L V B Diop

O Isnard

G Ballon

J P Itié

Investigation of the itinerant metamagnetic system Hf0.75Ta0.25Fe2 under extreme conditions of pressure or magnetic field

Keywords: neutron powder diffraction, spin reorientation, volume magnetostriction, spin fluctuations

The magnetic and structural properties of Hf0.75Ta0.25Fe2 have been investigated under the duplicate conditions, i.e., high magnetic fields and high pressures, by combining magnetization measurements up to 26 T, neutron powder diffraction experiments up to 14.5 T, and angle dispersive synchrotron x-ray diffraction up to 10 GPa. The intermetallic compound Hf0.75Ta0.25Fe2 exhibits an antiferromagnetic (AFM) ground state below the Néel temperature TN = 342 K. In the AFM structure, the Fe-6h magnetic moments (1.02  at 2 K) align along the c axis while Fe atoms at 2a sites are not magnetically ordered. We further show that the AFM ground state gets transformed into a ferromagnetic (FM) state via a field-induced metamagnetic phase transition. Magnetic-field dependent neutron diffractograms demonstrate that FM phase sets in with a change of the easy magnetization direction from axial to basalplane and an extremely anisotropic lattice expansion. The unit cell expands drastically in the basal plane upon increasing the applied magnetic field, while its dimension along c-axis shrinks slightly; resulting in a huge positive volume magnetostriction V/V = 0.78 %. The application of hydrostatic pressure leads to the continuous decrease of the cell parameters. The shrinkage of the hexagonal cell is anisotropic with a larger compression along the six-fold symmetry axis c. A bulk modulus of K0 = 247 GPa has been deduced from the pressure-volume relationship.

Introduction

The itinerant electron metamagnetism (IEM), i.e., field-induced first-order transition from a low magnetic moment state to a high magnetic moment state, is an essential physical phenomenon, describing magnetic properties of intermetallic compounds, demonstrating technologically important giant magnetocaloric, magnetoelastic and magnetoresistive effects [1][2][3][4]. The discovery of giant magnetocaloric effect in the itinerant metamagnetic system La(Fe,Si)13 [1], has opened new perspectives for room temperature magnetic refrigeration. This has triggered a spectacular upsurge of interest in magnetic cooling and the development of many researches from the viewpoints of both fundamental physics and technological applications.

One interesting example of IEM system is the Laves phase compounds Hf1-xTaxFe2. In the whole concentration range of x, the pseudo-binary system Hf1-xTaxFe2 adopts the hexagonal MgZn2-type structure (C-14 Laves phase, space group P63/mmc). There are two inequivalent positions for Fe atoms, 2a and 6h, and one position, 4f for Hf/Ta atoms in this structure. The magnetic ground state changes at a Ta concentration of about x = 0.25 from ferromagnetic (FM) for Hf rich to antiferromagnetic (AFM) for Ta rich side . In the intermediate concentration range 0.1 < x < 0.25, the alloys exhibit temperature-induced magnetic transitions from ferromagnetic (FM) to antiferromagnetic (AFM) order and then to the paramagnetic (PM) state upon heating [5][6][7][8][9][10][11][12][13]. In the FM ground state, the magnetism comes from Fe atoms on the two crystallographically inequivalent sites, 2a and 6h, both carrying a sizable magnetic moment.

The high-temperature AFM phase has a complex spin configuration with two independent Fe sublattices: only 3/4 of the Fe atoms are involved in the AFM arrangement, namely those located on 6h crystal sites, whereas Fe atoms occupying 2a Wyckoff positions carry no ordered magnetic moment [5,8,10,13,20]. The FM-AFM magnetic transition is of first order and is accompanied by an isomorphic (without change of symmetry) lattice collapse, large magnetocaloric [14][15][16], magnetoresistive [6,10] and magnetovolume effects [6,12,[17][18][19] as well as a giant negative thermal expansion [8,9,11]. Staircase-like transitions and anomalous thermomagnetic irreversibility arising due to kinetic arrest have also been observed in Hf1-xTaxFe2 compounds (x = 0.225, 0.230, and 0.235) [21][22][23]. Interestingly, peculiar and unconventional thermal effects across the FM-AFM first-order transition were most recently discovered in Hf0.82Ta0.18Fe2 [7]. Such thermal effects result in warming (cooling) when heat is extracted from (supplied to) the material, thus providing indisputable evidence of metastable superheated FM and supercooled AFM states [7]. Changes from ferromagnetism to antiferromagnetism can be induced by external pressure; for instance, ferromagnetic Hf0.825Ta0.175Fe2 becomes AFM at pressures beyond 0.75 GPa ( critical pressure) that corresponds to the decrease of volume of only 0.5% [6]. An investigation of the interatomic Fe-Fe exchange coupling coefficients and their lattice dependence demonstrated that the ferromagnetic interaction with the first shell is remarkably reduced upon shrinkage of the a cell parameter [19]. Another fascinating result of the theoretical calculations in Ref. [19] is that the Fe6h-Fe6h coupling is always positive meaning that these interactions will favour a ferromagnetic spin configuration within the atomic plane containing the Fe6h positions with Fe6h as near neighbours. The magnetic phase diagram of this pseudo-binary compounds Hf1-xTaxFe2 (0.1 < x < 0.25) has been qualitatively explained on the basis of the spin fluctuations theory of itinerant-electron system developed by Moriya and Usami (M-U) [24].

Up to now, the extensive investigations devoted to the Hf1-xTaxFe2 compounds have mainly focused on the intermediate composition range 0.1 < x < 0.25. It appears there have been no direct measurements of the magnetic structure of the compounds with x ≥ 0.25, that is, the precise magnetic structure of the AFM ground state remains an open question.

The present work aims at answering the open questions on the basis of an experimental study carried out on purposefully produced compound of composition Hf0.75Ta0.25Fe2. To determine the magnetic structures and understand their coupling with the lattice, we conducted a detailed neutron powder diffraction study as a function of both temperature and applied magnetic field. In addition, a careful look at the literature data led us to notice that despite the extensive investigations noted above, there appears to have been no experimental works addressing the high-pressure behaviour of the crystallographic sublattice in the series of compounds (Hf,Ta)Fe2. To contribute to the understanding of the role of volume in the pseudobinary system (Hf,Ta)Fe2, we have investigated the effect of hydrostatic pressure (up to 10 GPa) on the structural properties of Hf0.75Ta0.25Fe2 by means of high-resolution synchrotron x-ray diffraction.

Experimental methods

The polycrystalline alloy of composition Hf0.75Ta0.25Fe2 was synthetized by arc melting high-purity constituting metals (at least 99.95%) under a purified argon gas atmosphere. The sample was melted several times with the button flipped over after each melting, which is usually sufficient to achieve compositional homogeneity. The so-obtained ingot was wrapped in tantalum foil, sealed in an evacuated quartz tube and subsequently annealed in a resistive furnace as follows. It was heated up to 1273 K at a rate of 5 K/minute and kept at this temperature for 7 days. Then, the power was switched off and the furnace quickly cooled down to room temperature. The room temperature crystal structure and phase purity of the synthetized material was checked by standard x-ray powder diffraction using a Siemens D5000 diffractometer in reflection mode with the Bragg-Brentano geometry and Co-Kα radiation.

Magnetization curves were measured in steady magnetic fields up to 10.5 T at temperatures ranging from 2 K to 500 K using extraction-type magnetometers [25]. The magnetization measurements were extended up to 26 T using a non-destructive pulsed-field magnet at the French National High Magnetic Field Laboratory (LNCMI) in Toulouse. A single 1.25 MJ capacitor bank was employed. When fully charged, this could produce a maximum field of 53 T with a rise time of about 25 ms and a total pulse duration of 150 ms. In our measurements the capacitor module was charged to about one-half. A detailed description of the pulsed-field facility can be found in Ref. [26]. All pulsed-field data were calibrated against the magnetization recorded in steady magnetic fields.

Neutron powder diffraction (NPD) experiments were performed at the Institut Laue Langevin (ILL) in Grenoble, France. About 3 g of fine powder were introduced into a cylindrical vanadium container (D = 6 mm, H = 5 cm) and mounted on the stick of a He cryostat, whose contribution to the diffraction patterns was eliminated using a radial oscillating collimator. Several diffractograms were collected at selected temperatures ranging between 2 and 375 K on the high-intensity two-axis powder diffractometer D1B with a detector angular range coverage 5° ≤ 2θ ≤ 128° which is especially suited for magnetic structure determination.

The data were collected using a 3 He multicounter containing 1280 detection cells with a step of 0.1° between neighbouring cells. A neutron incident wavelength of 2.52 Å was selected by a (002) Bragg reflection of a pyrolytic graphite monochromator, the take-off angle being 44.2° in 2θ. All measurements were performed upon heating after a stabilization time of 3 minutes with typical acquisition times of 20 minutes per isotherm. Due to the high flux available on the instrument, a second set of diffraction patterns was recorded in situ every 3.5 K while ramping the temperature from 2 K to 300 K. Rietveld analysis of the collected data was done using the FullProf suite software package [27], which allows the simultaneous refinement of structural and magnetic profiles.

High-field neutron diffraction experiments were carried out on the Extreme Environment Diffractometer (EXED) at the BER II research reactor at Helmholtz-Zentrum Berlin, Germany. Neutron measurements under applied magnetic fields of up to 14.5 T (superconducting magnet) were performed on pelletized sample, to avoid reorientation of the crystallites. For neutron detection, the time-of-flight (TOF) instrument EXED is equipped with 1/2" diameter position-sensitive 3 He detector tubes (GE Reuter Stokes). These tubes are combined into 4 detector banks (48 tubes/bank) that can be positioned at different angles and distances around the sample environment [28][29]. Our diffraction experiments using EXED focused on the low angle (large d spacing) region with the aim to concentrate on the magnetic signal.

Angle-dispersive x-ray powder diffraction data under hydrostatic pressures of up to 10 GPa were obtained using gasketed diamond anvil cells (DAC) at the PSICHÉ beamline (monochromatic radiation λ = 0.3738 Å) of the French Synchrotron facility (SOLEIL), at room temperature. The powder sample is loaded into the gasket hole along with small chips of ruby.

