Benjamin Bradu

Philippe Gayet

Silviu-Iulian Niculescu

Emmanuel Witrant

Modeling of the very low pressure helium flow in the LHC Cryogenic Distribution Line after a quench

Keywords: Helium (B), Heat transfer (C), Fluid Dynamics (C), Numerical Simulation (F), Transmission lines (F)

This paper presents a dynamic model of the helium flow in the Cryogenic Distribution Line (QRL) used in the Large Hadron Collider (LHC) at CERN. The study is focused on the return pumping line, which transports gaseous helium at low pressure and temperature (1.6 kPa / 3 K) over 3.3 km. Our aim is to propose a new real-time model of the QRL while taking into account the non-homogeneous transport phenomena. The flow model is based on 1D Euler equations and considers convection heat transfers, hydrostatic pressure and friction pressure drops. These equations are discretized using a finite difference method based on an upwind scheme. A specific model for the interconnection cells is also proposed. The corresponding simulation results are compared with experimental measurements of a heat wave along the line that results from a quench of a superconducting magnet. Different hypotheses are presented and the influence of specific parameters is discussed.

Introduction

In 2008, the European Organization for Nuclear Research (CERN) started the most powerful particle accelerator of the world, the Large Hadron Collider (LHC). The LHC accelerates proton beams which are driven by superconducting magnets maintained at 1.9 K over a 27 km ring. To cool-down and maintain superconductivity in different magnets, large helium refrigeration plants are used [START_REF] Ph | Cryogenics for the Large Hadron Collider[END_REF].

The LHC cryogenic systems are divided into 8 equivalent cryoplants around the LHC ring and each of them supply helium to superconducting magnets over 3.3 km via a cryogenic distribution line (called QRL) installed underground in parallel to magnets [START_REF] Erdt | The cryogenic distribution line for the LHC : functional specification and conceptual design[END_REF]. The dynamic behavior of helium flows in the QRL is not well understood, especially during transients as after a quench. To the best of the authors knowledge, relevant dynamic models for such a cryogenic line do not exist, due to the system specificities (length, interconnections, gaseous helium at cryogenic temperature and at low pressure). Thus, this paper provides a numerical dynamic model Email addresses: benjamin.bradu@cern.ch (Benjamin Bradu) for gaseous helium flows in long cryogenic lines that contain interconnections with heat transfers. Previously, a real-time dynamic simulator for cryogenic systems called PROCOS and using the modeling software EcosimPro c was developed at CERN [START_REF] Bradu | A process and control simulator for large scale cryogenic plants[END_REF]. In this context, the model of a 1.8 K refrigeration unit for the LHC achieved to simulate the pumping over the QRL performed by the cold-compressors [START_REF] Bradu | Dynamic Simulation of a 1.8 K Refrigeration Unit for the LHC[END_REF].

The model of the QRL flow is embedded in a larger model that aims at operator training, diagnostic and control improvements. Thus, the QRL model must be computed fast enough to obtain simulations faster than realtime in case of operator training.

First, the paper presents briefly the cryogenic distribution line of the LHC and its interconnections to the main cooling loops. Then, the numerical model of the return pumping line at very low pressure is detailed: the model of the flow, the heat transfers and the pressure drops are described with their numerical implementations in Section 3. Next, steady-state and dynamic simulations during a quench are presented in Section 4. The presented results are discussed and compared with real data obtained in 2008.

The Cryogenic Distribution Line

The QRL is composed of five different headers (called B-F) with temperatures ranging between 4 K and 75 K to supply magnets, beam screens and thermal shields of the magnet cryostats, see Figure 1. All main QRL elements as inner headers and internal fixed points are built in stainless steel AISI 304 L.

A LHC sector of 3.3 km is divided into different cryogenic cells that constitute the elementary cooling loops. A sector is composed of 23 standard cells of 106, 9 m (composed of 6 dipoles and 2 quadripoles) plus other specific cells according to the sector considered, see Figure 2 where the sector 5-6 of the LHC is represented. The main cooling loops are supplied in supercritical helium by the header C and the helium flows are returned back to the header B through sub-cooled heat exchangers.

Modeling of the header B

Helium is considered as a perfect fluid : viscosity and thermal conductivity are neglected as helium has a very low viscosity (µ ≈ 10 -6 Pa.s) and a very low thermal conductivity (λ ≈ 10 -3 W.m -1 .K -1 ). Moreover, gaseous helium is considered as a compressible gas.

Modeling of the Flow

First, the main flow line is modeled, neglecting interconnections. According to previous assumptions, an inviscid flow in the header B is considered. Thus, the flow can be described by Euler equation which is obtained from Navier-Stokes equation neglecting the viscosity and the thermal conductivity [START_REF] Hirsch | Numerical Computation of Internal and External Flow[END_REF]. The conservative form of Euler equations is the following:

∂ ∂t           ρ M E           + ∇ •             ρ • V ρ • V T ⊗ V + P • I ρ • V • u + P ρ             =           0 0 q           (1)
where the different variables are described in Table 1. These three equations represent mass, momentum and energy balances. The momentum is defined as M = ρ • V and the total energy per unit volume is Perfect Gaz constant J.mol -1 .K -1 R He = 2078 Helium specific constant

E = ρ • u + 1 2 • (V 2 x + V 2 y + V 2 z ) .
J.kg -1 .K -1 C v = 3148
Helium specific heat J.kg -1 .K -1 γ = 1.66 Specific heat ratio,

C p C v -
The following assumptions on the flow are assumed :

• flux according to the x direction only (the main flow direction) :

V = V x and M = ρ • V x ;
• straight line. The QRL curvature (radius of curvature of 4.3 km) has a negligible impact on the flow;

• in operational conditions, the kinetic component can be neglected:

ρ • | V| 2 << P, which implies that ρ • V T ⊗ V + P • I ≈ P • I.
Considering the above approximations, Euler equation (1) can be expressed in 1D as: where X = [ ρ M E ] T is the state vector, F is the Jacobian flux matrix and Q = [ 0 0 q ] T is the source vector.

∂X(x, t) ∂t + F(X) • ∂X(x, t) ∂x = Q(x, t) (2

Modeling for an ideal gas flow

To compute the Jacobian flux matrix, it is necessary to include an equation of state to link pressure, density and internal energy. First, an ideal gas is considered: it is assumed that u = C v • T , which leads to the following equation of state:

P = ρ • R • T = ρ • u • R C v = ρ • u • γ (3)
where the constant γ = (γ -1). For this case, the Jacobian matrix was calculated in [START_REF] Hirsch | Numerical Computation of Internal and External Flow[END_REF][START_REF] Toro | Riemann Solvers and Numerical Methods for Fluid Dynamics[END_REF] as:

F =             0 1 0 (γ-3)V 2 2 (3 -γ)V γ γV 3 -γV E ρ γE ρ -3γV 2 2 γV             (4)
and the eigenvalues of the Jacobian are:

         λ 1 = V + c λ 2 = V λ 3 = V -c (5)
where c is the speed of sound, defined by:

c = γRT = γP ρ = γ γ( E ρ - V 2 2 ) (6)
All the eigenvalues of the Jacobian are real and distinct. This means that Eq. ( 2) is a strictly hyperbolic system of equations. Moreover, for subsonic flows (V < c) there are two positive eigenvalues and one negative eigenvalue: informations are propagated forward and backward and the eigenvalues represent the different speeds of propagation.

Modeling for a gaseous helium flow

The previous equations cannot be directly applied to helium flows at low temperature as the equation of state (3) is not valid anymore. Hence, the following empirical formulation for the helium internal energy is considered:

u = u 0 + C v • T (7)
where u 0 = 14950J.kg -1 is a constant. This equation remains valid for gaseous helium at low pressure (P < 10 kPa), with a relative error less than 1 % between 1.8 K and 300 K. For higher pressures, this equation is still valid for gaseous helium far from the saturation line and far from the critical point. We are interested in simulating the helium flow during the final cool-down of magnets from 4.5 K to 1.8 K. During this phase, the pressure in the header B decreases from 0.1 MPa to 1.6 kPa and temperatures are between 2 K and 10 K. Within these ranges, Eq. ( 7) remains valid and the equation of state and the sound velocity for helium become respectively:

P = ρ • R He • T = ρ • (u -u 0 ) • γ (8) c = γ γ E ρ - V 2 2 -u 0 (9)
A new Jacobian flux matrix is computed with additional terms containing the constant u 0 :

F =             0 1 0 (γ-3)V 2 2 -u 0 γ (3 -γ)V γ γV 3 -γV E ρ γE ρ -γ( 3V 2 2 + u 0 ) γV             (10)
and the eigenvalues are the same as for an ideal gas, see Eq. ( 5). Thus, the system of equations is still a strictly hyperbolic system with the same propagation speeds than for ideal gases but the dynamics is different.

