T A Zaleski
email: t.zaleski@int.pan.wroc.pl

T K Kopeć

Effect of next-nearest-neighbour hopping on Bose-Einstein condensation in optical lattices

Keywords: numbers: 67.85.Hj, 67.85.Bc, 03.75.Hh

We study the influence of the various hopping elements on the ground state properties of the bosons in the 2D optical square lattice. Using the quantum rotor approach for the Bose-Hubbard model with nearest t and next nearest-neighbours t hopping, we quantify to what extent changing the ratio t /t affects the superfluid to Mott-insulator transition.

Introduction

Optical lattices offer remarkably clean access to a particular Hamiltonian and, thereby, serve as a model system for testing fundamental theoretical concepts and providing an exemplar of quantum many-body effects [1]. It is well known that the ground state of a system of repulsively interacting bosons in a periodic potential can be either in a superfluid (SF) state or in a Mott-insulating (MI) state, characterized by integer boson densities [2]. A periodic lattice potential with tightly confining potential wells can be created by realising a dipole trap with superimposed counter propagating laser beams [3]. These beams interfere and the interference pattern results in a periodic potential landscape. For example, to form a two-dimensional lattice potential two standing waves can be superimposed orthogonal to each other. The strength of the tunnel coupling is characterized by the tunnel matrix element

t ij = -drw 0 (r -r i ) Ĥ0 w 0 (r -r j ) , ( 1
)
where Ĥ0 is the Hamiltonian of the system, which includes the periodic lattice potential. This element can be calculated by considering two Wannier functions of neighbouring lattice sites r i and r j :

w n (r -R i ) ≡ 1 N q e -iq•R i ψ nq (r) , (2)
where ψ nq (r) = u nq (r) e iq•r , with u nq (r) = u nq (r + a i ) being a periodic function, n is the band index and a i is a lattice vector. If a sample of sufficiently cold bosonic atoms is loaded into a deep enough lattice (e.g. from a Bose-Einstein condensate), then only states of the lowest band will be populated. Since, the Wannier state is well localized, t ij rapidly decreases with increasing relative separation |r ir j |. Usually, the second nearest neighbours have tunnelling rates that are 2-4 orders of magnitude smaller than those of the nearest neighbours and the tight-binding can be expected to provide an excellent description of the system in this regime. This represents the so-called tight binding limit, where a state at any given site only couples to a few neighbouring sites. However, it turns out that the Wannier state of the two-dimensional (2D) square lattice can be weakly localized [4] (see, Fig. 1) depending on the intensity of the lattice. We note that, even in 1D one can have regimes where next nearest-neighbour coupling is or is not relevant [5]. Therefore, it appears reasonable to quantify the influence of the next nearest-neighbour sites on the ground state properties of the bosons in the 2D optical lattice. Furthermore, due to the separability of the potential for the 2D square lattice, the tunnelling to sites lying on diagonals is identically equal to zero. As a consequence, the tunnelling matrix arrangement including nearest and next to nearest neighbours hopping is restricted to the lattice axes, as shown in Fig. 1.

Model

We start from a second quantized purely bosonic Hubbard Hamiltonian:

H = -t r,r a † (r) a (r ) + a † (r ) a (r) -t r,r a † (r) a (r ) + a † (r ) a (r) + U 2 r n 2 (r) -µ r n (r) , (3)
where, a † (r) and a (r ) are bosonic creation and annihilation operators defined on sites r and r of a regular square lattice. The summations are performed over the nearest . . . and the next-nearest neighbours . . . , respectively with hoppings t and t (see, Fig. 1). A dispersion relation of the lattice is:

ε (k) = 2t (cos k x + cos k y ) + 2t (cos 2k x + cos 2k y ) . (4)
Furthermore, U is the on-site repulsion energy, n (r) = a † (r) a (r) represents the boson number operator, while µ = µ + U 2 , with µ being a chemical potential controlling the average number of bosons. The Bose-Hubbard model in Eq. (3) has been studied intensively for a number of years [6,[START_REF] Sachdev | Quantum Phase Transitions[END_REF] and recently has been applied to bosons in optical lattices [START_REF] Jaksch | [END_REF]9]. A number of approaches has been applied to investigate the physics of the model like mean-field theories [10], strong-coupling expansion [11,12], coarse graining [13], methods based on a systematic strong-coupling approach, of the Bose-Hubbard model, but going beyond mean field model [15,16], or numerical once, e.g. quantum Monte Carlo [17,18,19].