A mixture of methanol and ethanol was employed as a pressure-transmitting medium. Accurate determination of sample pressure was made by the ruby fluorescence method. In order to improve the powder statistic, the sample was rotated ±5° during the exposure. The sample-todetector distance was calibrated using a polycrystalline powder CeO2 standard. Twodimensional diffraction images were recorded by means of a MAR detector and the resulting diffraction data were azimuthally integrated with the FIT2D program [30], yielding one dimensional intensity versus diffraction angle 2θ patterns. A precise determination of the unitcell parameters was obtained by a least-squares refinement of the diffraction patterns, including all the observed Bragg reflections. T. One remarks that, when the external magnetic field is reduced to zero, the system recovers the initial state; this indicates that the forced FM phase loses its stability when the applied field is brought back to zero. In the investigated temperature interval 1.5 -100 K, we observed a clear hysteresis against the field scan, which is characteristic of the first-order nature of the phase transition. The width of the magnetic hysteresis becomes smaller upon heating. The /f.u.)

Results and discussion

Magnetic susceptibility and magnetization

 0 H (T) 1.5 K 20 K 40 K 60 K 85 K 100 K
saturation magnetization Msat of the field-induced FM phase amounts to 3.02 B/f.u. at 1.5 K; leading to an average magnetic moment of 1.51 B per Fe atom. The saturation magnetization of the forced FM state of our sample Hf0.75Ta0.25Fe2 compares very well with the value of the spontaneous magnetization MS = 3.10 B/f.u. obtained at 4 K for the ferromagnetic parent compound HfFe2 [6].

 0 H up  0 H down  0 H cr (T) T (K)
compounds with the competing FM and AFM phases [24]. They proposed magnetic phase diagrams in which both FM and AFM components of spin fluctuations are present. The M-U theoretical model takes into account the equilibrium state of the free energy displaying a double minimum and assumes that the thermal evolution of 𝜒 Q 𝜒 0 ⁄ is monotonic, where 𝜒 0 and 𝜒 Q represent respectively the uniform and staggered magnetic susceptibilities. In the high temperature region, the behaviour of the critical field 0Hcr for Hf0.75Ta0.25Fe2 which exhibits an AFM ground state is similar to that for the compounds on the intermediate composition range 0.1 < x < 0.25 having a FM ground state (the increase of 0Hcr upon heating being correlated to the monotonic decrease in 𝜒 Q 𝜒 0 ⁄ ). By contrast, the critical field increases with decreasing temperature at low temperatures in Hf0.75Ta0.25Fe2 system against the M-U theoretical model.

This observed behaviour indicates that the thermal variation of 𝜒 Q 𝜒 0 ⁄ in the low temperature region is different from that of 𝜒 Q 𝜒 0 ⁄ at high temperatures. The M-U theory does not take into account interactions arising from magnetovolume coupling in the free energy expansion.

Consequently, we suggest that the thermal evolution of the critical field of the AFM-FM metamagnetic phase transition is explainable by considering the contribution from both the magnetic and elastic energies resulting from magnetovolume effects.

To further investigate the magnetic behaviour of Fe sublattice in the paramagnetic phase, magnetization isotherms have been measured at temperatures well above TN. The thermal evolution of the inverse susceptibility is shown in Figure 4. The magnetic susceptibility follows Curie-Weiss law but two distinct regions are observed involving different paramagnetic Curie temperatures p and Curie constants C. The determined values are summarized in Table 1. The paramagnetic susceptibility presents a break at T * = 400 K. The Curie constant above T * is found to be about 1.5 times smaller than that measured below. The effective moment eff derived from the Curie constant is 4.25 /Fe and 3.55 /Fe below and above T * , respectively.

This study of magnetic properties of Hf0.75Ta0.25Fe2 intermetallic in both paramagnetic and magnetically ordered states has led to remarkably different values of Fe magnetic moments in the ordered state and in the paramagnetic phase. We calculated the Rhodes-Wohlfarth ratio r for both paramagnetic domains, namely, below and above T * and the corresponding values are listed in Table 1. The determined r values are larger than 1 and point to the itinerant character of magnetism in this compound. Following Moriya and co-workers [31][32][START_REF] Kübler | Theory of Itinerant Electron Magnetism[END_REF][START_REF] Moriya | [END_REF], the observed bend at T * of the reciprocal magnetic susceptibility of Hf0.75Ta0.25Fe2 is due to the temperature variation of the local amplitude of spin fluctuations.

Neutron powder diffraction (NPD) A. Zero-field NPD measurements

In order to establish the magnetic structures of this complex system Hf0.75Ta0.25Fe2, we have performed NPD studies. So far, no member of the (Hf,Ta)Fe2 series of compounds with an AFM ground state has been investigated using neutron diffraction. Rietveld refinements have been performed on diffractograms recorded at selected representative temperatures in the PM and AFM states, in order to refine the crystal structure and magnetic ordering -Figures 5a and 2-Theta (degrees)

T = 2 K
The analysis of the NPD pattern obtained at 2 K shows that the crystallographic structure is retained at low temperature. The 2 K diffractogram presents the same Bragg peaks as the spectra collected in the PM phase. No additional magnetic reflections are detected, which implies that the corresponding AFM structure is defined by the magnetic propagation vector k = (0, 0, 0). In other words, the magnetic unit cell coincides with the crystallographic one. At 2 K, the best refinement is achieved for a magnetic arrangement in which Fe-6h spins align ferromagnetically within the same layer but opposite orientations on successive Fe-6h layers.

In the AFM structure, the Fe magnetic moments align along the six-fold symmetry axis c. The refined moment for Fe-6h atoms at 2 K is 1.02 ; the Fe-2a atoms are not magnetically ordered.

Since the 2a Wyckoff position constitutes an inversion centre for the 6h sublattice, the molecular field acting on the 2a crystal site from the Fe-6h magnetic moments is cancelled by symmetry in this AFM spin configuration. Therefore, Fe-2a atoms display null magnetic moment. The obtained Fe magnetic moment (1.02 ) is remarkably decreased in comparison with the pure Fe moment of 2.22 μB. This experimental result is consistent with the presence of non-magnetic Hf and Ta atoms as near neighbours. The AFM magnetic structure of Hf0.75Ta0.25Fe2 is similar to that established elsewhere for isostructural compound TiFe2 [20].

Not knowing a priori the orientation of the magnetic moments in the AFM ground state, Rietveld refinements of the 2 K pattern were also conducted considering an alignment of the moments in the basal-plane of the hexagonal lattice. The two principal basal-plane directions,

[100] and [120], were tested. The magnetic reliability factors (Rmag) obtained for the two directions are very close: 14.7% and 15.3% for [100] and [120] orientation, respectively. The orientation of the magnetic moments in the basal-plane leads to much worse fits compared to that along the c-axis [001] which leads to much smaller reliability factor 7.11%. Consequently, at 2 K, Rietveld refinement gives a significantly better fit when the magnetic moments are along the [001] direction giving confidence to such orientation of the magnetic moments. However, it should be noted that the magnitude of the magnetic moment for Fe-6h atoms differs only very slightly between the two orientationsc axis and basal-planeand Fe-2a atoms carry no ordered moment. Fitting the 2 K magnetic pattern with Fe-6h moments oriented along the caxis yields the results presented in table 2.

From now on, it is clear that the magnetic structure of the AFM ground state in Hf0.75Ta0.25Fe2 is different from the spin configuration of the high-temperature AFM phase found in compounds when 0.1 < x < 0.25, whose ground state is the FM state. It is recalled that in the high-temperature AFM the Fe magnetic moments are confined to the basal-plane of the hexagonal structure [13].

The thermal evolutions of the lattice parameters a and c, the c/a ratio, and the unit-cell volume V are plotted in Figures 6a,6b, and 6c, respectively. The results of the Rietveld analysis of the thermodiffraction data reveal that the lattice constants, a and c, decrease as the temperature is lowered, thus causing the overall reduction in the unit-cell volume upon cooling.

On the contrary, the c/a ratio decreases with increasing temperature. Below 25 K, the unit-cell parameters remain essentially constant; and then present a normal behaviour by increasing as a result of thermal expansion, with a rate of 1.03×10 -4 Å K -1 , 1.56×10 -5 Å K -1 , and 4.53×10 -3 Å 3 K -1 for a, c, and V, respectively. Hf0.75Ta0.25Fe2 intermetallic exhibit positive thermal expansion (PTE) over the investigated temperature interval. Based on the refined values of cell volume, we calculated the volumetric thermal expansion coefficient V = (1/V)(dV/dT) and a value of 26.9×10 -6 K -1 is obtained at room temperature. The observation of a metamagnetic phase transition by magnetization measurements prompted us to perform neutron experiments in applied magnetic field to determine the magnetic-field dependence of the structural and magnetic state of Hf0.75Ta0.25Fe2. For this NPD experiments we used the EXED instrument in the high-intensity mode, and focused on the large d-spacing lines, that is, maximizing the neutrons flux for peaks displaying large d-spacing which appear at low diffraction angles where the generalized increase of the magnetic signal in the Bragg positions is more evident. The normalized neutron diffraction patterns at 2 K and 20 K under various applied magnetic fields are depicted in Figures 7a-7b. Our choice of these (1 0 0)

T = 20 K
particular temperatures is due to the fact that we aimed to observe the field-induced phase transition (see Figure 2) within the maximum available magnetic field of 14.5 T in this experiment. All the magnetic intensities are observed on top of the nuclear Bragg peaks and the contribution of the magnetic order to the NPD diagrams is reflected in the intensities of the (100), ( 101) and (002) reflections. It can be clearly seen on the diffraction patterns in figures 7a and 7b that there is a large difference between the spectra at zero field and under applied magnetic fields. This is most reflected in the dramatic change of intensity of the (002) peak.

The intensity of the (002) line is much more susceptible to magnetic fields than that of the ( 100) and ( 101) lines. As expected the application of a magnetic field supresses the AFM phase and a FM state is induced, as can be observed in figure 7a looking especially at the diffractograms collected at 2 K at zero field and 14.5 T.

T = 2 K (0 0 2)
FM phase c/c (2 K, 14.5 T) = -0.059 %. Consequently, the forced volume magnetostriction is estimated to be V/V (2 K, 14.5 T) = 0.78 %. The AFM-FM metamagnetic transition is accompanied by large magnetovolume effects. Our results demonstrate that the effect of the applied field is to induce a magnetic phase transformation from a low-volume AFM state to a high-volume FM state through a first-order transition at a certain critical field. The obtained value for the field-induced volume magnetostriction compares very well with the volume change V/V = 0.75 % derived from dilatometric magnetostriction measurements for isostructural compound Hf0.825Ta0.175Fe2 [12].