Discretization Scheme

To solve the system of partial differential equations, the flow is discretized in N nodes, see Figure 3. A finite difference method was chosen to compute space and time derivatives using a first order upwind scheme, as suggested in [START_REF] Toro | Riemann Solvers and Numerical Methods for Fluid Dynamics[END_REF]. The time discretization is performed according to an implicit discretization scheme based on a backward Euler method. As the cryogenic line is pumped by cold-compressors, natural Dirichlet boundary conditions are set as follows:

• the input energy E(0, t) = E 1 (t);

• the input density ρ(0, t) = ρ 1 (t);

• the output momentum M(L, t) = M N (t).
Hence, mass and energy are propagated forward whereas momentum is propagated backward, which is in agreement with the signs of eigenvalues obtained in Eq. ( 5) and with the physics of the flow. As the coldcompressors are 'pulling' helium atoms in the line, the momentum propagation is backward whereas the mass (helium atoms) and energy (heat) are naturally transported with the flow. Thus, in the framework of the upwind scheme, ∂ρ/∂x and ∂E/∂x are approximated by a first-order backward finite difference and ∂M/∂x is approximated by a first-order forward finite difference. The following algebraic system is then obtained:

Ẋi (t) + A i (X i ) ∆x X i (t) + B i (X i ) ∆x X i-1 (t) (11)
+ C i (X i ) ∆x X i+1 (t) = Q i (t)
where i denotes the value at x i and:

X i =           ρ i M i E i           A i =              0 1 0 - (γ-3)V 2 i 2 + u 0 γ -(3 -γ)V i -γ γV 3 i -γV i E i ρ i γE i ρ i -γ( 3V 2 i 2 + u 0 ) γV i              B i =            0 -1 0 0 0 0 -γV 3 i + γV i E i ρ i -γE ρ i + γ( 3V 2 i 2 + u 0 ) γV i            C i =            0 0 0 (γ-3)V 2 i 2 -u 0 γ (3 -γ)V i γ 0 0 0            Q i =           0 0 q i           E i-1 ,r i-1 E i+1 ,r i+1 E i ,r i M i M i-1 M i-2 M i+1
...

Dx

...

Modeling of interconnections

Additional helium fluxes coming from LHC magnets enter in the header B every 106.9 m in standard cells as depicted in Figure 2. These additional fluxes are perpendicular to the main flux but their contribution to the momentum is considered as additive, since it is small compared to the total flow.

Interconnections are included in the discretization scheme at regular intervals every N sub nodes. Thus, for N inter interconnections, the line is discretized in N sub • N inter nodes, see Figure 4.

i=(j-1)*Nsub+1 The source term Q i of interconnection nodes is now augmented to include the incoming mass, momentum and energy, and consequently writes as:

Q i =               M ext i ∆x (3 -γ)V ext i M ext i ∆x q i + γE ext i M ext i ρ ext i •∆x -(γ -1)( 3V ext2 i 2 + u 0 ) M ext i ∆x               (12)
for i = ( j -1) • N sub + 1 with j = 1 . . . N inter . Superscripts ext refer to external inputs, namely the input fluxes at interconnections. As LHC magnet cooling circuits are not the input fluxes variables are the boundary conditions, determined from mass-flow, pressure and temperature measurements on the real plant.

Heat transfers

The term q i in the energy balance represents heat transfers between the fluid and its environments per unit volume in a node. The QRL is well insulated inside a vacuum jacket using a Multi Layer Insulation (MLI) and thermal shields to minimise radiation and conduction losses.

Heat losses can be divided in two parts: static losses q stati due to radiation, conduction and vacuum barriers and dynamic losses q convi due to convection between the stainless steel pipe and the fluid:

q i = q stati + q convi (t, ρ i , M i , E i ) (13)
In a former CERN study, the static heat losses were evaluated at 1.92 W/m 3 in the header B, this corresponds to 0.1 W/m along the line [START_REF] Riddone | Cryogenic distribution line heat loads[END_REF].

To compute dynamically the convection heat transfer between the pipe and the fluid, supplementary variables are introduced to characterize the pipe properties: its temperature T w , its specific heat C pw , its mass M w and its internal surface S w . The headers are built in stainless steel 304L and the specific heat can be computed from an empirical logarithmic polynomial of the 8 th order valid between 2 K and 300 K [START_REF] Marquardt | Cryogenic Material Properties Database[END_REF]. The convective heat transfer coefficient h c is computed by:

h ci = Nu i • k i D ( 14
)
where the Nusselt number Nu is computed from the Colburn formulation [START_REF] Colburn | A method of correlating forced convection heat transfer Data and a comparison with fluid friction[END_REF]:

Nu i = 0.023 • Pr 1/3 i • Re 0.8 i ( 15
)
because the flow regime in the header B of the QRL is always turbulent (Reynolds number Re > 10 5 ). Finally the convection heat transfer per unit volume is computed by the Newton law:

q convi = h ci • S wi • (T wi -T i ) S • ∆x = M wi S • ∆x •C pwi • dT wi dt (16)

Pressure drops

The QRL is not perfectly horizontal, the slope varies between -1.5 % and +1.5 % along the LHC ring, and it results a hydrostatic pressure. This hydrostatic pressure is not taken into account in previous equations and should be considered.

Moreover, friction phenomena are not embedded (the flow is considered inviscid) whereas frictions can lead to pressure drops not negligible in the case of a very low pressure flow as in the QRL. Thus, we propose to replace the momentum partial differential equation (∂M/∂t + ∂P/∂x = 0) by an algebraic equation to compute the mass-flow ṁi in each cell (compartmental approach).

The pressure in a cell is computed using the equation of state [START_REF] Marquardt | Cryogenic Material Properties Database[END_REF] and the total pressure drop in a cell can be computed as the sum of the hydrostatic pressure difference and the friction pressure drop :

∆P i = P i -P i+1 = ρ i •g•dz i + f r i • ∆x D • ṁ2 i 2 • ρ i • S 2 (17)
where dz i is the elevation over the cell i and f r i is the Darcy-Weisbach friction factor. The line B of the QRL can be considered as a smooth pipe and the flow is always turbulent with a Reynolds number Re > 2 • 10 4 . In this case, the friction factor can be computed from the empirical formulation of Kakac, Shah and Aung [START_REF] Kakac | Handbook of Single-Phase Convective Heat Transfer[END_REF] :

f r i = 0.184 • Re -0.2 i = 0.184 • D • ṁi S • µ i -0.2 (18)
Combining ( 17) and (18) the mass flow is calculated as function of the total pressure drop:

ṁi =         (P i -P i+1 ) -ρ i • g • dz i 0.184 • ( D S •µ i ) -0.2 • ∆x 2•D•ρ i •S 2         1/1.8 (19)
Finally, the momentum is simply deduced from ṁi adding the interconnection momentum :

M i = ṁi S + M ext i (20)

Numerical implementation

The modeling and simulations are performed on a commercial software called EcosimPro c , which is able to simulate systems of differential and algebraic equations (DAEs) [START_REF]Ecosimpro mathematical algorithms and simulation guide[END_REF]. Ecosimpro c uses a DASSL algorithm to solve the DAE system [START_REF] Petzold | A description of DASSL: A Differential/Algebraic System Solver[END_REF]. This method is based on an implicit time discretization scheme:

ẏ(t) = y n -y n-1 t n -t n-1 (21)
The non-linear system dynamics is then solved by iterations for time t n using an implicit Newton-Raphson method. The iteration matrix required by Newton-Raphsons method calculates a Jacobian matrix numerically using finite differences. Note that the use of an implicit scheme renders the time integration robust with respect to numerical errors but its main drawback is the associated lack of precision.

Results and discussion

Simulation in steady-state

The first task consists in checking that the steadystate reached by the model under constant boundary conditions agrees with the measured values on the QRL. The header B is equipped with temperature sensors every 200 m but there is only one pressure sensor and one mass-flow sensor at the end of the header. Each interconnection also has a virtual flow meter allowing to estimate the mass-flow in each cell and the electrical heater power in the phase separator S970 allows us to estimate the mass-flow at the entrance of the header B.

Each of the 8 LHC sectors have the same configuration but they present some differences at their boundaries. A lot of experimental data were collected in one of these sectors during the hardware commissioning in 2008, namely sector 5-6 (chosen for this study). Constant boundary conditions were set in the model, in agreement with the values measured in the sector 5-6 during April and May 2008, when the sector was cold in steady-state. Pressure and temperature at the entrance were set respectively at 1.630 kPa and 1.8 K, the output mass-flow was set to 58 g/s and the different external input fluxes are listed in the Table 2.

Static heat losses were evaluated theoretically at 1.92 W/m 3 on the header B but after measurements on the real line, those losses were reevaluated at 1.1 W/m 3 , which represent a loss of 0.07 W/m along the line. Q8/Q9/Q10 3.8 3.5

Figure 5 shows the simulated temperatures, pressures and mass-flows along the line obtained in steady-state (curves) in comparison with sensor values (crosses).