Quantum rotor representation

In the following, we switch from the particle-number representation to the conjugate phase representation of the bosonic degrees of freedom which constitutes an effective bosonic description of the strongly correlated system [16]. To this aim the secondquantized Hamiltonian of the model is translated to the phase representation with the help of the topologically constrained path integral formalism. A similar method that is based on quantum rotor formulation was recently employed by one of us in the fermionic Hubbard model [20]. The best merit of this approach is that it is capable of handling spatial and quantum fluctuation effects properly and complying with Mermin-Wagner theorem [21]. It also gives the results that are in a very good agreement with Quantum Monte Carlo method (see, Fig. 2 in Ref. [16]).

The statistical sum can be written in a path integral form with use of customary complex fields, a (rτ ) depending on the "imaginary time" 0 ≤ τ ≤ β ≡ 1/k B T , (with T being the temperature) that satisfy the periodic condition a (rτ ) = a(rτ + β):

Z = [DaDa] e -S[a,a] , (5)
where the action S is equal to:

S [a, a] = r β 0 dτ a (rτ ) ∂ ∂τ a (rτ ) + H (τ ) . (6)
Next, we performing a local gauge transformation to new bosonic variables:

a (rτ ) a (rτ ) = e iφ(rτ ) 0 0 e -iφ(rτ ) b (rτ ) b (rτ ) . (7)
The U(1) group governing the phase field φ (rτ ) is compact, i.e. φ (rτ ) has the topology of a circle S 1 . The chief merit of the transformation in Eq. ( 7) is that we have managed to cast the strongly correlated bosonic problem into a system of weakly interacting bosons, submerged in the bath of strongly fluctuating U(1) gauge potentials on the high energy scale set by U . In the phase-only representation, the statistical sum reads:

Z = [Dφ] e -S ph [φ] (8)
with the action:

S ph [φ] = β 0 dτ r φ2 (rτ ) 2U + i µ U φ (rτ ) -J r,r cos [φ (rτ ) -φ (r τ )] -J r,r cos [φ (rτ ) -φ (r τ )] (9)
where φ (rτ ) = ∂φ (rτ ) /∂τ . We parametrize the boson fields as follows

b (rτ ) = b 0 + b (rτ ) ( 1 0 )
and restrict our calculations to the fixed amplitude b 0 thus, dropping the amplitude fluctuation term b (rτ ). This is justified by the observation that in the strong coupling limit t/U 1 the physics of the problem under study is dominated by the behaviour in the U(1) sector. As a consequence, J = |b 0 | 2 t, J = |b 0 | 2 t and the integration over phase variables is restricted over the compact U(1) group manifold [START_REF] Schulman | Techniques and Applications of Path Integration[END_REF]. In performing this integration one should take phase configurations that satisfy the boundary condition

φ (rβ) -φ (r0) = 2πm (r) ( 1 1 )
and m (r) = 0, ±1, ±2, . . .. By the definition, for the superfluid transition the order parameter reads:

Ψ B = a (rτ ) = b (rτ ) ψ B . (12)
Its non-vanishing signals a bosonic condensation (we identify it as superfluid state). Note that a nonzero value of the amplitude b (rτ ) is not sufficient for superfluidity.

To achieve this, also the phase variables, must become stiff and coherent, which implies that ψ B defined as:

ψ B = e iφ(rτ ) ph ( 13
)
is non-vanishing. The averages appearing in Eqs. ( 12) and ( 13) are as follows:

. . . = DξDξ . . . e -S[ξ,ξ] DξDξ e -S[ξ,ξ] , (14)
where ξ denotes a or b fields, respectively,

. . . ph = [Dφ] . . . e -S ph [φ] [Dφ] e -S ph [φ] (15)
and the action S ph [φ] given in Eq. ( 9).