We have found a different effect of the magnetic field in the intensities of the three peaks at low temperature. This is illustrated in Figure 9, where the integrated intensities of the (100), ( 101) and (002) reflections are displayed as a function of the magnetic field at 2 K. The difference in the influence of the applied magnetic field is remarkable. While the external applied magnetic field leads to a very fast rise of the intensity associated to the (002) peak, the intensities of the (100) and (101) lines become almost field independent below 9 T and then increase rapidly as the sample undergoes a metamagnetic transition. This phenomenon is in agreement with the macroscopic magnetic measurements, where the onset of significant change in the magnetization is found at 9.25 T. Upon application of an external magnetic field, a continuously increasing magnetic scattering is observed on (002) peak indicating that a ferromagnetic component is progressively induced in the material. The magnetic contribution to (002) reflection is proportional to the square of the in-plane magnetization component since neutrons are sensitive to the component of the magnetic moment perpendicular to the scattering vector. In the forced FM state, the presence of magnetic contribution on (002) peak reveals that the Fe moments are not aligned along the c-axis. One could conclude that Hf0.75Ta0.25Fe2 features a first-order field-induced spin-reorientation transition; the easy magnetization direction changes from axial to basal-plane upon application of magnetic field.

High-Pressure synchrotron x-ray diffraction (SXRD)

To probe the effect of external pressure on the crystal structure, angle dispersive synchrotron x-ray diffraction patterns have been recorded at room temperature for various applied pressures ranging from 0 to 10 GPa. In Figure 10 we compare selected SXRD patterns during compression of the sample. SXRD diffractograms collected at various pressures reveal that the hexagonal symmetry of the atomic arrangement remains unaltered within the investigated pressure range; no indication of a structural phase transition was found up to the highest applied pressure. All the observed Bragg reflections could be identified to arise from the original C-14 Laves phase structure (space group P63/mmc). The crystal symmetry is stable up to the highest applied pressure of 10 GPa. Analysis of the diffraction patterns shows that the diffraction peaks shift to higher angles due to the contraction of the lattice. The lattice constants (a and c), deduced from least-squares refinements, are plotted in Figure 11 as a function of applied pressure. Both cell parameters decrease continuously upon compression, and values of da/dP = -5.90×10 -3 Å GPa -1 and dc/dP = -9.51×10 -3 Å GPa -1 are obtained. The contraction of the hexagonal cell is anisotropic and more pronounced along the c direction. In the crystal structure, the arrangements of atoms and bonds perpendicular to c-axis direction are dense, while, along c-axis direction the atoms arrange loosely, which points out it has considerable relaxation space on compression. The pressure dependence of the unit-cell volume is illustrated in Figure 12. The experimental data were fitted to a third-order Birch-Murnaghan equation of state [35] where 𝐾 0

´ was fixed at 4. The fit yielded V0 = 168.56 Å 3 for the zero-pressure unitcell volume and K0 = 247 GPa for the bulk modulus. For comparison purposes we note that, TiFe2 and LaFe12B6 materials have a bulk modulus of 201 GPa [36] and 204 GPa [37],

respectively. The bulk modulus of Hf0.75Ta0.25Fe2 is ≈1.2 times larger than that of the antiferromagnetic itinerant-electron system LaFe12B6.

Conclusions

Competition between AFM and FM states in the itinerant-electron system Hf0.75Ta0.25Fe2 has been investigated by means of neutron powder diffraction and magnetizations measurements both performed under extreme conditions of low temperature and large magnetic fields. The studied compound Hf0.75Ta0.25Fe2 presents an antiferromagnetic order as a ground state below TN = 342 K. The AFM ordering is only due to the Fe atoms at 6h sites which carry a magnetic moment of 1.02  at 2 K. It is demonstrated that the AFM phase can be transformed into a FM phase via a field-induced metamagnetic transition accompanied with an in-plane lattice expansion and a change of the easy magnetization direction from axial to basal-plane. In the disordered phase, an unusual and intriguing transition from one paramagnetic state to another one involving two distinct Curie-Weiss behaviours has been observed at T * , a critical temperature at which the reciprocal magnetic susceptibility presents a bend. Thanks to synchrotron radiation, x-ray diffraction has been carried out upon application of a high external pressure. In the investigated pressure range, no change of crystal symmetry occurs. The unit cell shrinks continuously, but in an anisotropic way with a larger compression along the high-symmetry direction c.

Figure 1 .

 1 Figure 1. Temperature dependence of the magnetization for Hf0.75Ta0.25Fe2.

Figure 2 .

 2 Figure 2. Magnetization curves of Hf0.75Ta0.25Fe2 measured between 1.5 and 100 K in pulsed fields.

Figure 1

 1 Figure1presents the thermomagnetic curve, M(T), of the zero-field cooled (ZFC) sample, recorded upon heating in an applied magnetic field of 0.1 T. The peak at 342 K in the isofield magnetization curve of Hf0.75Ta0.25Fe2 corresponds to the Néel temperature TN reflecting the transition from an AFM state to a PM phase. Field-dependent magnetization curves M(H), at several fixed temperatures ranging from 1.5 to 100 K, are displayed in Figure2for the thermally demagnetized Hf0.75Ta0.25Fe2 sample. For the sake of clarity, just a selection of curves is shown in Figure2but all of the measured magnetization data were employed to plot the temperature dependence of the critical field illustrated in Figure3. The M(H) curves exhibit an interesting field-induced metamagnetic phase transition between the AFM and FM states. The metamagnetic phase transformation proceeds through a progressive conversion of the AFM phase into FM domains with increasing the applied magnetic field. At 1.5 K, the fieldinduced metamagnetic phase transition is remarkable in two ways: (i) by the huge magnetization change and (ii) by the presence of a particularly large magnetic hysteresis of 3

Figure 3 .

 3 Figure 3. Thermal evolution of the critical field 0Hcr upon increasing (up) or decreasing (down) the applied magnetic field for Hf0.75Ta0.25Fe2.

Figure 4 .

 4 Figure 4. Thermal variation of the reciprocal magnetic susceptibility for Hf0.75Ta0.25Fe2.

Figure 5 .

 5 Figure 5. Refinements of the neutron diffraction patterns recorded for Hf0.75Ta0.25Fe2 at 375 K (a) and 2 K (b). In figure (a) the top row of Bragg markers is for the Hf0.75Ta0.25Fe2 nuclear contributions, the second row corresponds to the position of the vanadium sample environment contribution. In figure (b) the top and second rows of Bragg markers are referring to the nuclear and magnetic contributions of the Hf0.75Ta0.25Fe2 phase. The third row is corresponding to the nuclear contribution from the vanadium sample environment.

Figure 6 .Figure 7 .

 67 Figure 6. Thermal dependence of the lattice parameters a and c (a), c/a ratio (b), and unit-cell volume (c) obtained from Rietveld refinement of NPD data measured upon heating for Hf0.75Ta0.25Fe2.

Figure 9 .

 9 Figure 9. Integrated intensities of (100), (101) and (002) reflections at 2 K as a function of the applied magnetic field.

Figure 10 .

 10 Figure 10. Synchrotron x-ray diffraction patterns collected at room temperature for Hf0.75Ta0.25Fe2 at various applied pressures (λ = 0.3738 Å).

Figure 11 .

 11 Figure 11. Lattice constants of Hf0.75Ta0.25Fe2 as a function of applied pressure. The solid lines correspond to linear fits.

Figure 12 .

 12 Figure 12. Unit-cell volume of Hf0.75Ta0.25Fe2 as a function of applied pressure. The continuous red line represents the third-order Birch-Murnaghan fit for the data.

Table 1 :

 1 TN or TC, magnetic ordering temperature; T

		TN/TC	s 1.5 K	T *	qs	C	eff	qc	p	r
	Compound	(K)	(/Fe)	(K)		( /f.u.T)	(/Fe)		(K)
						T<T * 8.08	4.25	3.37	103	2.23
	Hf0.75Ta0.75Fe2	342	1.51	400 1.51
						T>T * 5.64	3.55	2.69	193	1.78
	-Fe	1043	2.22	-2.22	2.26	3.18	2.33 1093 1.05

* temperature of change of the Curie Weiss behavior, qs, number of carriers deduced from saturation magnetization s; qc, number of carriers deduced from Curie-Weiss constant C; eff, paramagnetic effective moment; θp, paramagnetic Curie temperature and ratio r = qc / qs, for the Hf0.75Ta0.25Fe2 compared to that of the pure Fe.

Table 2 .

 2 Rietveld refinement results and reliability factors obtained from the analysis of the powder

	neutron diffraction patterns recorded at 2 K and 375 K for Hf0.75Ta0.25Fe2.
		AFM	PM
	T (K)	2 K	375
	a (Å)	4.904(1)	4.925(3)
	c (Å)	8.040(3)	8.045(2)
	Hf /Ta (4f)
	z	0.062(1)	0.066(3)
	Fe (6h)
	x	0.832(2)	0.834(3)
	y = 2x	1.664(4)	1.668(6)
	Fe 2a moment (µB)	-	-
	Fe 6h moment (µB)	1.02(6)	-
	Moment-c-axis angle (°)	0	-
	c 2	6.39	4.01
	RBragg (%)	7.51	4.34
	Rmag (%)	7.11	-
	Rwp (%)	11.8	11.1
	Rp (%)	12.6	11.1
	Rexp (%)	4.69	5.52

Figure 8 .

 8 Unit-cell parameters of Hf0.75Ta0.25Fe2 as a function of applied magnetic field.

	168.96
	168.52
	168.08
	167.64
	8.0410
	8.0393
	8.0376
	8.0359
		4.925
	(Å)	4.915 4.920
	a
		4.910
		4.905
		4.900
		0	2	4	6	8	10	12	14

Acknowledgments

The authors gratefully acknowledge the Institut Laue Langevin and Helmholtz-Zentrum Berlin for access to the neutron facilities. We acknowledge the French Synchrotron facility SOLEIL for the allocation of beam time. We also acknowledge the support of the French National High Magnetic Field Laboratory (LNCMI), member of the European Magnetic Field Laboratory (EMFL).


François Contensou
email: con-tensou@essec.edu.z

Radu Vranceanu
email: vranceanu@essec.edu.