The simulated temperatures agree with the observed ones except for the temperature located at 700 m, due to a poor sensor calibration or to a bad approximation of the first external fluxes. The pressure drop calculated including the hydrostatic pressure and friction pressure drops gives a total pressure drop of 110 Pa over the 3.3 km. This pressure drop is pretty low in the case of the sector 5-6 because it has a negative slope of -1.54 % and hydrostatic pressure compensates friction pressure drops.

Dynamic simulation during a quench

When a quench occurs in a superconducting magnet, the resistive transition releases a huge amount of energy. A quench protection system was designed to discharge this energy in large electrical resistances in case of a quench, but an important heat flux is briefly induced in the headers B and D of the QRL (the return lines), thus creating a propagated heat wave.

In April and May 2008, 23 quenchs were provoked in sector 5-6 during the hardware commissioning of the LHC. It was chosen to reproduce a quench provoked on a dipole magnet located 774 m after the beginning of header B, inducing a heat wave on more than 2 km.

The quench is simulated by setting a peak at the interconnection boundary (external momentum and temperature) corresponding to the quench location. Due to the lack of instrumentation on the external fluxes, the momentum and temperature peaks are set in such a way that the temperature peak observed on the line B after 107 meters has the correct amplitude and shape. The temperature and mass flow coming from the flux at the quench location are approximated by a bell shape with a peak of 30 K and 2.6 g/s (the temperature of 30 K is approximatively the temperature reached by the magnet after the quench).

The first simulation is performed with N = 31 nodes of 106.9 m each (one node per cooling loop), the convective heat transfer and the pressure drops are neglected (h c = 0 and dP = 0). Therefore, ∆x = 106.9 m and a time step ∆t = 1s is chosen (no special condition is required as an implicit discretization scheme is used). The simulation with these parameters shows that the heat wave is much more dissipated (smaller amplitude and larger width) and much faster in simulation, see Figure 6, where simulation results are plotted in comparison with experimental measurements made on the real line after a quench at different locations.

Then, the convection heat transfer is included in the model. The simulation result is more in agreement with the measurements, as the heat wave is slowed down by the convection transfers. Nevertheless, the heat peak is still too much dissipated.

In order to reduce spatial and time dissipation, the number of nodes is increased by a factor 10 (N = 310). Therefore, the spatial step is reduced to ∆x = 10.69 m. This simulation shows that the dissipation of the heat peak was reduced and that the simulation result is close to the experimental measurements, but the wave is still a little bit faster in simulation. Note that there are no significant modifications when there are more than 310 nodes.

A fourth simulation was performed including hydrostatic pressure and friction pressure drops using the algebraic equation (19) instead of the partial differential equation to compute the momenta. The result obtained is very similar to the previous one but the heat wave is a little bit slowed and the peak is slightly reduced. This configuration is more in agreement with the measurements and we can consider this simulation as satisfying.

The temperature rise observed on measurements after the quench that is not observed in simulations mainly comes from the approximation of the external flux coming from the quench location. Simulations are performed on a classical computer, with a processor Pentium c D 3.4 GHz and 1GB of RAM. The heat wave generated by the quench is propagated over 20 minutes and for N = 31, the simulation is 3.4 times faster than real time (6 min), whereas for N = 310 the simulation takes 8 min. Moreover, the use of the algebraic equation including pressure drops instead of the partial differential equation to compute the momentum increases the simulation speed by a factor 1.5.

Conclusion

In this paper, a very low pressure gaseous helium flow model based on 1D Euler equation was used to describe the flow dynamics and heat transfers in a header of the LHC cryogenic distribution line over 3.3 km. Comparisons between real data obtained on the LHC and simulations after a quench have shown that the model is good enough to predict the dynamical behavior of the flow over the line.

These simulations highlight the importance to include the convection heat transfer between the fluid and the pipe in the model to obtain a correct propagation speed and to reproduce a good attenuation of the heat wave. In addition, pressure variations along the line due to hydrostatic pressure and friction pressure drops can be embedded in the model using an algebraic equation to compute momenta instead of the classical partial differential equation. These pressure variations can be significant to simulate the flow dynamics in long cryogenic line presenting a slope.

Moreover, it was shown that the numerical scheme directly impacts the results and deserves a dedicated analysis. The spatial step has to be small enough because the discretization induces dissipation phenomena.

This model can be applied to other gaseous helium flow in long cryogenic distribution lines if helium is far enough from the saturation line, where the equation of state (8) remains valid. For instance, this model works in the headers D, E and F of the QRL but not in the header C, where helium is in the supercritical state, just above the helium critical point.

Figure 1 :

 1 Figure 1: QRL cross-section with remarkable data

Figure 2 :

 2 Figure 2: Details of the LHC sector 5-6 with the main cooling loops for the superconducting magnets

Figure 3 :

 3 Figure 3: Discretization of the QRL main flow

Figure 4 :

 4 Figure 4: Discretization of the QRL with interconnections for N inter = 3 and N sub = 5

Figure 5 :

 5 Figure 5: Temperatures, pressures and mass-flows along the header B in steady-state compared with experimental measurements

Figure 6 :

 6 Figure 6: Temperatures along the header B after a quench (real measurements compared to 4 simulations)

Table 1 :

 1 Main variables and physical constants

	Symbol	Description	Unit
	c	Sound velocity	m.s -1
	D	Diameter	m
	E	Energy per unit volume	J.m -3
	f r	Darcy-Weisbach friction factor -
	h	Specific enthalpy	J.kg -1
	h c	Heat transfer coefficient	W.m -2 .K -1
	k	Thermal conductivity	W.m -1 .K -1
	L	Length	m
	ṁ	Mass flow	kg.s -1
	M	Momentum	kg.m -2 .s -1
	P	Pressure	Pa
	Pr	Prandtl number	-
	Re	Reynolds number	-
	S	Hydraulic cross section	m 2
	q	External heat inputs	W.m -3
	T	Temperature	K
	u	Specific internal energy	J.kg -1
	V	Gas speed	m.s -1
	µ	Viscosity	Pa.s
	ρ	Density	kg.m -3
	g = 9.81	Gravity acceleration	m.s -2
	R = 8.31

1.8 K refrigeration unit 4.5 K Refrigerator

								Warm compressors		Cold compressors
					Sector 5-6 = 3351 m
	RM	S970	QRL -Header B								Gaseous helium , Very Low Pressure : 16 mbar / 4 K
	P 5	1.8 K EH970	QRL -Header C 3.5K 8g/s	3.5K 0.7g/s	3.5K 2.4g/s	3.5K 0.8g/s	3.5K 0.8g/s			3.5K 0.8g/s	3.5K 0.8g/s	3.5K 1.3g/s	3.5K Supercritical helium : 3 bar / 4.6 K 3.8g/s	P 6
	C M S		Q1/Q2/ Q3	Q4/D2/ Q5/Q6/ DFB	Q7/Q8	Q9/ Q10		ARC : 23 cells			Q11	Q8/Q9/ Q10	Jumper/DFB/ Q5Q4	Q U I
			LSS 5R =269m	DS 5R = 184m	ARC = 2459m					DS 6L=170m	LSS 6L=269m
			DS = Dispersion Suppressor LSS = Long Straight Section RM = Return module		D1	D2	D3	Q1	D1	D2	D3	Q2	D : Dipole magnet Q : Quadripole magnet DFB : Distribution Feed Box
		QUI = Cryogenic interconnection box			1standard cell = 106.9m
													)
								2

Table 2 :

 2 Boundary conditions for input fluxes in sector 5-6

	Location Description ṁ (g/s) T (K)
	LSS-5R	Q1/Q2/Q3	8	3.5
	DS-5R	Q7/Q8	0.7	3.5
	DS-5R	Q9/Q10	2.4	3.5
	ARC	23 cells	0.8	3.5
	DS-6L	Q11	1.3	3.5
	DS-6L


Thomas Barré
email: thomas_barre@yahoo.fr

Overemployment, Underemployment and the opportunity cost of time Building policy agendas for developing economies

Keywords: Market imperfections, Shadow wages, Allocative efficiency, Vietnam. JEL classification: D1, J2, O12, Q12

Focusing on individual labor market positions, this article proposes a new approach to elicit and measure constraints faced by rural households. Under market imperfections, individuals fail to equalize their hourly income to their shadow wage and become over-or underemployed. We estimate and explain this gap in a stochastic frontier framework for rural Vietnam. Both employees and farmers are found to fail in equalizing their hourly income to their shadow wage. Constraints faced by farmers are found to be stronger than that of employees: farmers' marginal revenue of labor is 3 times higher than their shadow wage while market wages earned by employees are 1.5 times higher than their shadow wages. Price risk is found to be the most important constraint faced by Vietnamese rural farmers while employees would benefit from the development of the road network.

Introduction

Even in the most advanced economies, people face market imperfections that constraint their behavior and reduce economic efficiency. In developing economies, where these failures are the most important, they can even prevent people from escaping out of poverty by limiting the range of opportunities they can grab. From an economic policy point of view, [START_REF] Janvry | Peasant Household behaviour with missing markets: some paradoxes explained[END_REF] showed that market imperfections can make peasants irresponsive to price incentives, limiting the impact of standard development tools. Hence, a central question for economic development is to elicit the market failures that households face and to solve them.