Effect of next-nearest-neighbour hopping on Bose-Einstein condensation in optical lattices5

The value of the boson amplitude b 0 can be obtained from saddle-point condition: Using the parametrization in Eq. ( 10) we obtain the action :

S b 0 , b 0 = Nβ -(zt + z t ) |b 0 | 2 -µ |b 0 | 2 + U 2 |b 0 | 4 . ( 17
)
So that, the amplitude of the bosonic field reads:

|b 0 | 2 = 2 zt + z t U + µ U , (18)
where z and z are number of nearest neighbours and next-nearest neighbours, respectively, for a given lattice (in our case, z = z = 4). The definition in Eq. ( 13) of the order parameter suggests that the description in terms of the unimodular complex field ψ (rτ ) = e iφ(rτ ) (|ψ (rτ )| 2 = 1) would be more desirable than working a with trigonometric action involving the phase variable φ (rτ ). Furthermore, we substitute the rigid unimodular constraint |ψ (rτ )| 2 = 1 for each site r by the weaker spherical closure relation, which maps the problem onto the soluble quantum spherical (QS) model [16]. The convenient way to enforce the spherical constraint is the functional analog of the δ-function representation δ (x) = +∞ -∞ (dλ/2πi) e λx , which introduces the Lagrange multiplier λ. The evaluation of the effective action in terms of the ψ (rτ ) fields may be organized using the loop method [START_REF] Kopeć | [END_REF]. To the second order in ψ (rτ ) we obtain the partition function of the effective QS model:

Z = 1 2πi +i∞ -i∞ dλ DψDψ e -S[ψ,ψ] , (19)
with the Fourier transformed action:

S ψ, ψ = - 1 Nβ k J (k) -γ -1 (ω ) -λ ψ (kω ) ψ (kω ) , (20)
where the Bose-Matsubara frequencies are ω = 2π /β with = 0, ±1, ±2, . . .. Here, the dispersion for the nearest and the next-nearest neighbours phase stiffness parameters is given by

J (k) = 2J (cos k x + cos k y ) + 2J (cos 2k x + cos 2k y ) , (21)
where J = t |b 0 | 2 and J = t |b 0 | 2 with b 0 given in Eq. (18). Furthermore, γ (ω ) is Fourier transformed phase correlation function:

γ (rτ ; r τ ) = e -i[φ(rτ )-φ(r τ )] ph , (22)
which explicit form is given in Ref. [16].

Effect of next-nearest-neighbour hopping on Bose-Einstein condensation in optical lattices6

Results

Considering the problem at the mean-field level, it is obvious from Eq. ( 18) that the effect of t is a simple correction to the direct hopping t. However, as already mentioned in a strongly correlated case t U the actual superfluid order is determined by the phase coherence among bosons rather than the non-vanishing amplitude of the Bose field. Therefore, going beyond the mean-field result using quantum rotor method it is not a priori clear what the influence of t will be, especially for a two-dimensional system, where phase fluctuations are significant. As usual, in the quantum rotor model calculations the critical boundary is determined from the condition of vanishing of the order parameter susceptibility G -1 (k = 0, ω = 0) = 0, which defines the value of the Lagrange multiplier λ 0 at criticallity. Here,

G -1 (kω ) = λ 0 -J (k) + γ -1 (ω ) . (23)
The "equation of state" with the order parameter |ψ B | 2 reads:

1 -|ψ B | 2 = 1 βN k G (kω ) . (24)
Because in two-dimensions a long-range order is expected only in the ground state we have to specialize in the zero temperature limit β → ∞ in Eq. ( 24). We obtain:

1 -|ψ B | 2 = 1 N +∞ -∞ dερ (ε) 2 λ 0 -t|b 0 | 2 ε U + 1 4 , (25)
where the critical value of the Lagrange multiplier is:

λ 0 = J (0) + γ -1 (0) = |b 0 | 2 ε (0) - U 4 + v 2 µ U (26)
with a periodic function v [x] = x -[x] -1/2 resulting from a boundary condition in Eq. (11), where [x] is the floor function giving the greatest integer less than or equal to x. In order to perform summation over wave vectors in equation of state in Eq. ( 24), it is desirable to introduce a density of states of the 2D lattice with the nearest t and next-nearest t neighbours hopping:

ρ ( ) = 1 N k δ - ε (k) t , (27)
which can be conveniently written as a convolution involving one dimensional densities of states:

ρ ( ) = +∞ -∞ dxΛ (x) Λ ( -x) , (28)
where Lobes for various values of the nearest neighbours t to the next-nearest neighbours hopping t ratios (from the lightest to the darkest shading: t /t = 0, -0.016, -0.1, -0.2). MI denotes a Mott Insulator phase, while SF is a superfluid region. The lower inset shows details of the t = 0 and t /t = -0.016 lobes.