WORKING TIME AND WAGE RATE DIFFERENCES: A CONTRACT THEORY APPROACH

Keywords: Working hours, Wage gap, Labor market discrimination, Contract theory. JEL Classi…cation: D86, J31, J41, J71

In the labor economics literature, discrimination is often de…ned as occurring when identically productive workers, placed in the same working conditions, are assigned contracts involving, in particular, di¤erent hourly wage rates. This paper applies contract theory to explain how in some circumstances such di¤erences take place, even if contract discrimination and productivity di¤erences are strictly ruled out. It is assumed that worker types di¤er only in their consumption/leisure preferences and in their availability. A labor cost-minimizing …rm o¤ers a menu of labor contracts, and lets workers self-select. The model reveals external e¤ects between types and the possibility of a paradoxical situation in which less demanding workers obtain a higher wage rate. A mixed employment regime always requires a minimum number (a quantum) of most demanding workers.

Introduction

According to the standard labour market theory, wage rates tend to re ‡ect the marginal productivity of labour services. Consequently, wage rates should coincide for workers performing the same task with the same level of ability. Labour market observations however, in many instances, tend to challenge this view. Despite convergence in educational attainments and other relevant explanatory variables, unexplained di¤erences still exist between the full-time wages of men and women, particularly at the top of the wage distribution [START_REF] Gunderson | Male-female wage di¤erentials and policy responses[END_REF][START_REF] Blau | The gender wage gap: Extent, trends, and explanations[END_REF][START_REF] Cahuc | Labor Economics[END_REF]OECD, 2018;[START_REF] Neumark | Experimental research on labor market discrimination[END_REF].

To explain the di¤erences between the neoclassical expected equality and observed facts, economists have often resorted to the notion of discrimination [START_REF] Becker | The Economics of Discrimination[END_REF]. Labor market discrimination has been de…ned as "a situation in which persons who provide labor market services and who are equally productive in a psychical or material sense are treated unequally in a way that is related to an observable characteristic such as race, ethnicity, or gender" [START_REF] Altonji | Race and gender in the labor market[END_REF]. It is well known, however, that some observed di¤erences in wages obtained by equally competent workers may be explained without resorting to discrimination. The theory of compensating di¤erentials, originating in Adam Smith's celebrated seminal works, argues that wage gaps can be grounded in di¤erences in preferences for job attributes (Thaler and Rosen, 1975;[START_REF] Rosen | The theory of equalizing di¤erences[END_REF].1 Some other explanations resort to a productivity argument: in her presidential address to the 126th Annual Meeting of the American Economic Association, [START_REF] Goldin | A grand gender convergence: Its last chapter[END_REF] pointed out that in some quali…ed occupations (legal services, business, and …nance), longer hours provided by men in general, are rewarded at a higher wage rate. 2 Goldin explains this outcome based on productivity increasing with hours, as …rms and their clients value "temporal ‡exibility": on-site presence, intensive client contact, face-to-face time, etc. Workers who work longer hours and ac-cept fragmented schedules could be more valuable to a …rm since they can exchange information with their peers more e¢ ciently. If men are more available than women in such jobs, then a gender wage gap could result.

This paper provides a model of wage and hours di¤erentiation consistent with the abovementioned observations that is completely free of productivity di¤erences or discriminatory practices as de…ned by the labor market literature: in our model, all workers have equal access to all contracts proposed by the employer. The fundamental source of heterogeneity in the model is workers'disutility from work. For the sake of parsimony, we assume that workers are of only two types, and workers of the …rst type always demand less compensation for any amount of working time than workers of the second type. The other basic assumption of this model is that workers of the …rst type are in limited supply.

The principal seeks to hire workers for a predetermined number of working hours, as required to achieve a production target. The goal of the …rm is to minimize the total cost of labor. If workers of both types were available in any quantity, cost minimization would generally cause labour demand to concentrate on the "less expensive" type. The more demanding workers being crowded out, they do not appear in the data. Our analysis focuses on the nontrivial, and most plausible situation, where less demanding workers are in scarce supply and the demand for hours is great enough to prompt the employer to hire both types of workers. When contract discrimination is not possible, hiring workers of the more demanding type creates an externality in ‡uencing the contract o¤ered to the less demanding type, through an incentive compatibility constraint that ensures the e¢ cient self-selection of workers. The determination of optimal contracts builds on standard principles of contract theory (inter alia, [START_REF] Bolton | Contract Theory[END_REF][START_REF] Salanié | The Economics of Contracts: A Primer[END_REF]La¤ont and Martimort, 2009), to which we add, as an original theoretical contribution, an endogenous determination of the proportion of worker types. Despite the relatively complex structure of the problem, the solution is fully characterized for a quadratic compensation function.

A compensation function de…nes the level of consumption required by a worker to ful…ll his/her participation constraint. Our analysis reveals that ordering worker types by non-crossing compen-sation functions is not su¢ cient to predict the ordering of optimal working times, compensations and implicit hourly wages. In particular, the model points out the crucial role played by the speci…c assumption concerning the sensibility of the worker compensation di¤erential with respect to working time itself.

The analysis also reveals that the employer minimizes the total labour cost by o¤ering contracts generally exhibiting di¤erent (implicit) hourly wages. We refer to these wage di¤erences as forms of "pseudo-discrimination", since these wage rate di¤erences between workers doing the same job with the same level of skills exist, but are not explained by incorrectly perceived productivity di¤erences or by a biased objective function of the employer. In particular, the model reveals the possibility of a paradoxical situation in which less demanding workers are granted a higher wage rate, in sharp contrast to the case in which discrimination is possible.

Finally, in formalizing and interpreting the labour cost function, the model explains the demand for more exacting workers and predicts local discontinuity. For a threshold of needed working services, the cost minimizing policy switches from employing less demanding workers only, to the adoption of a mixed labour force, including a minimum number (quantum) of the more demanding type. This setting is compatible with some forms of type-speci…c mass redundancy in the case of an economic slowdown.

Our analysis can shed light on the topic of gender discrimination if consumption/leisure preferences are speci…c to the gender of the employee. Because women still have the responsibility of caring about their children in many countries, they might obtain more utility from out-of-job hours [START_REF] Cain | The economic analysis of labor market discrimination: A Survey[END_REF]OECD, 2018). Women can therefore be represented as the group of more demanding workers. Men would then represent the group of less demanding workers. If "less expensive" male labour is in short supply, the model shows possible gender wage di¤erentials in the absence of any biased information about productivity or a nonconventional objective of the employer involving contract discrimination. As such, the tools used to study hours/wage o¤ers are quite standard (contract theory); however, including the demanded hours constraint in the optimization problem and solving it explicitly with quadratic compensation functions can be seen as a contribution to this literature.

The paper is organized as follows: In section 2, we introduce our main assumptions and de…ne the mixed employment regime. In sections 3, 4 and 4 the model is analyzed in three increasingly detailed sets of assumptions. Section 3 analyses the cost minimization problem for the general case. In section 4, we provide a more precise de…nition of optimal contracts when the compensation di¤erential is a linearly increasing function of working hours. In section 5 more precision is achieved by adopting a quadratic compensation function, and numerical simulations are used to support the analytical results. The minimum number of type 2 employed workers is endogenous. Section 6 concludes.

Main assumptions

We analyze the cost minimizing labor contracts designed by an employer who needs a given amount of labour services H to achieve its production target. 3 Technology is such that H can be expressed by a sum of hours considered as perfect substitutes in production. Working hours are homogeneous not only in terms of productivity but also in terms of working conditions, as applying to the same task. Di¤erences related to the intrinsic disutility of work, as considered by the theory of compensating di¤erentials, are therefore ruled out.

a/ Preferences

The utility function of a worker i is represented by u i = c i v i (h); where c i is consumption obtained from trading hours in the labor market, and v i (h) stands for the disutility of working h hours. In line with standard neoclassical assumptions, we assume that v i (0) = 0; v 0 i (h) > 0 and To keep the analysis simple, we assume that there are only two types of workers, i = (1; 2); with compensation functions v 1 (h) and v 2 (h) respectively. The two compensation functions cross at the origin, for h = 0. With convex and increasing compensation functions, two cases can be considered:

v 00 i (h) > 0. The participation constraint of a worker is u i = c i v i (h) 0 , c i v i (h): Therefore, v i (h)
1. The two compensation functions cross for some h > 0: This situation can occur, but it leads to trivial conclusions: there is one consumption/hours contract that is unambiguously preferred by each type of worker. The …rst-best solution prevails irrespective of worker availability.

2. The two compensation functions do not cross for h > 0: This assumption is tantamount to having type 2 workers always demanding more compensation than type 1 workers for any amount of working time:

v 2 (h) > v 1 (h); 8h > 0: (1)
The two types of workers are consequently unequivocally ordered in this respect. This case leads to nontrivial situations in which a nondiscriminating employer must take into account worker incentive compatibility constraints. If women receive a higher utility from out-of-employment hours, women might be considered representative of more demanding type 2 workers, with men being representative of less demanding type 1 workers.

Let us denote the compensation premium of type 2 workers by

&(h) = v 2 (h) v 1 (h), with &(h) > 0.
Letting aside other compensation pro…les, we further assume that the premium &(h) is monotonically increasing with h & 0 (h) > 0; 8h or:

& 0 (h) = v 0 2 (h) v 0 1 (h) > 0; (2) b/ Cost of labor
The cost of labour includes worker compensation and a per individual, …xed, nonwage expense (cost of a work place unrelated to worked hours), denoted by > 0 [START_REF] Rosen | Short-run employment variation on class-I railroads in the US, 1947-1963[END_REF][START_REF] Hart | Working Time and Employment[END_REF]Contensou and Vranceanu, 2001). This …xed cost is assumed to be independent of the worker type.

We de…ne ĥi as the contract hours that minimizes the hourly cost of each type i regardless of any other constraint:

ĥ1 = arg min h v 1 (h) + h (3) ĥ2 = arg min h v 2 (h) + h (4)
We can refer to these hours as "hourly-cost minimizing" (HCM).

c/ Labor contracts

The employer can o¤er labor contracts de…ning hours of work and compensation (h; c): We make the basic assumption that the employer cannot discriminate, i.e., cannot prevent workers from choosing their preferred contract. This can be interpreted as a consequence of either legal or social constraints banning discrimination when types can be observed (the case of gender for instance), or just of imperfect information with the employer not being able to distinguish the types. When contract discrimination is impossible, the principal must o¤er two labor contracts P 1 = (h 1 ; c 1 ) and P 2 = (h 2 ; c 2 ) and let workers self-select for their preferred one. The employer must keep in mind that hiring type 2 workers with contract P 2 introduces a costly constraint on contract P 1 ; which must now be at least as attractive as P 2 for type 1 workers. This supplementary constraint is equivalent to a …xed cost that must be paid no matter how small is the number of type 2 workers hired.

d/ Availability of workers

Let n 1 be the number of type 1 workers and let n 2 be the number of type 2 workers hired by the …rm. A second important assumption is that there is a relative shortage of type 1, less demanding workers. Let n 1 be the number of type 1 workers available in the labour market: Type 2 workers are abundant, i.e., they are available in any number demanded by the …rm: e/ The employment regime If total demand for hours is relatively low, there is no shortage of type 1 workers (n 1 < n 1 ) and the employer would minimize the labor cost by employing only the less expensive type 1 workers.

