total number of trips is equally divided between the two roads. Thus, travel time increases linearly with increasing traffic volume. Both roads exhibit the same congestion functions.

We define a household's utility function 14 separable in two terms, the utility of consumption/leisure and the disutility of travelling:

(1')

We choose =0.33 and =0.4 to be consistent with values of consumption/leisure elasticity of the related literature 15 . We set =1 16 and give no particular weight to any of the roads, so that the marginal disutility of traveling for any of the routes is the same 17 . In other words, preferences with respect to the use of the two roads are symmetric, such that households do not systematically prefer one road to the other.

We introduce two groups of households (in equal proportion in the economy) that only differ in their labor productivity. Labor productivity factor of high-income households is 13 The algorithm was written in Wolfram Mathematica 7.0.

14 Similar functions are used in Parry and Bento (2001) and Van Dender (2003). 15 See Parry and Bento (2001) p. 658 for a discussion of empirical evidence of these parameters. 16 In the basic setting, =1 implies a disutility of traveling around 7% (3.5%) of the total utility for low-income consumers (high-income consumers). Increasing (decreasing) this parameter raises (drops) the disutility of traveling (see the sensibility analysis). 17 This implies that the roads are perfect substitutes from the consumer perspective. This reflects the consumer taste and has no relation with the characteristics of the roads.

around five times the productivity factor of low-income households 18 . This implies that the gasoline expenditures represent around 20% of the after tax wage of a low-productivity household and around 5% of the after tax wage of a high-income household. We assume a wage tax rate of 20% and 8 hours of work per day.

The constraints of this maximization problem are those described in ( 2) and (3). Thus, each household individually chooses, with perfect knowledge of the travel conditions on the road network, the route and the number of commuting trips.

When we use a toll, the toll revenues can be recycled in two ways: (1) recycling through poll transfers, and (2) recycling through labor tax reductions.

Figure 2 depicts the number of trips that a representative household of each type makes in a period on each of the roads in function of the toll when toll revenues are recycled through labor tax cuts (recycling through poll transfers gives similar results in terms of road use). The point where the toll is zero in Figure 2 gives the no-toll equilibrium. At the no-toll equilibrium, the tolled road is intensively used by both types of consumers. Low-income consumers make around 70% of their trips by road T, whereas high-income consumers make around 60% of their trips by this road. If the toll increases, low-income consumers reduce the number of trips they make and exchange some of the trips on road T by trips on road U. On the contrary, high-income consumers increase the number of trips they make and exchange trips on road U by trips on 18 We consider total disposable income, without distinguishing the source of income. Income data were taken from INSEE (2009); according to them, the best-off of households have five times as much disposable income as the most modest. Low-income consumers benefit only with head transfers. Recycling through labor-tax cuts is welfare reducing for them. On the other hand, high-income consumers benefit from both measures but they gain more when revenues are redistributed through labor-tax cuts. There are two reasons for these effects. First, an equal head transfer represents a higher percentage of the after-tax income of a low-productivity consumer than of a high-productivity one.

Second, the reduction that high-productivity workers obtain from a labor tax cut is higher than the reduction obtained by low-productivity workers.

High-income households

Labor-tax cuts Poll transfers

The social welfare of both kinds of recycling measures, when the distributive weight to the low-income group is higher (the revealed policy preference in the basic setting gives θ=0,6), is shown in Figure 4. Increasing the toll and reducing labor taxes is not necessarily welfare optimal because poor can gain more from poll transfers.

Conclusion

In this paper, we have derived an optimal toll rule that can be used for a government that, in order to reduce traffic congestion, charges commuters for the use of a road and uses the revenues to redistribute wealth among all commuters. The tolling rule goes beyond the simple external congestion cost. Issues as the governmental distributive concerns, who uses the tolled road, who benefits from redistribution, how easily consumers switch to other alternatives, move away the toll from the marginal external congestion cost.