Empirical literature has extensively shown that rural households are very likely to be constrained and theoretical literature provides many possible sources for market imperfections, but which are the most relevant remains unanswered. The objective of this article is to find which are the most important in order to propose an agenda for economic policy based on microeconomic empirical evidence of the constraints faced by rural households in Vietnam.

As noted by [START_REF] Barrett | Shadow wages, allocative inefficiency and labor supply in smallholder agriculture[END_REF] "the principal asset of the poor is their labor". And it is also a very flexible input for agricultural production. So, the main hypothesis in this article is that if a household is constrained by some market failure, this latter can be recovered when studying household members' labor market position. This idea was initiated by [START_REF] Skoufias | Using shadow wages to estimate labor supply of agricultural households[END_REF] who argued that differences between market wage and marginal revenue of labor indicate a labor market constraint. Recently, [START_REF] Barrett | Shadow wages, allocative inefficiency and labor supply in smallholder agriculture[END_REF] mitigated this result1 by demonstrating that price uncertainty can also explain the gap between market wage and marginal revenue of labor, so that the same stylized fact (the inequality between market wage and marginal revenue of labor) can be explained by two phenomenons. The cause of this puzzling result lies in assumptions made for the identification of individual cost of time (also called opportunity cost of time, implicit wage or shadow wage): While [START_REF] Skoufias | Using shadow wages to estimate labor supply of agricultural households[END_REF] assumes that farmers can equalize their marginal revenue of labor to their shadow wage and that they are labor market constrained, [START_REF] Barrett | Shadow wages, allocative inefficiency and labor supply in smallholder agriculture[END_REF] consider that output price uncertainty prevents farmers from adjusting their marginal revenue of labor to their shadow wage but put the labor market constraint aside, so that employees can equalize market wage and shadow wage. Both market failures highlighted by the authors surely exist and this article introduces an econometric method for retrieving shadow wages in a framework where both constraints bind.

Assuming that other market failures could cause this gap, this article will test for many possible sources of inefficiency in a "horse race" approach. The argument is that interpretations of econometric results are often limited by mathematical formalization of the theoretical framework. In this case, the methodology should be reversed: econometric results should draw the way we are building theoretical models. A general econometric framework is proposed in which both farmers and employees can be constrained and in which we can test for possible sources of market failures including, among others, labor market constraint and price uncertainty.

The approach consists in measuring individual shadow wages given that market imperfections prevent households from equalizing it to market wage and marginal revenue of labor.

Basic microeconomic theory emphasizes that the shadow wage is the marginal rate of substitution from consumption to leisure valuated at the market price of consumption. So, the objective is to measure and explain the gap between this marginal rate of substitution and observed prices of labor (market wage or marginal revenue of labor) by some variables representing possible market failures proposed by the economic literature presented in the following section. The main difficulty is that the marginal rate of substitution is not observed. Section 3 will demonstrate that the problem can be handled by the estimation of a mixture of stochastic frontier models when we assume Cobb-Douglas utility functions. The method is very instructive as it allows us to determine whether people are constrained or not, who is constrained, why and how much, answering the four basic questions that must be asked when studying market imperfections.

Econometric application to Vietnamese data for 2004, presented in section 4, shows that most of Vietnamese rural workers are under-employed (96% of the sample): their shadow wages are 2.5 times lower than observed labor prices (market wage or marginal revenue of labor). Hence, reducing market imperfections could generate large efficiency gains and may be welfare improving.

Econometric results demonstrate that the development of roads for employees and price stabilization policies toward peasants are very promising ways to increase economic efficiency in Vietnam. Finally, constraints faced by farmers are found to be stronger than that of employees, confirming the intuition developed by Barrett (1996).

Literature Review

The approach developed in this article is essentially based on two related literatures: the analysis of household's constraints and the measurement of shadow wages. We present them in two distinct sub-sections despite the fact that some references belong to the two literatures in order to highlight the advances proposed in this paper.

Household constraints

The problematic of the literature about households' market imperfections has slightly changed in the recent years: While first approaches mainly questioned the presence of a constraint within the population, following articles clarified the problem, focusing on three main topics: Who is constrained? How much are they constrained? Why are they constrained? 2.1.1 First approaches: Is there a constraint? Many studies have analyzed the existence of market failures by testing the separation hypothesis: if a constraint binds, then consumption and production decisions are taken simultaneously (they are not separable) so that household characteristics (number of children, age of household head, educational level, etc.) affect production decisions such as market participation, input choices, level of production, etc. Econometric tests have provided contrasting results: While [START_REF] Lopez | Estimating labor supply and production decisions of self-employed farm producers[END_REF] and [START_REF] Grimard | Rural Labor Markets, Household Composition, and Rainfall in Cote d'Ivoire[END_REF] reject the separation hypothesis for Canada and Côte d'Ivoire, [START_REF] Benjamin | Household composition, labor markets, and labor demand: Testing for separation in agricultural household models[END_REF] cannot reject the hypothesis that household structure does not affect labor allocation decisions in rural Java. [START_REF] Bowlus | Moving toward markets? Labor allocation in rural China[END_REF] do not reject separability only for localities benefiting from sufficiently developed movements of resources, but separability is rejected overall in their sample, indicating that "factor markets remain underdeveloped" in China.

Who is constrained?

As noted by [START_REF] Carter | Local versus Global Separability in Agricultural Household Models: The Factor Price Equalization Effect of Land Transfer Rights[END_REF], "Global tests for separability […] are theoretically inappropriate when the market failures […] differentially constrain some, but not all households". Hence, the central question is not whether a constraint binds or not but whose constraint binds? Two main advances have been made in this sense by [START_REF] Lambert | Implicit prices and recursivity of agricultural household's decisions[END_REF] and [START_REF] Vakis | Testing for Separability in Household Models with Heterogeneous Behavior: A Mixture Model Approach[END_REF]. [START_REF] Lambert | Implicit prices and recursivity of agricultural household's decisions[END_REF] extend [START_REF] Skoufias | Using shadow wages to estimate labor supply of agricultural households[END_REF]'s method demonstrating theoretically that "a necessary and sufficient condition for recursivity to hold is that implicit prices […] are equal to market prices". Computing standard errors for implicit prices, they can test individually the equality between shadow wage2 and market wage for each household and show that "non recursivity is a common case in their sample" of households in Côte d'Ivoire.

Considering that this test is too sensitive to the specification and the estimation of a production function for agriculture, [START_REF] Vakis | Testing for Separability in Household Models with Heterogeneous Behavior: A Mixture Model Approach[END_REF] propose another approach based on mixture models: Given that household characteristics affect production decisions only for constrained households, they prefer to measure the probability that a household is constrained (household characteristics affect production decisions) or not. Their study focuses on on-farm labor supply and estimates the probability that a household is constrained (household characteristics affect on-farm labor supply). They find that, on average, a household has a probability of 0.51 of being constrained in rural Peru, confirming that market failures affect a large part of the population in rural areas.

2.1.3 Why are they constrained? Many possible constraints have been proposed in the literature (see for example: Sadoulet, De Janvry and Benjamin (1998) for price bands, Barrett (1996) for price uncertainty, [START_REF] Carter | Local versus Global Separability in Agricultural Household Models: The Factor Price Equalization Effect of Land Transfer Rights[END_REF] for land transfer rights, De Janvry, Sadoulet, Fafchamps and Raki (1992) for credit constraint).

Most of the time, a theoretical model is introduced in order to focus on the constraint analyzed and interpretations of econometric results are driven by the hypothesis made when building the theoretical model. Because different models can explain the same phenomenon (see the introduction), policy implications can become confused: Why should we believe one interpretation more than others? Is the constraint under scrutiny the most important? Vakis, Sadoulet, De Janvry and Cafiero ( 2004) try to answer this question by putting some explanatory variables in the switcher equation of their mixture model so that they can test for some possible origins of non-separability and find that both demand (education, ethnicity, etc) and supply (availability of job opportunities) variables affect the probability of being constrained. This is a clear value added as compared to standard approach: while sources of non-separability are usually proposed by theory, Vakis, Sadoulet, De [START_REF] Vakis | Testing for Separability in Household Models with Heterogeneous Behavior: A Mixture Model Approach[END_REF] propose an econometric approach which could help identifying the most important failures faced by rural households and build more relevant theoretical models.

How much are they constrained?

Once one knows that households are constrained and why, a question remains: how much are they constrained? How much do they behave differently as compared to unconstrained ones? This is a central question because if constrained household do not really differ from unconstrained ones, then, there is no reason for a specific development policy toward them and standard policy tools should apply. This problem was questioned by [START_REF] Janvry | Peasant Household behaviour with missing markets: some paradoxes explained[END_REF] who showed that market failures reduce the price elasticity of cash crop supply from 0.99 with no market failure to 0.18 when labor and food markets fail. It means that if a government wants to increase cash crop supply, it will have to increase the price of cash crops 5.5 times more if households are constrained than if they are not in order to obtain the same result. So the same economic policy would cost 5.5 more if households are constrained. This is a good argument for making government more sensitive to the problem of market failures.