In this case the optimal contract is P 1 = ( ĥ1 ; v 1 ( ĥ1 )); it can be easily veri…ed that the more demanding type 2 workers would not choose it. If all type 1 workers are employed (n 1 = n 1 ), the employer might still want to work with only this type of workers. The employer might ask theses workers to work longer hours, and compensate them more, in line with their convex compensation function. However, if the demand for hours is great enough, the high cost of the longer hours for a type 1 worker can justify the hiring of the more demanding type 2 workers. In the following, the analysis will focus on the most interesting case in which the demand for hours is great enough to justify the employment of both types of workers, which we refer to as a mixed-employment regime.

We denote this threshold volume of hours by H M :

3 The general problem

The goal of the …rm is to minimize the cost of a total number of hours H to be provided by a limited number of type 1 workers and a ‡exible number of type 2 workers. Contract discrimination is not allowed. We follow the standard resolution steps applied in contract theory analysis (e.g., La¤ont and Martimort, 2001); we …rst determine feasible allocations taking into consideration both participation and incentive compatibility constraints, and then analyze the …rm's optimization problem within the set of feasible allocations. As a benchmark, online Appendix 1 analyzes cost-minimization in the context of perfect discrimination (the employer can make type-speci…c take-it-or-leave it o¤ers to each type of worker).

Participation and incentive compatibility constraints

If both types are employed, taking the hour constraint into consideration, total cost minimization does not involve independently minimizing the cost of hours supplied by the two types since the existence of a contract P 2 = (c 2 ; h 2 ) modi…es the terms of the incentive compatibility constraint applied to type 1 workers.

The set of constraints includes:

1) Participation constraints (PC):

In the case where there are only two types of workers, i = (1; 2), the participation constraint of each type of worker can be written:

u 1 = c 1 v 1 (h 1 ) 0 (5) u 2 = c 2 v 2 (h 2 ) 0 (6)
2) Incentive compatibility (IC) constraints:

Type 1 workers are eligible to contract P 2 = (c 2 ; h 2 ); they prefer or accept the contract

P 1 = (c 1 ; h 1 ) only if: c 1 v 1 (h 1 ) c 2 v 1 (h 2 ) (7)
Similarly, type 2 workers prefer or accept contract P 2 to contract P 1 only if:

c 2 v 2 (h 2 ) c 1 v 2 (h 1 ) (8)
We can then identify a …rst property of the contracts.

Proposition 1 Since the compensation premium increases with working hours h, type 1 workers are o¤ ered at least as long hours than type 2 workers, h 1 h 2 :

Proof. Adding ( 7) and ( 8) :

v 2 (h 1 ) v 1 (h 1 ) v 2 (h 2 ) v 1 (h 2 ): (9) Because the compensation premium [v 2 (h) v 1 (h)] increases with h; the inequality (9) is not compatible with h 1 < h 2 :
This rule is a consequence of the incentive constraints, and it applies not only to optimum values of working hours (cost minimizing values), but also to all feasible values.

The general set of constraints allows us to infer a set of rules:

1) From the IC condition (7), c 1 v 1 (h 1 ) c 2 v 1 (h 2 ): The participation constraint for type

2 (6) requires c 2 v 2 (h 2 ): Thus c 1 v 1 (h 1 ) v 2 (h 2 ) v 1 (h 2 )
; and from our basic assumption (1)

v 2 (h 2 ) v 1 (h 2 ) > 0. Consequently: c 1 v 1 (h 1 ) v 2 (h 2 ) v 1 (h 2 ) > 0: (10)
This arrangement makes the participation constraint for the type 1 workers redundant and indicates the need for a positive surplus

[v 2 (h 2 ) v 1 (h 2 )]
for type 1 workers.

2) Type 2 workers who are supposed to be available in any quantity obtain no surplus in the solution; their participation constraint must be saturated:

c 2 = v 2 (h 2 ): (11)
Limiting c 2 to its minimum value of v 2 (h 2 ) decreases the cost of workers of type 2 directly, and the cost of workers of type 1 indirectly by mitigating the cost of the relevant incentive compatibility constraint.

3) If c 2 = v 2 (h 2 ); the incentive compatibility constraint for type 1 workers (condition 7) becomes :

c 1 v 1 (h 1 ) v 2 (h 2 ) v 1 (h 2 ): (12)
where v 2 (h 2 ) v 1 (h 2 ) is the surplus obtained by type 1 worker choosing contract P 2 : Therefore, in its saturated form, the type 1 incentive compatibility constraint implies a positive surplus for the type 1 worker:

c 1 = v 1 (h 1 ) + [v 2 (h 2 ) v 1 (h 2 )] : (13)
This situation contrasts with the perfect discrimination case (see online Appendix 1), in which type 1 workers obtain a zero surplus.

4) It must be noted that in our assumptions, the incentive compatibility constraint for type 2 workers is satis…ed. Type 2 workers prefer contract P 1 = (h 2 ; c 2 ) to contract

P 1 = (h 1 ; c 1 ) if: c 2 v 2 (h 2 ) = 0 c 1 v 2 (h 1 ):
Replacing c 1 and c 2 by their expressions in ( 11) and ( 13), IC2

becomes

v 2 (h 1 ) v 1 (h 1 ) v 2 (h 2 ) v 1 (h 2 )
; which is true given the rule stated by Proposition 1.

This allows us to omit the explicit treatment of condition (8) in the constrained cost minimization problem.

Cost minimization: …rst order conditions

In the case where the demand for hours is large enough to justify the hiring of the both types of workers (H > H M ) the cost function can be written as:

C(n 1 ; n 2 ; h 1 ; h 2 ; H) = n 1 (c 1 + ) + n 2 (c 2 + ): (14)
The limited supply of type 1 workers imposes n 1 n 1 : We show in online Appendix 2 that when the two types of workers are employed, this constraint must be saturated. After applying the substitutions introduced in equations ( 11) and ( 13), the cost function becomes:

C( n 1 ; n 2 ; h 1 ; h 2 ; H) = n 1 [v 1 (h 1 ) + v 2 (h 2 ) v 1 (h 2 ) + ] + n 2 [v 2 (h 2 ) + ] : (15)
The total hours constraint is:

n 1 h 1 + n 2 h 2 H = 0: (16)
The …rm seeks to minimize the cost C( n 1 ; n 2 ; h 1 ; h 2 ; H) under the hour constraint, by choosing h 1 ; h 2 and n 2 . To solve this problem, we introduce the Lagrangian:

L = n 1 [v 1 (h 1 ) + v 2 (h 2 ) v 1 (h 2 ) + ] + n 2 [v 2 (h 2 ) + ] [ n 1 h 1 + n 2 h 2 H] : (17)
The …rst-order conditions applied to interior solutions are as follows:

@L @n 2 = v 2 (h 2 ) + h 2 = 0 (18) @L @h 1 = n 1 v 0 1 (h 1 ) n 1 = 0 (19) @L @h 2 = n 1 [v 0 2 (h 2 ) v 0 1 (h 2 )] + n 2 v 0 2 (h 2 ) n 2 = 0 (20) @L @ = n 1 h 1 + n 2 h 2 H = 0: (21)
where ( ; n 2 ; h 1 ; h 2 ) denotes the solution of the cost minimization problem.

Properties of the solution

The compensations involved in the two contracts are

c 1 = v 1 (h 1 )+v 2 (h 2 ) v 1 (h 2 ) and c 2 = v 2 (h 2 ):
From our assumptions, type 2 workers receive a positive surplus v 2 (h 2 ) v 1 (h 2 ) > 0:

We can further remark that condition (19) implies: v 0 1 (h 1 ) = : Thus conditions ( 18) and ( 19) imply:

v 2 (h 2 ) + h 2 = v 0 1 (h 1 ): (22)
Equation ( 22) indicates the equality of the marginal cost of hours obtained from the two possible sources: increasing the number of type 2 workers with constant working time (extensive margin)

or increasing the working time of type 1 workers, their number being constant (intensive margin).

a/ Ordering working times

We know from Proposition 1 that h 1 h 2 : We can show that h 1 > h 2 and h 2 < ĥ2 :

First, conditions ( 19) and (20) imply:

n 1 [v 0 2 (h 2 ) v 0 1 (h 2 )] + n 2 [v 0 2 (h 2 ) v 0 1 (h 1 )] = 0: (23) From condition (2), v 0 2 (h 2 ) > v 0 1 (h 2 ); thus condition (23) implies v 0 1 (h 1 ) > v 0 2 (h 2 ): Joined, the two conditions imply v 0 1 (h 1 ) > v 0 1 (h 2 ): Since v 0 1 (h) is assumed strictly increasing, h 1 > h 2 : Conse- quently, v 2 (h 2 ) v 1 (h 2 ) > 0 and c 1 > c 2 :
Second, we prove that h 2 < ĥ2 . From ( 22) and v 0