Λ (x) = Θ [ζ (x)] ρ 1D 1- √ ζ(x) 4r + ρ 1D 1+ √ ζ(x) 4r ζ (x) (29
The upper inset shows the lattice density of states calculated from the dispersion in Eq. ( 4) and Eq. ( 28) for the respective ratios of t /t.

with η = t /t, furthermore ζ (x) = 1 + 8r (x + r), Θ(x) is a step function and the one-dimensional density of states is given by ρ 1D (x) = (1/π) Θ (1 -|x|) / √ 1 -x 2 . For t /t = 0 we recover the usual density of state for the 2D lattice with the logarithmic singularity at = 0. For t > 0, the density of states given by Eq. ( 28) is rather uncommon e.g. in solid state physics applications, because usually the term next nearestneighbours hopping refers to the interactions along diagonals of the square lattice, which are vanishing in the present case. In the MI phase, bosons are incompressible ∂n B /∂µ = 0 and localized, which means that the total energy is minimized when each site is filled with the same number of atoms. Increasing fluctuations in the phase system reduces fluctuations in the boson number on each site according to the Heisenberg uncertainty relation ∆n B ∆φ ≥ 1/2. By crossing the boundary line, bosons can move from one lattice site to the next. The order parameter Ψ B has a non-vanishing value, and the system exhibits a long-range phase coherence. This is opposite to the case of the Mott insulator, where phase coherence is lost. In Fig. 2 we plotted zero-temperature phase diagram as a function of normalized hopping t/U vs. normalized chemical potential µ/U for various values of t /t hopping parameter. The phase boundary is periodic with the chemical potential µ/U and the filling is fixed to the integer values within the lobes. For square lattice, numerical investigation of the band structure and explicit calculation of the Wannier states give the negative sign of t /t ratio with value of t /t = -0.016 (t being 1.6% of t, see, Table 1 in Ref. [4]. In our calculations, it leads to an increase of the size of the lobes by approximately 3% with respect to the pure 2D case with the nearest neighbours interactions only. Also, the shape of the lobes changes with increasing t hopping. In general, the inclusion of the next-nearest neighbours suppresses the superfluid state (see, Fig. 2). From Ref. [4] it follows that for other type of lattices like hexagonal or rectangular the respective absolute value of t /t is even smaller than in the present case, which suggest that the prospective effects on the phase diagram will be even less pronounced.

Conclusions

In this work, we have investigated two-dimensional lattices with nearest and next-nearest neighbours hopping that may be used in experiments with Bose condensation of cold atoms. We have shown that lattices with next-nearest neighbours hopping could affect the ground state interaction -chemical potential phase diagram of the system, which consist of Mott insulating lobes and superfluid region by suppressing the superfluid state. In a view towards promising future developments, it would be interesting to consider experimental realization of the presented scenario by means of two-dimensional optical lattices with changing lattice potential depth.

  energy per lattice site f = -1 βN ln DbDbe -S[b,b] .

Figure 2 .

 2 Figure 2. (Colour online) Zero-temperature phase diagram of the system as a function of hopping and chemical potential.Lobes for various values of the nearest neighbours t to the next-nearest neighbours hopping t ratios (from the lightest to the darkest shading: t /t = 0, -0.016, -0.1, -0.2). MI denotes a Mott Insulator phase, while SF is a superfluid region. The lower inset shows details of the t = 0 and t /t = -0.016 lobes. The upper inset shows the lattice density of states calculated from the dispersion in Eq. (4) and Eq. (28) for the respective ratios of t /t.

Confidential: not for distribution. Submitted to IOP Publishing for peer review 17 February 2010

  Figure 1. (Colour online) Two-dimensional square lattice with the nearest neighbours t and the next-nearest neighbours t hopping. Shaded areas denote Wannier state density distribution based on data from Ref. [4].

	t' t'
	t t

Acknowledgments

Present work is supported from scientific financial resources in the years 2009-2012 as a research grant.


Arnaud Z Dragicevic
email: arnaud.dragicevic@cirano.qc.ca.