Other studies focused on the output gap generated by market failures as a measure of the importance of inefficiencies: Estimating a "generalized indirect production function" in which a parametric function of other prices, cost and quasi-fixed inputs is substituted for shadow prices, Bhattacharyya and Kumbhakar (1997) show that market imperfections induce an output loss of almost 12% for rice growers in West Bengal, India. Using an ex ante classification between credit constrained and unconstrained households, Boucher and Guirkinger (2008) estimate that "agricultural production could increase by 26% in Piura, Peru, if all credit constraints were eliminated".

The method proposed by Vakis, Sadoulet, De [START_REF] Vakis | Testing for Separability in Household Models with Heterogeneous Behavior: A Mixture Model Approach[END_REF] could easily be used to measure the "labor gap" generated by market imperfections. Hence, the method could answer the four basic question mentioned before (Is there a constraint? Who is constrained? Why? How much?) in a unified framework. However, in this framework, market failure variables influence the probability of being constrained but this gives no information about the inefficiency cost of each constraint.

An alternative approach based on factor price equalization as in [START_REF] Skoufias | Using shadow wages to estimate labor supply of agricultural households[END_REF] or Lambert

and Magnac (1998) is preferred here. However, while [START_REF] Lambert | Implicit prices and recursivity of agricultural household's decisions[END_REF] argue that non rejection of factor price equalization on one market does not imply that recursivity holds because non-recursivity could arise because of another good, we argue that if a household is constrained, this constraint will always prevent him from reaching his optimal level of utility, so that the shadow price of labor will differ from observed labor prices whatever the source of inefficiency.

The strategy relies on the idea that the gap between the shadow wage and observed labor prices (market wage or marginal revenue of labor) can be explained by market imperfections. A central issue is thus the measurement of individual shadow wage.

How does the literature (mis)measure the opportunity cost of time?

The opportunity cost of time is the price that one wants to be paid in order to work one more hour. As everybody has his own unobserved preferences over leisure and consumption, it is difficult to measure this opportunity cost directly. So, the economic literature uses to derive this shadow wage by assuming that it equals some other observed or easy-to-compute value. Here, I will distinguish between works devoted to the analysis of employed and self-employed people.

Employees

As long as one considers that people do not face labor constraints, she can observe individual shadow wages simply by retrieving individual hourly wages of employees3 . This assumption has been recently criticized by [START_REF] Feather | Estimating the Cost of Leisure Time for Recreation Demand Models[END_REF] who consider that employees can be over-employed or under-employed because they are not free to choose their working time. In their framework, an individual is considered as under-employed (over-employed) if she would prefer to work more (less) for the same hourly wage. Using specific questions allowing them to distinguish ex-ante who is over or underemployed, they show that over-employed (under-employed) earn an hourly wage significantly lower (higher) than their shadow wage. So, observed market wages are not good measures of the shadow wage. The main drawback of this approach is that we need to know if people are over or under-employed ex-ante and this information is rarely available. The method needs to be extended to the case of unknown sample separation in order to be widely used.

Self-Employed

Market wages have been the only measure of the shadow wage for a long period of time during which self-employed people were considered as simple selection problem just like unemployed people. Thus, when [START_REF] Heckman | Shadow Prices, Market Wages, and Labor Supply[END_REF] developed his technique which allows recovering shadow wages for unemployed people, the method was also applied to self-employed. In 1993, Jacoby showed that we have much more information about self-employed than about unemployed: in his framework, the marginal revenue of labor equals the shadow wage. Thus, one can retrieve selfemployed shadow wages by simply estimating a production function. [START_REF] Skoufias | Using shadow wages to estimate labor supply of agricultural households[END_REF] used this result and tested the equality between market wage and marginal revenue of labor (considered as equating the shadow wage) and concluded that agricultural households are constrained on the labor market.

Recently, [START_REF] Barrett | Shadow wages, allocative inefficiency and labor supply in smallholder agriculture[END_REF] mitigated this result: They assumed no labor market imperfection but introduced allocative inefficiency due to uncertainty on output price for agricultural households and found the inequality between market wages and marginal revenue of labor. Hence, the same result (the inequality between market wage and marginal revenue of labor) can be explained by two possible imperfections: labor market imperfections and price uncertainty. Policy implications become confused.

As noted earlier, the source of the problem lies in the identification of the true shadow wage: if it differs from both market wage and marginal revenue of labor, how can we measure the opportunity cost of time?

What really is the "shadow wage"? Turning back to basic microeconomic theory

Standard neoclassical framework

Consider an individual with utility ܷ = ܷሺܺ, ݈; ݖሻ where "X" is the level of consumption, "l" the number of hours spent for leisure activities and "ݖ" some individual characteristics determining preferences over consumption and leisure. Each individual wants to maximize utility with respect to a budget constraint:

 ܺ = ݕ + ܮݓ +  ொ ܳ - ூ ܫ
Where " ொ ܳ =  ொ ܳሺℎ, ܫሻ" is the value of household production, "h" is on farm working time, "I" is a vector of inputs (with a price vector " ூ ") used in the production process, "y 0 " is non labor income, "ܮ" is off-farm working time and "ݓ" is hourly wage paid on the labor market.

If no other constraint binds, we obtain the standard result that the marginal rate of substitution from consumption to leisure equals the price ratios:

ܷ ܷ = ݓ  =  ொ  ߲ܳ ߲ℎ
In this setting, the "shadow wage" is simply the marginal rate of substitution from consumption to leisure ሺܷ ܷ ⁄ ሻ valued at the market price of the consumption good ሺ ሻ which equals the market wage and the marginal revenue of labor at equilibrium.

ݓ * =  ܷ ܷ = ݓ =  ொ ߲ܳ ߲ℎ
Assuming that the shadow wage ݓ * equals the market wage ݓ or the marginal revenue of labor  ொ ߲ܳ/߲ℎ simplifies the analysis because we don't need to account for unobserved preferences. However, if some constraint binds (labor market constraint, price uncertainty, etc.) then the shadow wage doesn't equal any observed value anymore and we have to focus on the problem of individual preferences over consumption and leisure.

Deviations from the neoclassical equilibrium

Assuming a Cobb-Douglas utility function ܷ = ܺ ఈ ݈ ఉ , we know that

ݓ * = ߚ ߙ  ܺ ݈
Our problem is that we do not observe individual relative preference for leisure ሺߚ ߙሻ ⁄ and we only know that  ொ . ሺ߲ܳ/߲ℎሻ ≠ ݓ * ≠ ݓ is likely to occur. However, this issue can be overcome by posing ݇ ொ .  ொ . ሺ߲ܳ/߲ℎሻ = ݓ * = ݇ ெ . ݓ (where ݇ ொ > 0 and ݇ ெ > 0ሻ . To make things clearer, consider the case of employees.

The case of employed people

Assume that ݓ * = ݇. ݓ where ݇ = 1 if our individual is not labor constrained. Following the terminology of [START_REF] Feather | Estimating the Cost of Leisure Time for Recreation Demand Models[END_REF], if ݇ ≥ 1 (0 < ݇ ≤ 1ሻ, then our worker is overemployed (underemployed): he would prefer to work less (more) for the same hourly wage.

Substituting ݓ * by its expression, we obtain the following relation:

 ܺ ݈ݓ = ߙ ߚ ݇
Our problem is to distinguish between preferences heterogeneity ߙ ߚ ⁄ and individual labor market constraint ݇. Note that ݇ bears two informations: it tells us whether people are constrained or not (if ݇ ≠ 1 or not) and how much they are constrained (how far is ݇ from 1). It already answers two of the questions highlighted earlier: who is constrained and how much?

The identification strategy relies upon Feather and Shaw (1999)'s classification which allows splitting employees into two groups: overemployed and underemployed. Considering underemployed workers, we know that ݇ ≤ 1. Assume that the heterogeneity of preferences can be modeled as: ߙ ߚ ⁄ = ݂ሺݖሻ. ߝ where ߝ is unobserved heterogeneity.

We obtain: ሺ ܺሻ ሺ݈ݓሻ ⁄ = ݂ሺݖሻ. ߝ. ݇

Taking logs, we obtain a stochastic "production" frontier model (see [START_REF] Aigner | Formulation and estimation of stochastic frontier production function models[END_REF] and Meeusen and Van den Broeck (1977)) so that ߝ and ݇ can be separated by maximum likelihood estimation:

ln ൬  ܺ ݈ݓ ൰ = ln ݂ሺݖሻ + ݒ -ݑ
Where ݒ = ln ߝ, is random and ݑ = lnሺ݇ሻ ≥ 0.