1 (h 1 ) > v 0 2 (h 2 ); we obtain [v 2 (h 2 ) + ] =h 2 > v 0 2 (h 2 ) or v 2 (h 2 ) h 2 v 0 2 (h 2 ) > ; whereas v 2 ( ĥ2 ) ĥv 0 2 ( ĥ2 ) = : Since the convex function '(h) v 2 (h) hv 0 2 (h) is monotonously decreasing, h 2 < ĥ:
As a consequence of these shorter hours (h 2 < ĥ2 ), the principal agrees to pay an extra cost per hour for the type 2 workers (compared to the …rst best choice), to achieve a reduction in the surplus he/she must grant to type 1 workers. 5Intuitively, when the number of target hours H increases, the importance of the type 2 workers also increases, and the principal has lower incentives to accept a higher hourly cost per type 2 worker. The optimal hours h 2 should increase, and tend toward ĥ2 as H ! 1:

b/ Ordering wage rates

To each consumption-hours contract is associated an implicit hourly wage. Since no surplus is needed for the participation of type 2, c 2 = v 2 (h 2 ) and the wage rate can be written

w 2 = v 2 (h 2 )=h 2 : From (22), [v 2 (h 2 ) + ] =h 2 = v 0 1 (h 1 ) , v 2 (h 2 )=h 2 = v 0 1 (h 1 ) =h 2 : Thus w 2 = v 0 1 (h 1 )
=h 2 : The wage rate of the type 1 is w 1 = c 1 =h 1 ; where according to his IC constraint (13),

c 1 = v 1 (h 1 ) + &(h 2 ); where &(h 2 ) = v 2 (h 2 ) v 1 (h 2 ) is the compensation premium. The wage rate is w 1 = [v 1 (h 1 ) + &(h 2 )
] =h 1 : Therefore, using the elasticity of compensation with respect to working time (h 1 ) =

h 1 v 0 1 (h 1 )
v1(h 1 ) > 1, we can write:

w 1 > w 2 if h 1 h 2 + &(h 2 ) > v 1 (h 1 ) [ (h 1 ) 1] ; (24)
where h 1 and h 2 are endogenous variables. It turns out that the …rst order conditions (18) to ( 21) are a priori compatible with any wage rate ordering. We will be able to provide additional insights to this important question in Section 5, for a speci…c compensation structure.

c/ Cost function

The (optimal) cost function, to be obtained by introducing the optimal values n 2 (H); h 1 (H) and h 2 (H) in equation ( 15) can be written in a compact form as:

C (H) = C(n 2 (H); h 1 (H); h 2 (H)) (25)
It can be con…rmed, after relevant substitutions, that Lagrangian multiplier re ‡ects the marginal cost of hours, as obtained at the optimum from either type 1 or type 2 workers.

dC (H) dH = = v 0 1 (h 1 ) = v 2 (h 2 ) + h 2 : (26)
4 The case of a linearly increasing compensation premium

Additional insights can be obtained for a simple preference structure that ful…lls the condition

(2):

v 2 (h) = v 1 (h) + h; (27)
where is a positive constant. Under this assumption, not only is the type 2 worker more demanding than the type 1 (see condition 1), but the compensation premium

[v 2 (h) v 1 (h)] is
linearly increasing with h.

We have denoted by ĥ1 and ĥ2 the hours of work that minimize the hourly cost in employing each type of worker: The …rst order conditions for minimizing the hourly cost are:

v 0 1 (h) = v 1 (h) + h and v 0 2 (h) = v 2 (h) + h :
In the linear case, the …rst order condition for the type 2 become:

v 0 1 (h) + = v 1 (h) + h + h , v 0 1 (h) =
v 1 (h) + h : Therefore, for this speci…c function, HCM hours are identical for the two types: ĥ1 = ĥ2 ĥ: (28)

Working time ordering

Proposition 1 shows that if the compensation di¤erences [v 2 (h) v 1 (h)] increase with working hours h, then type 1 workers are o¤ered more contract hours than type 2 workers, h 1 h 2 : We can verify that this general property of the model applies to optimal hours. From assumption

(27): v 0 2 (h 2 ) = v 0 1 (h 2 ) + .
Then condition (23) becomes:

n 1 + n 2 [v 0 1 (h 2 ) v 0 1 (h 1 ) + ] = 0 (29)
or equivalently:

v 0 1 (h 1 ) v 0 1 (h 2 ) = ( n 1 + n 2 ) n 2 (30) Since > 0; v 0 1 (h 1 ) > v 0 1 (h 2 ): From strict convexity of v 1 (h); we have v 0 1 (h 1 ) > v 0 1 (h 2 ) ) h 1 > h 2 :
We have shown that h 2 < ĥ in the general case. In the case of the linear increasing di¤erence in compensation we can further show that the optimal hours of type 1 workers exceed their hourly cost-minimizing hours, ĥ < h 1 : We obtain that:

Proposition 2 Optimal contract hours of the type 2 workers are shorter than their hourly-cost minimizing hours, and the optimal contract hours of the type 1 are higher than their hourly-cost minimizing hours. h 2 < ĥ < h 1 :

Proof. See online Appendix 3.

The consequence of increasing demand for hours

The macroeconomic business cycle leads to ‡uctuations in the number of hours required by …rms.

Strong economic expansion is tantamount to a substantial increase in H: From necessary …rst order conditions applied to (17), it is possible to predict the e¤ect of inde…nitely increasing target hours H on optimally contracted working hours.

Proposition 3 When the total demand for hours H inde…nitely increases, for constant n 1 , hours h 2 speci…ed in the optimal contract P 2 tend to the hourly-cost minimizing value ĥ.

Proof. From equation ( 16), we …nd that when the demand for hours increases (H ! 1), since n 1 n 1 is …xed and since h 1 and h 2 cannot increase inde…nitely, necessarily n 2 ! 1: From equation (30

) v 0 1 (h 1 ) v 0 1 (h 2 ) = ( n 1 + n 2 ) n 2 : If n 2 ! 1; then [v 0 1 (h 1 ) v 0 1 (h 2 )] ! : From equation (22), v 2 (h 2 ) + h 2 = v 0 1 (h 1 ) and since by assumption, v 2 (h 2 ) = v 1 (h 2 ) + h 2 , v 2 (h 2 ) + h 2 = v 1 (h 2 ) + h 2 + h 2 = v 1 (h 2 ) + h 2 + = v 0 1 (h 1 ): Since v 0 1 (h 1 ) ! v 0 1 (h 2 ) + ; equation (22) implies: v 1 (h 2 ) + h 2 ! v 0 1 (h 2 ).
The limit of this equation is only compatible (uniqueness) with the hourly cost minimizing working time for type 1 workers (and for type 2 workers in our special assumption).

Therefore: H ! 1 ) n 2 ! 1 ) h 2 ! ĥ2 = ĥ:

As is intuitively predictable, when the proportion of type 2 workers inde…nitely increases, the external e¤ect on their contract by type 1 worker preferences dwindles, and type 2 workers' working time tends towards its HCM value.

Proposition 4 When the total demand for hours H inde…nitely increases, hours h 1 in optimal contract P 1 always exceed the hourly-cost minimizing hours ĥ.

Proof. With respect to the evolution of contract P 1 we note that from equation (30

) [v 0 1 (h 1 ) v 0 1 (h 2 )] !
and simultaneously, h 2 ! ĥ = ĥ1 implying: v 0 1 (h 1 ) ! v 0 1 ( ĥ) + and therefore h 1 > ĥ:

The working time for type 1 is kept above its HCM value under the in ‡uence of , the parameter that captures the higher sensitivity of type 2 workers with respect to hours (the compensation premium).

The case of quadratic compensation functions

The speci…c compensations functions used above allowed us to better characterize the optimal contract hours. Introducing an even more precise structure for the compensation functions enables us to fully characterize the solution and numerically solve the model and reveal a possible paradoxical ordering of hourly wage rates. We also resort to a numerical simulation to provide additional intuition about the properties of the solution.

Quadratic function v(h) = h 2 has the required property of strict convexity. Our compensating consumption functions then become:

v 1 (h) = h 2 and v 2 (h) = h 2 + h: (32)
We have shown that for a linear increasing compensation premium, the hourly cost-minimizing hours ĥ1 and ĥ2 are identical: ĥ1 = ĥ2 = ĥ. With a quadratic compensation function, ĥ = p :

The solution to the cost minimization problem

With this assumption, the equations ( 23) and ( 22) become:

(h 2 ) 2 + h 2 + = 2h 1 h 2 (33) ( n 1 + n 2 ) + 2n 2 (h 2 h 1 ) = 0 (34)
To which we add the hours constraint:

n 1 h 1 + n 2 h 2 = H (35)
Equations ( 33), ( 34) and ( 35) form a nonlinear system with three endogenous variables.

The linearity of the subsystem (35 and 34) for a given value of n 2 enables the expression of h 1 and h 2 as functions of the endogenous employment of type 2 workers n 2 , yielding a feature of the in comparison to the average working time h = H= ( n 1 + n 2 ) :

h 1 = h + 2 (36) h 2 = h 2 n 1 n 2 (37)
We emphasize a …rst property of the solution. Equation ( 33) has distinct real roots only if its determinant ( 2h 1 ) 2 4 > 0; implying 2h 1 > 2 p or 2h 1 < 2 p : Since h > 0; the …rst case is not compatible with (36). The second case is consistent with (36) for h > p . Since hourly-cost minimizing hours for both types are de…ned by ĥ = p , …rst order conditions imply h > ĥ:

Introducing optimal hours as de…ned in equations ( 36) and (37) in equation ( 33), it turns out that:

Proposition 5 The optimal number n 2 of P 2 contracts o¤ ered by the employer (n 2 ) is the superior root of:

H 2 ( n 1 + n 2 ) 2 = + 2 n 1 n 2 2 : (38)
Proof. See online Appendix 4 for the full proof. In brief, we show that there is a value H = H o for which equation ( 38) has a single root,

n o 2 = n 1 3 r 2 4
: We also show that for H < H o , the equation ( 38) has no real root, and for H > H o it has two positive roots, n l 2 and n h 2 ; with n l 2 < n o 2 < n h 2 : Furthermore, online Appendix 5 reveals the second order necessary condition for cost minimization:

n 2 n 1 3 2 4 , n 2 n o 2 (39)
We conclude that only the superior root n h 2 satis…es the second order necessary conditions for cost minimization.

Corollary 6 For n 2 < n o 2 ; the cost minimization problem has no solution. Any mixed employment regime involves a minimum number of type 2 workers determined by parameters in inequality (39), as associated to a minimum amount of hours H o :

The threshold hours H o is obtained from equation (38) evaluated for n o 2 :

Numerical simulations and graphic representations of the relevant functions in Appendix 4 con…rm that the superior root corresponds to the minimum of the cost function (i.e., our solution n 2 ).

Figure 1: Contract hours as a of H

For n 2 as de…ned by equation ( 38), equations ( 36) and ( 37) allow us to determine h 1 and h 2 , thus fully characterizing the solution to the cost-minimization problem.

Comparative statics

It is interesting to study how h 1 ; h 2 and n 2 evolve when the target hours H increases. We can show that:

Proposition 7 For interior solutions, n 2 i.e., the optimal number of P 2 contracts (demand for type 2 workers), is increasing with H:

Proof. See online Appendix 6.