Jim Engle-Warnick

Jason Shogren

Bernard Sinclair

Anne Boring

Damien Bosc

Loss of Substitutability: A Note On Disparity between WTP and WTA

Keywords: Disparité CAP-CAR, effet de substitution, aversion aux pertes WTP-WTA gap, substitutability, reference-dependence. Codes JEL : D61, D81, Q51

sans but lucratif constitué en vertu de la

I. INTRODUCTION

Experiments have given evidence of large disparities between willingness-to-pay (WTP) and willingness-to-accept (WTA) and laboratory markets confirmed persistency in disparities [START_REF] Knetsch | Willingness to Pay and Compensation Demanded: Experimental Evidence of an Unexpected Disparity in Measures of Value[END_REF][START_REF] Brookshire | Measuring the Value of a Public Good: An Empirical Comparison of Elicitation Procedures[END_REF][START_REF] Shogren | Resolving Differences in Willingness to Pay and Willingness to Accept[END_REF]. To justify the persistence of the disparity, theorists invoke either the substitution effect [START_REF] Hanemann | Willingness to Pay and Willingness to Accept: How Much Can they Differ?[END_REF] or the endowment effect [START_REF] Kahneman | Prospect Theory: An Analysis of Decisions Under Risk[END_REF][START_REF] Thaler | Toward a Positive Theory of Consumer Choice[END_REF]. The effects work with changes in the qualities or changes in the fixed quantities of both public and private goods. The substitution effect results from the agent's imperfect trade-off between the goods. When a good has an imperfect substitute, the disparity between indices increases as the degree of substitution decreases [START_REF] Shogren | Resolving Differences in Willingness to Pay and Willingness to Accept[END_REF]. The endowment effect makes agents value losses higher than equivalent gains. Morrison (1997) tries to reconcile the effects by combining them. [START_REF] Hanemann | The Economic Theory of WTP and WTA[END_REF] suggests that a quasi-concave utility function with inversely proportional disparity to the substitution effect can both explain the standard WTP and WTA disparity as well as the gain and loss disparity.

This paper considers the debate on substitutability between public and private goods (or income). We define the substitution effect as an opportunity loss and the endowment effect as a loss of substitutability. We make both effects depend on substitutability. This demarche unfolds an amalgam. Substitutability and loss aversion do not consider the same type of WTA. As we transpose them into a common setting of reference-dependence, we find that compensation demanded for the loss of substitutability is respectively unbounded in the neoclassical framework and bounded with loss aversion. The concepts of opportunity loss and loss of substitutability are introduced in Section 1. We show the properties of boundedness in Section 2. Concluding comments are given in Section 3.

II. OPPORTUNITY LOSS AND LOSS OF SUBSTITUTABILITY

Given a preference ( , ) u x q where q and the x'sbought by the income yrepresent the public and private goods (or income), WTP + for q is defined as the maximum amount an agent is willing to pay to guarantee the improvement of the public goods' level from 0 q to 1 q 01 () qq  ; likewise, WTA + for q is the minimum amount an agent is willing to receive to forego the improvement from 1 q to 0 q . The symmetric reasoning is WTP -to avoid the public goods' degradation and WTA -to tolerate it. Combinations of the x's (via y) and q along the indifference curves being imperfectly substitutable, agents display convex preferences and WTP + < WTA + (Figure 1).

Fig. 1 Substitutability between q and the x's

In the usual neoclassical framework, preferences are independent of initial endowments. That is why [START_REF] Hanemann | Willingness to Pay and Willingness to Accept: How Much Can they Differ?[END_REF] points out that [START_REF] Kahneman | Prospect Theory: An Analysis of Decisions Under Risk[END_REF] loss aversion differs from Hicksian preferences (footnote 25). To address this issue, [START_REF] Bateman | A Test of the Theory of Reference-Dependent Preferences[END_REF] incorporate reference points into the neoclassical framework. Their work permits identifying the equivalent gain (WTA + ), which is the minimum amount an agent is willing to receive to sacrifice a foregone gain 10 (qq    where 0  is the change from the status quo level 0 q ) and the compensating q q 1 q 0 0 y + WTP + WTA y 3 loss (WTA -), which is the minimum amount an agent is willing to receive to tolerate a net loss 10 () qq    . Unlike the standard disparity where the changes go in the same direction ( or )  , the gain and loss disparity is the study of changes that depart in opposite directions ( and )  . An endowed agent introduces a reference point and shifts her position on the map, such that the shape of her indifference curve is altered. The income or wealth effectthe spacing between the indifference curvesdoes not count, for the gain and loss perspective involves a single curve observed from some positive or negative shift. Figure 2 illustrates the welfare indices observed from the reference point coordinates 0 ( , ) qy of the utility arguments. The grey curve depicts some pre-endowed utility in q and the x's (via y). The inclusion of contextdependence changes the initial utility to either a gain in utility ()  or a loss in utility ()  .