The same result applies to overemployed people with a small (but important) difference, so that we obtain a stochastic "cost" frontier model:

ln ൬  ܺ ݈ݓ ൰ = ln ݂ሺݖሻ + ݒ + ݑ
Our problem now reduces to the estimation of a mixture of two stochastic frontier models:

one for underemployed people (the "production" model) and one for overemployed people (the "cost" model).

The case of self-employed

If one wants to adopt this approach for self-employed people, she needs to measure the marginal revenue of labor that should replace the market wage "ݓ" in our equation. The standard approach [START_REF] Skoufias | Using shadow wages to estimate labor supply of agricultural households[END_REF], [START_REF] Lambert | Implicit prices and recursivity of agricultural household's decisions[END_REF] or [START_REF] Barrett | Shadow wages, allocative inefficiency and labor supply in smallholder agriculture[END_REF])

relies on the estimation of a production function that is often subject to critics (dealing with zeros in input quantities, missing variables, reduction of a long decision process to a simple equation, etc.). In order to avoid estimating such function, we follow [START_REF] Le | Shadow wages and shadow income in farmers' labor supply functions[END_REF] and assume that ܳ = ܳሺℎ, ܫሻ = ݍሺܫሻ. ℎ ఊ , so that the marginal revenue of labor is ܮܴܯ =  ொ . ߛ. ሺܳ ℎሻ ⁄ .

We obtain:

ln ቆ  ܺ ݈ ℎ  ொ ܳ ቇ = ln ݂ሺݖሻ + ln ߛ + ݒ ± ݑ
where ߛ is labor elasticity of output.

Hence, the final econometric specification is:

Incorporating explanatory variables for inefficiency scores

In Barrett (1996)'s framework, farmers' labor position is determined by household net food buyer or net seller status which is related to land endowment: If the household owns a big cropping land, he is a net seller of the production good and labor is underemployed. However, if land is scarce for the household, he is a net buyer and labor is overemployed. It is thus necessary to include household land endowment as an explanatory variable of inefficiency scores of farmers in the model.

Furthermore, the objective of this article is to provide a framework allowing us to design a policy agenda for rural households. Other explanatory variables that could influence "݇", the inefficiency score will be introduced in order to perform a Horse Race for market imperfections. To do so, we adopt the stochastic frontier formulation proposed by [START_REF] Huang | Estimation of a Non-Neutral Stochastic Frontier Production Function[END_REF] and [START_REF] Battese | A Model for Technical Inefficiency Effects in a Stochastic Frontier Production Function for Panel Data[END_REF] which assumes that ݑ = lnሺ݇ሻ = ܼߣ + ݓ where ݓ ≥ -ܼߣ.

The final specification is:

݈ℎݏ = ܺߚ + ܵߜ + ݒ -ݏሾܼߣ + ݓሿ
Where ݓ ≥ -ܼߣ, ݏ = -1 if overemployed, and ݏ = 1 if underemployed.

Method, data and estimation results

Estimation method

The identification strategy relies on the estimation of a mixture of two stochastic (truncated normal) frontier models (a "production" and a "cost" frontier) that share the same frontier.

Following [START_REF] Huang | Estimation of a Non-Neutral Stochastic Frontier Production Function[END_REF], we assume ࣨ~ݒሺ0; ߪ ௩ ሻ and ࣨ~ݓሺ0; ߪ ௪ ሻ truncated below at -ܼߣ. [START_REF] Battese | A Model for Technical Inefficiency Effects in a Stochastic Frontier Production Function for Panel Data[END_REF] note that this specification is consistent with that of [START_REF] Kumbhakar | A Generalized Production Frontier Approach for Estimating Determinants of Inefficiency in U.S. Dairy Farms[END_REF] which simply states that ࣨ~ݑ ା ሺܼߣ; ߪ ௨ ሻ. The corresponding likelihood is:

ܮ = ෑሼ ݈ + ሺ1 - ሻ݈ ௨ ሽ ݈ ௨ = 1 ߪ ௌ . ߶ ൬ ߦ + ܼߣ ߪ ௌ ൰ . Ф ൬ ܼߣ ߪ ௌ ߬ - ߦ߬ ߪ ௌ ൰ . Ф ൬ ܼߣ ߪ ௨ ൰൨ ି ݈ = 1 ߪ ௌ . ߶ ൬ ߦ -ܼߣ ߪ ௌ ൰ . Ф ൬ ܼߣ ߪ ௌ ߬ + ߦ߬ ߪ ௌ ൰ . Ф ൬ ܼߣ ߪ ௨ ൰൨ ି Where ߪ ௌ = ඥߪ ௩ ଶ + ߪ ௨ ଶ , ߦ = ݕ -ܺߚ -ܵߜ, ߬ = ߪ ௨ /ߪ ௩
,  is the share of the population overemployed and ݈ ௨ (݈ ) is the likelihood associated to under(over)employed workers.

Estimation is performed via an Expectation -Maximization (E-M) algorithm [START_REF] Dempster | Maximum likelihood from incomplete data via the EM algorithm[END_REF]) in order to deal with mixture. One of the main problems with this algorithm lies in the choice of initial values that must be good enough to catch the global maximum. Our strategy is to get a random value for  , then get initial values for the other parameters given this probability and switch to the EM algorithm and run 10 iterations4 . Then, get another  and maximize again, and so on. The operation is repeated 30 times and we finally got the best parameters obtained (those that maximize the log-likelihood) and used it as initial values for a "long run" E-M algorithm5 . This last run of the EM algorithm comprises a maximum 500 iterations6 , followed by a final maximization of the incomplete log-likelihood.

Estimation sample and Data

The method is tested on Vietnamese Household Living Standard Survey (VHLSS) data for the year 2004. Even if the approach should be applicable to an industrialized country [START_REF] Feather | Estimating the Cost of Leisure Time for Recreation Demand Models[END_REF] show that the employees in the United States suffer from over and underemployment), a developing country is supposed to be a best framework because constraints should be more likely to occur. Furthermore, as long as the objective of the article is to propose policy tools against poverty, a developing economy is more appropriated.

Individual data

VHLSS 2004 collects data from 46,000 households and include detailed questions about household members (age, sex, education, health, etc.), employment, agriculture, aquaculture, forestry, non-farm activities and household expenditures. In order to narrow the sample under scrutiny and to remain in the scope of the literature about market imperfections, we only keep employed workers and farmer in rural areas and drop information about urban areas, aquaculture, forestry and non-farm business. We only keep information about the main job (multiple job holding is widespread in Vietnam). The final sample contains 14 316 individuals. For each observation, information about age, sex, education and household size is collected and will be used as control variables for preferences heterogeneity. Mean expenditure in the household, leisure time, on-farm working time, hourly wage if employed and agricultural production value were used when computing the left hand side variable. Finally, following Barrett (1996)'s argumentation, land per head endowment in the household is collected as a determinant of farmers inefficiency scores.

Finding constraints in the data

The objective of the article is to elicit binding constraints of households in rural Vietnam. So we have to include variables representing possible constraints in the model. VHLSS data propose three sources for such variables: household questions about the constraints faced during production activities, community level questions about the most important constraints faced by farmers, and questions about "infrastructures" in the commune.

Household level questions

Household level questions should be preferred as they better capture individual situations.

However, the questionnaire only proposes few possible constraints (access to capital, lack of knowledge about new technologies, disasters, prices variations or demand/supply balance) limiting the scope of the study (see table 1). Furthermore, only 42% of interviewed farmers declare themselves as constrained in their production activities and this tends to show that individuals are not aware of the constraints they face7 .

Community level questions

The commune questionnaire contains information about « the main difficulties faced by commune's farmers in agricultural production ». Constraints highlighted concern access to capital, lack of new varieties, lack of technical knowledge, prices or demand/supply impact, market access, irrigation network, disasters, pest disease, access to agricultural services and inappropriate agricultural policy. As in the case of household questionnaire, the main constraint cited by community leaders is access to capital (see table 2). This is not surprising as long as if farmers are declared capital constrained, the leader can expect to obtain public subsidies from the Central Government. Above all, this kind of information is not very useful for economic policy: For example, if commune leaders point out that "lack of knowledge" is a great constraint for farmers, what should be done? Does the development of information centers could solve the problem? Do farmers really apply advices given by staff members? Building a good policy for reducing households' constraints requires an ex-ante evaluation of the impact of policy tools.

Infrastructure variables

The main critics against information given by the household and the commune leader are subjectivity and the impossibility of ex ante evaluation of development policies. Infrastructure data correspond much better to our objectives. The choice of relevant infrastructure data follows household and commune questionnaires: we focus on output and input market access, credit constraint, meteorological and epidemic disasters, access to information and new technologies and quality of infrastructures such as irrigation and electricity networks.

Market access is represented by the distance from the hamlet to the nearest market and we distinguish between input and output markets. Credit constraint is approximated by the distance to the nearest institutional lender (State Bank, Private Bank or Credit Organization).

Irrigational network is measured by the share of irrigated land in the commune and access to electricity is a dummy variable equal to one if the commune is connected to the national electricity network and zero otherwise.