A numerical illustration shows that, in general, the implementation of a toll makes highincome consumers to win and low-income consumers to lose when other transport options (free roads or public transit) are costly and limited. On the other hand, if toll revenues are to be distributed among commuters, low-productivity commuters take better advantage of head transfers, while high-productivity commuters take better advantage of labor tax reductions. However, the choice of the policy largely depends on the characteristics of the economy and the weight that the decision maker gives to redistribution.

We conclude our analysis by drawing attention to some caveats. A differentiated labor tax between income groups could for instance change the results. Implementation costs of tolling systems can also be important. Other interesting considerations include the introduction of environmental externalities related to the use of motor fuels as well as the differentiation of fuel or car technology according to consumers groups. A more comprehensive analysis would also include tolling leisure-related traffic.

Revenues used to make lump-sum transfers to the individuals:

Differentiating the social welfare function with respect to τ, when affects , we have:

Revenues used to cut labor taxes:

The welfare impact when the incremental toll revenues are used to cut labor tax rates . Differentiating ( 13) with respect to τ and solving for , gives:

B.9

Inserting B.9 into B.8, dividing by , setting , and using we get:

Setting and using we get the optimal toll in (14).

Revenues used to cut labor taxes:

Differentiating the social welfare function with respect to τ, when

Figure 1 .

 1 Figure 1. The two-route problem Households choose which route to use to commute to work, U or T. Total number of worked days in a period (D) is the sum of the number of days the household commutes by the untolled road D U and by the tolled road D T . Thus, the budget constraint is:

( 4 '

 4 ) and (5') yields the Wardrop equilibrium condition 5 in which the two roads have equal generalized prices 6 :(6) This expression indicates that households are indifferent between taking either of the two roads when the toll imposed on road T equals the extra-cost of commuting by road U. That is, the extra-gasoline and the extra travel time costs plus the difference between the marginal disutility of commuting by each road 7 (per distance traveled a day).

( 4 '

 4 ) and (5') we can get the expressions to compare the generalized cost of commuting by each road per type of household: conditions compare total generalized cost of commuting by T (left hand-side) with the cost of commuting by U (right hand-side) for a high-productivity household (7) and for a low-productivity household (8). When a household takes the decision to commute by one of the roads, it already knows the cost of time it will face: total time required times its own value of time 9 . Time required (per unit of distance) by each road is an increasing function of total traffic volume.is the time required by road T when all high-income households commute by T, and is the time required by road U when all low-income households use road U. and represent the value of time for high and low-productivity households, respectively.

  High-income households commuting by T and low-income households byU: If equations (7') and (8') hold both with inequality, high-income households will strictly commute by road T and low-income households by road U. In this case, road T should be less expensive than road U for high-income consumers. That is, the cost of an extra-trip on T should be lower than the cost of an extra-trip on U for a high-income commuter, .Moreover, as road U should be less expensive than road T for low-income commuters, the cost of an extra-trip on T should be higher than the cost of an extra-trip on U for a low-income commuter, . These equations can hold simultaneously since the cost of time is different for each type of consumer and normally . Equilibrium 2. High-income households commuting by T and U, and low-income households commuting by U: If equation (7') holds with equality and (8') with inequality, high-income households will be indifferent towards the two roads and low-income households will strictly commute by road U. In this equilibrium the cost of an extra-trip on road T for a high-income consumer equals the cost of an extra trip on road U, and . Equilibrium 3. High-income households commuting by T, and low-income households commuting by T and U: If equation (7') holds with inequality and (8') with equality, highincome households will strictly prefer road T and low-income households will be indifferent towards taking either of the two roads. In this equilibrium the cost of an extra-trip on road T for a low-income consumer equals the cost of an extra-trip on road U, and . Equilibrium 4. High-income households commuting by T and U, and low-income households commuting by T and U: If both equations hold with equality, both types of households will be indifferent towards taking either of the two roads. In this equilibrium the cost of an extra-trip on road T for both kinds of consumers equals the cost of an extra-trip on road U, and .

Figure 2 .