This comparative static rule could appear intuitively trivial; however, it is not, since for an increased hours constraint, the employer also controls the two working times h 1 and h 2 :

A numerical simulation allows us to study how the optimal working hours vary when H increases. The parameters used are n 1 = 10; = 0:10 and = 0:20: We apply H 7, as we numerically verify (in 5.4) that for H < 7 the employer prefers to hire only the type 1 workers.

Figure 1 displays h 2 as the red, lower curve and h 1 as the blue, upper curve: The horizontal line corresponds to the hourly cost minimizing hours of type 2 workers, ĥ = p = 0:447: In line with Proposition 3, when H ! 1; h 2 ! ĥ: In line with Proposition 4, when H ! 1;

v 0 1 (h 1 ) ! v 0 1 ( ĥ) + or h 1 ! ĥ + 0:5 = 0:497:

Wage rate inequality

We analyze conditions to obtain w 1 > w 2 ; i.e., a paradoxical situation in which the less demanding type of worker obtains a higher wage rate than the more demanding type of worker.

The wage rate being

w 1 = v 1 (h 1 ) + v 2 (h 2 ) v 1 (h 2 ) h 1 and w 2 v 2 (h 2 ) h 2
; the inequality can be written as:

v 1 (h 1 ) + v 2 (h 2 ) v 1 (h 2 ) h 1 > v 2 (h 2 ) h 2 () v 1 (h 1 ) v 1 (h 2 ) h 1 h 2 > v 2 (h 2 ) h 2 : (40)
In the quadratic case v 1 (h 1 ) = h 2 1 and v 2 (h 2 ) = h 2 2 + h 2 so:

w 1 > w 2 , h 2 1 h 2 2 h 1 h 2 > h 2 2 + h 2 h 2 , h 1 + h 2 > h 2 + , h 1 > : (41)
From ( 36),

h 1 = h 1 = h + 2 ; therefore w 1 > w 2 , h > 2 ; (42)
where h H n 1 + n 2 is a dependent variable.

We can reveal two su¢ cient conditions for this speci…c wage ordering to be observed.

a/ From equation ( 38)

h 2 = + n 1 2n 2 2 implying h 2 > ) h > p : It turns out that p > 2
is a su¢ cient condition in terms of parameters for w 1 > w 2 : Under our assumptions, chances to observe higher wages for the less demanding workers would be higher in sectors that involve a large …xed cost per worker.

b/ Remembering the second order condition (39), we have

n 2 n 1 3 > 2 4 or p > 2 n 1 n 2 3=2 :
Once again, we recall that equation (38

) implies h > p and p > 2 n 1 n 2 3=2 : Thus, another su¢ cient condition in terms of dependent variables is 2 n 1 n 2 3=2 > 2 , n 2 < n 1 :
The wage rate ordering w 1 > w 2 should be observed at least when n 2 < n 1 : The simulation con…rms that the wage rate of the less demanding workers (who also work longer hours) is higher than the wage rate the more demanding workers. The wage gap favorable This result is contrasting with the perfect discrimination situation as depicted in online Appendix 1. In that case, with quadratic compensation functions, the optimal contracts involve longer working hours and a lower wage rate for the less demanding workers compared to the more demanding workers. Furthermore, these wage rates are independent of the total number of hours H; while they increase with H in the nondiscriminating case.

As mentioned in the introduction, several scholars [START_REF] Goldin | A grand gender convergence: Its last chapter[END_REF][START_REF] Cortés | When time binds: Substitutes for household production, returns to working long hours, and the skilled gender wage gap[END_REF] provide evidence that in some high-skilled occupations long working hours are associated with higher hourly wages. The authors attribute this di¤erence to the higher productivity of employees working longer hours, associated to better serving of clients who require "temporal ‡exibility" and a better within-…rm circulation of information. In our model, the use of a speci…c compensation structure can generate a higher wage rate to the less demanding type 1 workers (who work longer hours)

while not resorting to any productivity argument.

Cost analysis

Finally, we would like to ensure that our simulations adequately match the case in which the employer prefers to use both type of workers rather than to use only the less demanding type 1.

precisely, we must verify that the hours H used in the simulation are great enough to justify the employment of both types of workers.

In the quadratic compensation case, v 1 (h) = h 2 ; the hourly cost minimizing working hours of these least demanding workers is ĥ1 = p : As long as H < H 1 = n 1 p (in our simulation, H 1 = 10 p 0:20 4:5); the …rm should o¤er only the contract (v 1 ( ĥ1 ); ĥ1 ) and employ only type 1 workers. For H > H 1 ; the cost of labor when employing only type 1 workers becomes convex,

C 1 (H) = n 1 H n1 2 + . 6
For a large H the …rm might want to hire both type of workers, provided that the cost of doing so is lower than using only type 1.

The cost of labor when using both type of workers, denoted by C (H); has been de…ned in (25). With quadratic compensation functions, the cost function becomes:

C (H) = ( n 1 + n 2 ) + n 1 h (h 1 ) 2 + h 2 i + n 2 h (h 2 ) 2 + h 2 i ( 43
)
where n 2 ; h 1 and h 2 are optimal values as resulting from the cost minimization problem (they all depend on H):

To determine the threshold total hours for which the mixed employment regime dominates the type 1 only employment regime, we compare the cost of using only type 1 workers, C 1 (H) with the cost of labor under the mixed regime C (H):

Our numerical simulations show that for H H M = 7 the mixed employment regime is indeed less expensive than employing only type 1 workers, as it can be seen in Figure 3, where C 1 (H) is displayed as the upper black, curve and C (H) as the lower, green curve. Below H = 7; the cost of hiring only type 1 workers is lower than the cost of the mixed employment.

We argued in corollary 6 that the cost minimization problem has a interior solution (for the mixed employment regime) only if H H o . We can verify that H o = 6:12 thus the solution exists for H 7:

The comparison of costs functions across employment regimes reveals the emergence of a discontinuity in the demand for type 2 workers. Indeed, for H < 7; n 2 = 0; for H = 7; n 2 = 5:32;

and for H > 7; n 2 > 5:32: This feature of the model can be extremely problematic in periods of

Conclusion

In being confronted with a wealth of data exhibiting frequent discrepancies in labour contracts, especially including compensation di¤erences for apparently equivalent workers, many researchers have provided relevant explanations involving discriminating policies. This paper does not contradict this literature, but suggests the existence of an alternative explanation based on standard contract theory.

In the proposed model, a …rm seeks a predetermined volume of homogenous working hours, to be provided by workers who value leisure time to greater and lesser degree. Our analysis focuses on a nontrivial case in which less demanding workers are in scarce supply. Discrimination is forbidden, i.e., the employer cannot o¤er a speci…c contract to each type of worker. In this context, the contract o¤ered to the more demanding workers performs as a …xed cost of hiring the less demanding workers, a mechanism that ensures e¢ cient self-selection of the workers. As a consequence, the scarce less demanding workers can actually be o¤ered higher wage rates than their more demanding peers.

These results reveal wage di¤erentials originating "from nothing", rooted in an invisible scarcity impossible to trace by econometrics from individual data. If one agrees that women have better work alternatives outside the labor market [START_REF] Cain | The economic analysis of labor market discrimination: A Survey[END_REF]OECD, 2018), and demand higher compensation than men for any given working time, in our model the (hourly) wage gender gap in favour of male workers could be grounded in these di¤erences in preferences and the scarcity of male workers, and not in di¤erences in productivity, in negative perceptions or in women penalizing stereotypes. This interpretation of course does not address the important question of why the burden of child care falls in a signi…cant way on women.

We have also shown how contract hours depend on the structure of preferences; for a time increasing compensation premium, the less demanding workers will do longer hours than the more demanding workers.

Finally, the model allowed us to analyze how working time, hourly wages and employment respond in a somewhat unconventional way to changes in the total demand for hours. In particular, we show that the demand for the most demanding, abundant workers is exposed to discontinuities.

From a macroeconomic perspective, the discontinuity in the demand for the most demanding workers can explain why small ‡uctuations in global demand are sometimes associated with large ‡uctuations in some types of employment.

can determine the optimal wage rates:

w a 1 = v 1 (h a 1 ) h a 1 = (h a 1 ) 2 h a 1 = h 1 = p + 2 : (54) w a 2 = v 2 ( ĥ2 ) ĥ2 = ( ĥ2 ) 2 + ĥ2 ĥ2 = p + : (55)
We verify that: w a 2 > w a 1 ; the wage rate of the more demanding type is higher than the wage rate of the less demanding type.

Binding limitation of type 1 workers

In the general case (as analyzed in the main text), considering the problem:

min n1;n2;h1;h2 fn 1 [v 1 (h 1 ) + v 2 (h 2 ) v 1 (h 2 ) + ] + n 2 [v 2 (h 2 ) + ]g (56)
with [n 1 h 1 + n 2 h 2 H] and n 1 n 1 : The corresponding Lagrangian is:

L(n 1 ; n 2 ; h 1 ; h 2 ; ) = n 1 [v 1 (h 1 ) + v 2 (h 2 ) v 1 (h 2 ) + ] + n 2 [v 2 (h 2 ) + ] [n 1 h 1 + n 2 h 2 H] : (57)
If 0 < n 1 < n 1 , …rst order necessary conditions imply: @L @n 1 = [v 1 (h 1 ) + v 2 (h 2 ) v 1 (h 2 ) + ] h 1 = 0 (58) @L @n 2 = v 2 (h 2 ) + h 2 = 0 (59)

@L @h 1 = n 1 v 0 1 (h 1 ) n 1 = 0 (60) @L @h 2 = n 1 [v 0 2 (h 2 ) v 0 1 (h 2 )] + n 2 v 0 2 (h 2 ) n 2 (61)
The …rst three FOCs allow us to write:

v 1 (h 1 ) + v 2 (h 2 ) v 1 (h 2 ) + h 1 = v 2 (h 2 ) + h 2 = = v 0 1 (h 1 ): (62)
This equality indicates that if the availability constraint is not binding, the cost of hours obtained from each type should coincide.

But (62) implies:

v 1 (h 1 ) v 1 (h 2 ) h 1 h 2 = = v 0 1 (h 1 ): (63)
(we substitute v 2 (h 2 ) + = h 2 v 0 1 (h 1 ) in the …rst term).