Vietnam has a tropical monsoon climate and experiences frequent weather-related natural disasters. Uncertainty generated by meteorological events can prevent farmers from predicting their marginal revenue of labor, so that they fail in equalizing it to their shadow wage. Recent disasters are included in the regression in order to measure this phenomenon.

Access to information is proxied by the distance to the nearest extension center and household level dummies recording meetings with extension center staff members. As noted by [START_REF] Munshi | Information Networks in Dynamic Agrarian Economies[END_REF], technological knowledge is an important issue in rural areas and numerous studies focus on social networks in order to understand technology diffusion across groups. Here, we try to determine the way political leaders should intervene to increase the rate and the speed of technological adoption. This issue is summed up into three variables: distance to the nearest extension center, having visited an information center, or having been visited by a promotion agent.

The best form of promotion of new technologies with farmers should be revealed.

Land market may severely constrain farm households. Considering that productivity of land is limited, the absence of a land market can make farmers underemployed. According to Barrett (1996), peasants are risk averse and their labor position (over or underemloyed) is determined by their land endowment: Net sellers (abundant land) are underemployed, while net buyers (scarce land) are overemployed. Fortunately, these two effects act with different signs: while price uncertainty makes farmer with a small land endowment overemployed, land market constraint makes them underemployed.

In the case of employees, we are interested in finding what could reduce job scarcity in rural areas. Indeed, employees are supposed to find a better match between shadow and market wages if many job offers are available. Distance from the hamlet to the nearest road accessible by car is found to be potential instrument for economic policy toward employees.

Building variables representing possible constraints

If no constraint binds, then households are supposed to succeed in equalizing labor prices and the inefficiency score ݇ equals one. Hence each "constraint variable" included in the model must be built so that it equals 0 if the constraint does not bind and a positive value otherwise. The construction of distance variables (to financial organizations, markets or information center) is thus natural: log transformations of recorded distances are used and distances inferior to 1km are recoded as 1. Natural disasters are also easily coded as 1 if a catastrophe occurred in the last year and zero otherwise. Finally, the percentage of non-irrigated land in the commune is used to control for the irrigational network.

Other variables must be constructed in a less natural way: Having visited or having been visited by agricultural promotion services must be coded so that farmers who didn't access to information are coded as one and others have zero. The same applies to the access to the national power grid.

Last but not least is the problem of land. In Barrett (1996)'s framework, there exists a land per head ratio for which farmers are self-sufficient (neither net sellers nor net buyers). Hence, low land per head endowments are related to overemployment and high land per head endowments are associated to underemployment. The expected relationship is thus:

݇ = ߙ 0 + ߙ 1 ln ሺ݈ܽ݊݀ݎ݁ℎ݁ܽ݀ሻ
If ߙ 1 < 0, then the price risk effect proposed by Barrett (1996) dominates. Farmers are price risk averse and price stabilization policies should be implemented. If ߙ 1 > 0, peasants are constrained by their land endowments. Development projects should focus on improving land productivity or the extension of cropping area.

Constraints on coefficients

As explained above, we estimate a mixture of two models sharing the same stochastic frontier.

This raises the question of the parameter estimated in the two groups (over or underemployed).

Should they be different, equal or opposite? The answer depends on the activity considered:

Employees only suffer from job scarcity which can make them both over and underemployed, while farmers may face many constraints at the same time and each of them will tend to make her over or underemployed.

Employees

Consider first the case of an employee: if a worker can choose between a large numbers of jobs, she can surely find a position where her wage equals her opportunity cost of time (݇ = 1ሻ.

However, if job offers are scarce, she may be overemployed (݇ → +∞ሻ or underemployed (݇ → 0ሻ.

Job scarcity will always move the worker away from the equilibrium as presented in figure 1: the expected relationship between job scarcity and our inefficiency variable "݇" is decreasing if our worker is overemployed and increasing if she is underemployed. Variables affecting access to employment have opposite effects on "݇" for overemployed and underemployed. Parameters of employees' determinants of inefficiency will be constrained to be symmetric in the two groups (ߣ = -ߣ ௨ for employees).

Figure 1 -Expected relationship between job scarcity inefficiency variable "k"

4.4.2 Self-Employed

In the case of self-employed workers, this symmetry disappears. In fact, whatever the position of the farmer with respect to his shadow wage (over or underemployed), a quantity constraint on an input which is complementary with household labor will induce underutilization of both the input and labor so that the worker will be underemployed. To make things clearer, consider a peasant A who is underemployed and a farmer B who is overemployed. If a new constraint on an input appears, then farmer A will become "more" underemployed, while farmer B will be "less" overemployed and may become underemployed too. There is no reason why over-or underemployed farmers would suffer differently from the same constraint. So we will constrain parameters of self-employed workers to be the same in the two groups (ߣ = ߣ ௨ for self-employed). 2008) is proposed and demonstrates that both farmers and employees fail in equalizing labor prices. However, it seems that constraints faced by farmers are stronger than that of employees. Then, a Horse race including other relevant variables representing possible constraints is performed and confirms the importance of income uncertainty as a major constraint to peasants' behaviors while employees would benefit from improved road networks.. 4.5.1 First test: Barrett (1996) versus [START_REF] Lambert | Implicit prices and recursivity of agricultural household's decisions[END_REF] Table 6 presents estimation results for a test based on [START_REF] Lambert | Implicit prices and recursivity of agricultural household's decisions[END_REF] and Barrett (1996). First, a comparison of the labor elasticity of the Cobb-Douglas production function with those found in the literature tends to support the approach proposed here: Estimated elasticities found in the literature are very heterogeneous: While Strauss (1986) finds an elasticity of 0.6 for rural Sierra Leone, Deolalikar (1988) estimates it at 0.35 for rural South India, Jacoby (1993) obtains 0.10 for Peruvian Sierra and Skoufias (1994) finds 0.13 for rural India. Finally, [START_REF] Le | Shadow wages and shadow income in farmers' labor supply functions[END_REF] estimates this elasticity for Vietnam and finds 0.0472 for males and 0.0782 for females. These last results seem to be very low as compared to other studies. Our method gives results closer to that found in the literature: estimated labor elasticity of production is 0.37, close to Deolalikar (1988) and comprised between 0.05 and 0.6 confirming its consistence.

Also, determinants of preferences present consistent results: preference for consumption increases with age; women have relatively lower shadow wages than men. More educated workers expect higher wages. Finally, household size decreases the preference for consumption, as in a model in which people enjoy spending time with relatives8 .

Econometric results confirm the intuition that individuals fail in equalizing shadow wages and hourly income. Furthermore, we find a large proportion of underemployment in the population:

shadow wages are lower than their observed hourly income. This result implies that an increase in labor demand could be absorbed by the labor force without strong pressures on wages. It confirms the consensus which usually considers that Vietnamese work few hours and that labor demand could increase without inflation.

Interpreting inefficiency parameters

Empirical results support our intuitions regarding the determinants of allocative inefficiency:

increasing farm size tends to decrease inefficiency scores of farmers and makes them underemployed as predicted by the model of Barrett (1996) where farmers are price risk averse.

Proximity of roads is found to have a significant impact on allocative efficiency of employees:

reducing the distance to the nearest road makes employees' inefficiency scores to tend towards zero (perfect allocative efficiency), while increasing this distance can make workers either overemployed or underemployed.

Table 6 shows that allocative inefficiency is present in our data and it proposes sources for such inefficiencies. However, it does not tell us how strong is the allocative gap generated by these failures. Answering this question needs some additional steps: first, individuals must be classified as over-or underemployed. Then, a prediction of individual inefficiency scores must be performed.

Classification is based on the following formulae:

݀ = 1 if  ݈ ሼ ݈ + ሺ1 - ሻ݈ ௨ ሽ ⁄ > 0.
5 and zero otherwise

݀ ௨ = 1 if  ݈ ሼ ݈ + ሺ1 - ሻ݈ ௨ ሽ ⁄ < 0.

and zero otherwise

Where ݀ = 1 (݀ ௨ = 1ሻ for individuals classified as overemployed (underemployed).

Then, inefficiency scores are predicted given our classification. Battese and Coelli (1988)'s formula for inefficiency scores is used to predict individual differences between shadow wage and market wage or marginal revenue of labor:

݇ = ܧሼexpሺ-ݑݏ ሻ |ߦ ሽ ݇ = 1 -Ф ൬ ߪ * -ߤ * ߪ * ൰ 1 -Ф ൬ -ߤ * ߪ * ൰ . exp ൬-ߤ * + 1 2 ߪ * ଶ ൰ Where ߤ * = ሺ-ߦݏ . ߪ ௨ ଶ + ܼߣ. ߪ ௩ ଶ ሻ ߪ ௌ ଶ ⁄ , ߪ * = ߪ ௨ . ߪ ௩ ߪ ௌ ⁄ , ݏ = -1 if overemployed, and ݏ = 1 if underemployed
Distinguishing between farmers and employees, we obtain the distributions for inefficiency scores presented in graphs 1 and 2.