In the behavioral framework of loss aversion, [START_REF] Tversky | Loss Aversion in Riskless Choice: A Reference-Dependent Model[END_REF] When the agent values a true loss and weights it against an equivalent gain, the loss in consideration is a net loss. The net loss is a critical change, for agents take account of the goods they can no longer substitute; they explicitly value this loss of substitutability by shifting preferences, which makes them switch to a steeper indifference curve. The lower the ex ante substitutability between the public and private goods (or income), the higher the slope of both the 1 In loss aversion, the isolated consideration of the nonlinear value curve reveals the opportunity loss. [START_REF] Randall | Consumer's Surplus in Commodity Space[END_REF] demonstrate that the standard disparity is bounded by the ratio between the price flexibility of income and endowment. [START_REF] Cook | The Demand for Insurance and Protection: The Case of Irreplaceable Commodities[END_REF] assert that compensation demanded for irreplaceable commodities, which we assume to be imperfectly substitutable, depends on the initial level of wealth or endowment. As the probability of loss approaches certainty, WTA -tends to infinity. [START_REF] Amiran | Willingness To Pay and Willingness To Accept: How Much Can They Differ? Comment[END_REF] suggest that the high level of wealth produces an infinite WTA + for irreversible losses of public goods (WTA -de facto) when the utility function is asymptomatically bounded. They show that the income elasticity of the inverse compensated demand is bounded above and below by positive values independent of the public goods' amount. Even if the substitution effect plays a key role, for it induces frictional trade-offs between the public and private goods (or income), their referenceindependent model can only speak for opportunity losses, but surely not for net losses.

III. LOSS OF SUBSTITUTABILITY AND BOUNDEDNESS

In the neoclassical framework with reference-dependent preferences, we claim that the loss of substitutability, which induces the shifting on the map, implicates unbounded compensation demanded (WTA - )  . In order to prove our claim, we replace the nonsatiation assumption by the following assumption of preferences over reference points.

ASSUMPTION. For any level of income y that buys the x's, the status quo q r is strictly preferred to the net loss of public goods

0 q rq   
where >0  . Formally, the assumption verifies the following strict inequality (2)

q 0 WTA + WTA - (1)
(2) q r 0 q y r y PROPOSITION 1. Within the neoclassical framework with reference-dependence, the loss of substitutability between the imperfectly substitutable public and private goods (or income) induces unbounded compensation demanded.

PROOF IN THE APPENDIX.

The limit of WTA -is due to the pivoting of the convex indifference curve.2 Contrary to [START_REF] Cook | The Demand for Insurance and Protection: The Case of Irreplaceable Commodities[END_REF] and similarly to [START_REF] Amiran | Willingness To Pay and Willingness To Accept: How Much Can They Differ? Comment[END_REF], our curve is asymptotic at critical levels of losses around 0 q .

Despite the preceding models which consider substitutability as the mainspring for infinite numéraire compensation, our design neither depends on the initial level of wealth, nor the initial endowment in market goods, nor on the boundedness of the utility function. It simply depends on the opportunity loss when goods are imperfectly substitutable and on the loss of opportunity to substitute between the goods.

The same rationale of context-dependence within loss aversion produces a different result.

The agent substitutes the loss of public goods with monetary compensation above the kink point; below is the opposite. At the kink point coordinates ( , ) qy rr the agent is equidistant to both references points and perfectly indifferent between the public and private goods (or income). Any other point along the nonlinear curve exhibits some opportunity loss. The loss of substitutability, expressed in terms of compensation demanded, is captured by the segment above the kink point.

From Figure 3, we see that the marginal valuation exhibits a diminishing sensitivity in loss aversion. We have ( ) / 0 q q q r r v r       as < qq rr  .

(3) When q r  goes further from q r , additional reductions in q lead to smaller changes in the value of the value function. In other terms, the diminishing sensitivity implies a diminishing effect of the loss of substitutability. In opposition to the increasing slope for greater losses in the neoclassical framework, the loss in value of the value function from the public goods' downgrading decreases as the agent moves away from the reference point; this implies a bounded WTA -value for the loss of substitutability.