Convexity of v

1 (h) implies v 1 (h 2 ) v 1 (h 1 ) > (h 2 h 1 )v 0 1 (h 1 ) ) v 1 (h 1 ) v 1 (h 2 ) < (h 1 h 2 )v 0 1 (h 1 )
or v 1 (h 1 ) v 1 (h 2 ) h 1 h 2 < v 0 1 (h 1 ); contradicting (63).

can be shown that the function '(h) v(h) hv 0 (h) is monotonously decreasing, thus equation (69) implies: h 2 < ĥ:):

1.4 The optimal number of type 2 workers

The computed values of working times for a given (endogenous) number of type 2 workers are:

h 1 = h + 2 (70)
h 2 = h 2 n 1 n 2 (71)
where h = H ( n 1 + n 2 )

. Introducing these expressions in the main text equation h 2 2 + h 2 + = 2h 2 h 1 we obtain:

H 2 ( n 1 + n 2 ) 2 | {z } L(n2) = + n 1 2n 2 2 | {z } R(n2) (72)
The demand for type 2 workers, n 2 ; is the implicit solution to the former equation. The model has no solution if H is too low since the two curves do not cross. To determine this critical threshold, denoted by H o ; we acknowledge, from (graphic) analysis of L(n 2 ) and R(n 2 ), 5: Cost as a function of n 2 ; for optimal hours and wage rates

Second order conditions for cost minimization

We use the hours constraint to reduce the cost-minimization problem to a form of free variables minimization. Since for n 1 = n 1 , n 2 is explicitly determined by the hours constraint and the choice of h 1 and h 2 ; and the objective function to be minimized is:

m(h 1 ; h 2 ) = n 1 [v 1 (h 1 ) + v 2 (h 2 ) v 1 (h 2 ) + ] + H n 1 h 1 h 2 [v 2 (h 2 ) + ] : (78)
First order derivatives are:

m 1 (h 1 ; h 2 ) = n 1 v 0 1 (h 1 ) n 1 h 2 [v 2 (h 2 ) + ] (79)
m 2 (h 1 ; h 2 ) = n 1 [v 0 2 (h 2 ) v 0 1 (h 2 )] H n 1 h 1 h 2 2 [v 2 (h 2 ) + ] + H n 1 h 1 h 2 v 0 2 (h 2 ): (80)
After substitutions, since v 1 (h) = h 2 , v 2 (h) = h 2 + h and H n 1 h 1 h 2 = n 2 ; …rst order conditions are:

m 1 (h 1 ; h 2 ) = 2 n 1 h 1 n 1 h 2 n 1 n 1 h 2 = 0 (81) m 2 (h 1 ; h 2 ) = n 1 + n 2 1 h 2 = 0: (82)
where H n 1 h 1 h 2 = n 2 : From ( 81) and ( 82), the Hessian matrix is:

0 B B @ m 11 m 12 m 21 m 22 1 C C A = 0 B B @ 2 n 1 n 1 h 2 2 1 n 1 h 2 2 1 2n 2 h 2 2 1 C C A : (83)

  can be interpreted as the compensation function, indicating the minimum compensation required by an individual i to supply h hours of work. 4

Figure 2

 2 Figure 2 shows the evolution in hourly wages as the demand for hours H increases (n 1 = 10; = 0:10 and = 0:20; the condition p > 0:5 holds). The upper, blue curve indicates the wage rate of type 1 workers, and the lower, red curve indicates the wage rate of the type 2 workers.

Figure 2 :

 2 Figure 2: Wage rates by worker depending on H

Figure 3 :

 3 Figure 3: Cost functions

Figure 4

 4 Figure 4 represents the functions L(n 2 ) and R(n 2 ) for H = 7, which is the downward bound for existence of the mixed employment regime. The smaller H; the lower the curve L(n 2 ) is. The other parameters are similar to those used in the main text ( = 0:10; = 0:20; n 1 = 10). As we can see, in this case, equation (72) has two positive roots, n l 2 = 1:13 and n h 2 = 5:32:

As a recent illustration,[START_REF] Cook | The gender earnings gap in the gig economy: Evidence from over a million rideshare drivers[END_REF] study the wages earned by Uber drivers, and …nd a 7% gap favoring men. Since longer working schedules apparently have in this case little impact on the wage rate, it is suggested that male drivers earn a compensating di¤erential for their willingness to drive in areas with higher crime rates and more drinking establishments. It must be noticed that in such a case, male and female labour services are not perfect substitutes.

[START_REF] Cortés | When time binds: Substitutes for household production, returns to working long hours, and the skilled gender wage gap[END_REF] o¤er empirical evidence documenting a positive relationship between the demand for long hours and the wage gap for high-skilled employees, while[START_REF] Denning | The return to hours worked within and across occupations: Implications for the gender wage gap[END_REF] …nd no such relationship.[START_REF] Mas | Valuing alternative work arrangements[END_REF] …nd from experimental data that women are relatively averse to employer imposed hour ‡exibility.

The cost minimization assumption is quite general, as any …rm, be it a pro…t maximizer or not, should address it as a …rst-stage decision.

For a general form of the utility function, the compensation function would be de…ned by solving U (c; h) = 0 for an explicit c = v(h): If U (c; h) has standard neoclassical properties, then v(h) is increasing in h and is convex.

This result is in line with the damaged goods problem introduced by[START_REF] Deneckere | Damaged goods[END_REF].

At some point, it might even exceed the cost of employing only type 2 workers.

APPENDIX 1.1 The case of perfect discrimination

We study a cost-minimization problem similar to the problem addressed in the main text. There are two types of workers characterized by non-crossing compensation functions, with v 2 (h) > v 1 (h); and an increasing compensation premium v 2 (h) v 1 (h). Type 1 workers are in scarce supply, let n 1 be their maximum number. The demand of hours H is large enough to justify the employment of the type 2: However, in this case, the employer, who knows the worker's type, can make takeit-or-leave-it job o¤ ers to each of them. The type-speci…c contract includes hours of work and the compensation. In the perfect discrimination condition the …rm can o¤er zero-surplus contracts to both types (c 1 = v 1 (h 1 ) and

Let us denote the solution to this di¤erent cost minimization problem by h a 1 ; h a 2 and n a 2 :

The total cost is:

and the hour constraint:

The relevant Lagrangian for the cost minimization under the hour constraint is:

First order conditions are:

From the third condition we have a = v2(h a 2 )+ h a

2

: Combined with the second one, this involves:

This condition is identical to FOC hourly cost minimization for type 2. The …rst-best optimal hours of the type 2 correspond to the hourly-cost minimizing hours, h a 2 = ĥ2 ; a result which is in line with intuitive reasoning. Since these workers are available in any amount, and the contract o¤ered to type 1 is independent of the contract o¤ered to type 2, the …rm o¤ers to them the contract that minimizes their hourly cost.

From equations ( 47) and ( 48) we obtain:

At the optimum, the marginal cost from increasing the hours provided by the type 1 (intensive margin) must be identical to the marginal cost of hiring more of the type two workers (extensive margin) (at the lowest hourly cost for the later).

The …rst-best contract hour for the type 1, denoted by h a 1 are larger than the cost minimizing hours ĥ1 : Indeed, for any convex functions v 2 (h); v 1 (h) and v 2 (h) > v 1 (h); a simple graphical analysis can show that:

Finally, the hours constraint n a 2 h a 2 + n 1 h a 1 = H allows us to determine n a 2 as a function of H:

An example

To bring more intuition to this analysis of perfect discrimination, we study the speci…c com-

The associated hourly cost minimizing hours are ĥ1 = ĥ2 = p :

In this case, expression (51) becomes

2 + leading to …rst-best contract hours for the type 2 workers h a 2 = ĥ2 = p :

To determine contract hours of the type 1 workers, we set

1.3 Proof of the inequality h < ĥ < h 1

We have de…ned by ĥi the working time that minimizes the hourly cost of each type i regardless of any other constraint, ĥ1 = arg min v 1 (h) + h and ĥ2 = arg min v 2 (h) + h : The …rst order condition for minimizing the hourly cost are:

to hourly-cost minimizing ĥ1 and ĥ2 : In the linear case v 2 = v 1 (h) + h; the …rst order conditions become:

Therefore, in the case of the linear increasing di¤erences, the hourly-cost minimizing hours are identical for the two types: ĥ1 = ĥ2 ĥ:

Let us recall here the main text equation ( 22):

and main text equation ( 30):

a/ To show that h 1 > ĥ we start from the de…nition of ĥ2 and use equality ĥ1 = ĥ2 = ĥ: Uniqueness of ĥ2 implies:

Given that (equation 65)

ĥ) and (from strict convexity), h 1 > ĥ.

b/ It also can be shown that h 2 < ĥ. From equation (66

v 1 ( ĥ) + ĥ ; it implies:

and v 1 ( ĥ) ĥv 0 1 ( ĥ) = : (69) that the positive real root of is unique if L(n 2 ) = R(n 2 ) and L 0 (n 2 ) = R 0 (n 2 ): This unique root is obtained for H o :For H > H o the system has two roots (the superior being our solution), for H < H o there is no solution.

Derivatives with respect n 2 are:

leading to:

Substituting the …rst term in equation ( 72), we obtain:

The possible unique positive root, denoted by n 0 2 ; is de…ned by ( 76)

We conclude that the mixed employment regime always requires a quantum of type 2 workers. H 0 is hour demand for which n 2 = n 0 2 :

For H > H 0 ; equation (72) has two real positive roots, n l 2 and n h 2 : However, the lowest root is

. This is contradicting the second order condition for cost minimization (85) as shown in Appendix 1.5.

As a last check, we acknowledge that the cost of producing a predetermined amount of hours can be written as a function of n 2 only, all other variables being set at their optimal level (depending on n 2 ):

where h 1 = h 1 (n 2 ) and h 2 = h 2 (n 2 ) are provided in equations ( 36) and (37).

From ( 81)

and from ( 81) and ( 82)

The solution to the system of …rst order conditions corresponds to a minimum of the cost function if the solution veri…es:

This condition amounts to a minimum relative participation of type 2, n o 2 = n 1 1.6 The relationship between the optimal number of type 2 workers and H

We prove in this Appendix that dn 2 dH > 0:

From …rst order conditions, we obtain equation (38) in the main text:

In the neighborhood of an interior solution, n 2 is a continuous and di¤erentiable function of H; noted g(H) and the equation can be written:

After derivating both members with respect to H :

The condition for dn 2 dH = g 0 (H) > 0 is: : From second order necessary condition (85), this condition is necessarily ful…lled.

In Figure 6 we show how n 2 varies with H, for n 1 = 10; = 0:10 and = 0:20: We apply H 7, as we numerically verify that for H < 7 the employer prefers the type 1 only employment regime.