Both farmers and employees appear as strongly underemployed even if farmers are more constrained than employees (table 7).Farmers' shadow wages appear as being 3 times lower than their marginal revenue of labor, while market wages are 1.5 times higher than employees shadow wages. Given their hourly income, most of the Vietnamese workers in our sample would prefer to work more. Policy implications for employees are straightforward: job opportunities should be made more accessible, and developing road access is a promising tool. Concerning peasants, the link with farm size emphasized by Barrett seems to prove correct: increasing farm size tends to make peasants underemployed. Following Barrett's framework, this result highlights the importance of policies aiming at stabilizing producer prices in Vietnam: In fact, underemployed farmers in Barrett's model are net sellers and income risk averse: price uncertainty leads them to underemploy inputs and reduces output. Our results highlight factors participating to the stabilization of peasants' incomes. Indeed, irrigational networks facilitate farmers' job and make output more predictable. Credit access allows investments in new technologies which often increase the stability of agricultural output. Farmers far from their output market lose market power and can hardly find buyers for their production so that selling prices are unpredictable. Finally, as noticed earlier, the sensibility of farmer to their land endowment reveals their price risk aversion.

Price uncertainty is still the main difficulty faced by peasants in rural Vietnam even after controlling for other possible constraints. Allocative inefficiencies generated by price uncertainty significantly reduce the amount of opportunities peasants can grab. In a sense, price uncertainty prevents Vietnamese farmers to escape out of poverty. This result should bring policy leaders and economists to pay attention to price and income stabilization policies. Recently, Ghosh and Whalley (2004) demonstrated that price stabilization policies may be welfare improving for Vietnam. Other policy options such as the implementation of futures markets should be analyzed in order to propose the best policy mix for Vietnam. These questions are of particular importance in light of recent evolutions of agricultural prices on world markets.

Conclusions and policy implications

The implementation of development policies requires important adaptations to the specificities of the country under scrutiny. The aim of this paper is precisely to recover what are the main constraints faced by workers living in rural areas in Vietnam in order to propose a policy agenda focusing on these failures. Our point of view is that difficulties faced by workers will always influence their labor market equilibrium so that one can retrieve these market imperfections when analyzing individual labor market positions. Precisely, workers can be under or overemployed: they would like to work more or less respectively given their hourly income (market wage for employees and marginal revenue of labor for farmers). This article proposes to measure the gap between hourly income and individual shadow wages (also called opportunity cost of time or implicit wage) and explain it by variables representing possible difficulties that workers could face. The main difficulty of the approach is to obtain a measure of shadow wages against which we can compare observed hourly incomes (market wages or marginal revenue of labor). Assuming Cobb-Douglas preferences over consumption and leisure, we show that individual opportunity costs of time can be though as a stochastic frontier around which observed hourly incomes are distributed. Overemployed workers are located over the frontier while underemployed people stay below the frontier.

Estimation on a sample of 14,000 individuals in rural Vietnam shows that employees could benefit from the development of road network that help them to catch better job opportunities while farmers primarily suffer from price uncertainty. Risk averse farmers underemploy labor as demonstrated by Barrett (1996). Priority should thus be given to policies aiming at reducing income volatility in the agricultural sector. Different policy options must be considered. Ghosh and Whalley (2004) emphasize the welfare gains that could be obtained from a price stabilization policy.

Other instruments such as the implementation of agricultural futures markets (or the use of existing markets as proposed by [START_REF] Morgan | Agricultural futures markets in LDCs: a policy response to price volatility?[END_REF])) should be analyzed. Finally, the large share of underemployment found in our sample (96%) confirms the idea that labor demand could increase in Vietnam without strong pressures on wages, so that inflation could be limited. Also, given that most of Vietnamese workers are underemployed, the reduction of failures should increase the activity rate and be a source of growth.

  ݈ℎݏ = ܺߚ + ܵߜ + ݒ -ݑݏ Where ݈ℎݏ = lnሺ ܺሻ -lnሺ݈ݓሻ if our individual is an employee, ݈ℎݏ = ሾlnሺ ܺሻ -lnሺ݈ሻሿ + ൣlnሺℎሻ -ln൫ ொ ܳ൯൧ if our individual is self-employed, ܺߚ = ln ݂ሺݖሻ represents observed preferences heterogeneity, S=1 if the individual is a farmer and S=0 otherwise, ߜ = ln ߛ is the production function parameter which helps identifying the marginal revenue of labor without estimating a production function, ݒ = ln ߝ is random and represents unobserved preferences heterogeneity, ݑ = lnሺ݇ሻ ≥ 0 is our "inefficiency parameter" and ݏ = -1 if overemployed, and ݏ = 1 if underemployed.

  performed in two steps: First, a test between[START_REF] Lambert | Implicit prices and recursivity of agricultural household's decisions[END_REF] and Adesina, Barret and Sherlund (

  Inefficiency scores of employees were discovered in December 2003). Factors generating overemployment are the absence of national electricity network and low level of irrigation.

Table 6 : Barrett (1996) versus Lambert and Magnac (1998)

 6

	Preferences over
	consumption and
	leisure (Log of α/β)
	Production Function
	Parameter
	Employees'
	Inefficiency
	9,7%

Table 7 -

 7 Mean comparison test for inefficiency scores Robustness test: Include other possible constraintsThe objective of the paper is to propose and agenda of economic policies that could improve economic efficiency for rural areas in Vietnam. Other possible constraints are thus introduced. We follow the household and commune questionnaires and include output and input market access variables, credit constraint, meteorological and epidemic disasters, access to information and new technologies and quality of infrastructures such as irrigation and electricity networks.

		.8
		.6
	Density	.4
		.2
		0
		-6	-4	-2	0	2	4
				Log of inefficiency scores
						Underemployed	Overemployed
						Mean		Mean
					Obs erva tions	inefficiency	Standa rd Error Obs erva tions	inefficiency	Standa rd Error
						s core		s core
				Employees	3883	-0,458		0,004	183	0,873	0,172
				Fa rmers	10164	-1,121		0,007	86	1,415	0,052
				Di fference		0,663		-0,542
			H1: diff<0 (Pva lue)		1,0000		0,0000
	H0: diff=0	H1: diff≠0 (Pva lue)		0,0000		0,0000
			H1: diff>0 (Pva lue)		0,0000		1,0000
		4.5.2 Results presented in table 8 are consistent with those found in the previous section: the
	determinants of individual preferences still present the same coefficients, and estimated labor
	elasticity of production remains around 0.35. Inefficiency scores presented in graphs 3 and 4 remain
	similar to the previous estimations, and the weight attributed to price risk (land per head in our

estimations) remains the same. Furthermore, it remains the main contributor to underemployment of farmers. Other factors generating underemployment are credit constraint and meteorological or epidemic disasters (Vietnam hardly suffered from the avian flu in early twenties, and the first cases

Table 8 : Introducing determinants of allocative inefficiency

 8

	Dependent variable: lhs

Their approach is based onBarrett (1996) so that the two articles will be cited indifferently

Note that the shadow wage is supposed to equal the marginal revenue of labor as in[START_REF] Skoufias | Using shadow wages to estimate labor supply of agricultural households[END_REF]

The problem of unemployed people is usually handled by Heckman's selection technique.

Each maximization step comprises a maximum of 20 iterations and likelihood tolerance is fixed at 0.01.

Likelihood tolerance for maximization steps of the long run algorithm is fixed at 0.001 and the maximum number of steps is 25.

Convergence of the long run EM algorithm is declared if the improvement of the incomplete log-likelihood is less than 1.0E-06

This problem could be referred to a « nay-saying ». SeeCouch and Keniston (1960).

U-shape tests[START_REF] Lind | With or Without U? -The appropriate test for a U shaped relationship[END_REF] reject turning points for all variables.

Female 0,164 *** (0,015) hhold size -0,242 *** (0,016) hhold size ² 0,012 *** (0,001) nb of children -0,144 *** (0,014) nb of children ² 0,006 ** (0,003) nb of members aged 65+ 0,035 (0,037) nb of members age 65+ ² -0,019 (0,020) Attained Primary School -0,076 ** (0,032) Attained Lower Secondary School -0,022 (0,023) Attained Higher Secondary School -0,072 *** (0,021) Attained Post Secondary Education -0,076 (0,048) Vocational Training 0,022 (0,027) Constant -1,279 *** (0,074) Log of σ v ² -1,452 *** (0,076) Log of γ -0,988 *** (0,039) Labor Elasticity of Output "γ" Log of Distance to the nearest Road 0,255 *** (0,037) Log of σ u ² -0,947 *** (0,087) Inverse logit transformation of p o -2,228 *** (0,121) Proportion of overemployed workers Log of Land per hand -0,666 *** (0,030) Constant 4,938 *** (0,321) Log of σ u ² 0,204 *** (0,053) Inverse logit transformation of p o -3,970 *** (0,177)

Coefficient

Standard errors Age 0,045 *** (0,003) Age² -0,000 *** (0,000)

Appendix: Tables and Graphs