 2 Figure 2. Trips per household per road (labor tax cut case)

  As they can pay for the toll, they can take advantage of the reduction of congestion on road T caused by low-income commuters leaving this road.When the toll approaches 60 cents/Km, the cost of commuting by road T exceeds the cost of commuting by road U for low-income commuters. Consequently, from that point onwards they only use road U. On the other hand, high-income consumers, instead of decrease, increase the number of trips on road T until that point. From that point onwards they start switching to the other road since paying for the toll does not compensate the gain in time anymore as low-income consumers have completely abandoned that road.This exercise shows that if the tolled road allows faster and less fuel-consuming trips than the alternative roads, imposing a toll can reduce congestion on the tolled road. However, the negative impact of this reduction is mainly borne by low-income consumers that have to reduce their number of trips and bear the extra-time and extra-fuel costs implied by the alternative roads. The welfare effect of the two policies is depicted in Figure3. The vertical axis shows the change of individual welfare per policy (in units of utility), where zero corresponds to the welfare level at the no-toll equilibrium.

Figure 3 .

 3 Figure 3. Utility per type of household

Figure 4 .

 4 Figure 4. Social welfare

  with respect to τ gives the change in the transfer associated with a change in the toll : B.3 Inserting B.3 into B.2 and dividing by , we have: Setting , defining and the elasticity of demand of high-income consumers for the tolled road as , we get (11).

We consider that costs such as maintenance, insurance, vehicle ownership taxes, etc., are constant, since they do not vary with the level of congestion.

As road T attracts consumers who are willing to pay more for faster commuting.

It implicitly assumes consumers are informed about current traffic congestion conditions on both roads, by for example, electronic bulletin boards or services such as traffic forecast, and of course, by their own experience.

This is a purely utilitarian social welfare function where increases or decreases in individual utilities translate into identical changes in social utility. Aversion to income inequality is introduced via the social weight given by the government to each kind of household (with ).

This assumption will be relaxed in the numerical illustration.

This trade-off between roads affects the mecc as well as the second part of (14) since revenues collected from other taxes also depend on the road used.

Acknowledgements

We thank useful comments of Stéphane Tchung-Ming, Fay Dunkerley, André Decoster, and participants of the Lunch-Seminar at IFP Energies Nouvelles (October 2012), the Joint Public Economics & Environmental and Transportation Economics Workshop -KU Leuven (December 6 th , 2012) and the Friday Lunch Meeting of the Climate Economics Chair of Paris-Dauphine University and CDC Climat (January 11 th , 2013). Financial support of IFP Energies Nouvelles is gratefully acknowledged.

Appendix A. The households' problem

The household's problem defined by (1), ( 2) and (3).can be solved by maximizing the following Lagrangian function:

is the Lagrangian multiplier related to the income constraint, called the "marginal utility of income".

is the Lagrangian multiplier related to the time constraint, called the "resource value of time". For X>0, D U >0, D T >0 and t L >0, the system of first-order conditions can be written as:

A.1

Using these conditions and the budget constraint, we obtain the demand functions for and .

Replacing these functions into the utility gives the indirect utility function which allows rewriting the household's problem as:

A.2

Note that the marginal disutility of a toll increase ( ) is the marginal utility of income multiplied by the optimal number of trips (D T ). Similarly, the marginal disutility of an increase of travel time on road T ( ) is the resource value of time multiplied by the optimal number of trips.

Appendix B. The social planner's problem

B.1 High-income households commuting by T and low income households by U

In this case, since high-income households only use road T. And , since low-income households only use road U.

B.3 High-income households commuting by T, and low-income households commuting by T and U

Revenues used to make lump-sum transfers to the individuals:

Differentiating the social welfare function with respect to τ (when affects ) and replacing A.

B.4 High-income households commuting by T and U, and low-income households commuting by T and U

Revenues used to make lump-sum transfers to the individuals:

Following the same procedure, we obtain the two basic equations to get (19):

B.16

B.17

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