PROPOSITION 2. Given the constant diminishing sensitivity with loss aversion, compensation demanded for the loss of substitutability between the imperfectly substitutable public and private goods (or income) is bounded.

PROOF IN THE APPENDIX.

There is a difference between the neoclassical framework and loss aversion in the formal representation of loss of substitutability. If we superpose the segments illustrating WTA -, their respective curvatures reveal two types of behavior (Figure 5). Endowed neoclassical agents represented by the grey segment manifest an increasing marginal disutility as 0 ()

q rq    .
Inside loss aversion represented by the black segment, agents exhibit high loss aversion with small losses, but they turn out to be less sensitive with greater losses, for marginal valuation is subjected to diminishing sensitivity. The further something moves from a reference point, the less additional changes matter, which here means limited frictions in the loss of trade-off. Loss averse agents end up asking for bounded compensation. The implications of our results are new. In case of loss of substitutability between the public and private goods (or income)commodities that we know to be imperfectly substitutable or inclined to the effect of opportunity lossendowed neoclassical agents demand unbounded compensation, whereas loss averse agents ask for bounded compensation. This works as an anchoring effect or the retention of the status quo and occurs because neither adapt their reference points. While psychologists consider loss aversion to invalidate agents' rational preferences, we see that past some level of changes in the public goods' level, the perceptual sensitivity of loss of substitutability limits their proclivity toward exaggerated valuation of losses.

IV. CONCLUSIONS

The concept of opportunity loss clarifies the standard disparity from Hanemann's analysis. In case the scenario is a true loss instead of a foregone gain, the opportunity loss becomes a loss of substitutability, enlarging the initial disparity, for agents heavily value things they cannot substitute any longer. Experimental findings from [START_REF] Boyce | An Experimental Examination of Intrinsic Values as a Source of the WTA-WTP Disparity[END_REF] and [START_REF] Chapman | Similarity and Reluctance to Trade[END_REF] support this claim. Like in [START_REF] Hanemann | Willingness to Pay and Willingness to Accept: How Much Can they Differ?[END_REF] and [START_REF] Amiran | Willingness To Pay and Willingness To Accept: How Much Can They Differ? Comment[END_REF] our results show that q 0 WTA - y q r y r 0 q

  introduce a value function where agents have different preferences over gains and losses relative to their status quo. Figure3illustrates welfare measures from the loss aversion perspective. The reference point coordinates ( , ) qx rr stand for the initial use of public goods q and the initial consumption of private goods x (via y). Their model explains the source of the gain and loss disparity. Although the authors assert that the disparity between willingness-to-pay and willingness-to-accept is an implication of loss aversion, what they really compare is WTP + with WTA -. For that reason, their design cannot interpret the notion of foregone gain, nor can explain the suggested disparity between WTP + and WTA + .

Fig. 2

 2 Fig. 2 Reference-dependent preferences Fig. 3 Welfare measures in loss aversion

  indifference curves. InTversky and Kahneman's model, this pertains to the loss aversion segment of the value curve. In order to make a comparison between the models, one must introduce reference-dependence. The second definition comes out. DEFINITION 2. The shift in preferences, caused by the presence of reference-dependence, expresses a loss of substitutability between the goods.

Figure 4

 4 Figure4shows that the argument of substitutability intervenes twice. Outcome (1) lies in the

Fig. 5

 5 Fig. 5 Loss of substitutability

Although this result is comparable to that ofMorrison (1997), we consider the loss of substitutability rather than the endowment effect and focus on the properties of boundedness.

d'une subvention d'infrastructure CIRANO is a private non-profit organization incorporated under the Québec Companies Act. Its infrastructure and research activities are funded through fees paid by member organizations, an infrastructure grant from the Ministère du Développement économique et régional et de la Recherche, and grants and research mandates obtained by its research teams.

the unbounded compensation demanded arises from the impossibility to substitute public with private goods (or income). However, the results clarify the real stake of the nature of welfare measures, without making functional restrictions. At last, we observe that loss aversion produces bounded compensation demanded. This proposition should be tested in a laboratory. Whether agents have unbounded or bounded values for the public goods' losses could answer the longstanding question of which model best values the preferences over public goods.

APPENDIX

Proof of Proposition 1.

for a level of monetary compensation such that the utility remains constant. For each 0 q  we have ( ) ( , ) z q u q y  . The supremum () zq is increasing in q.

This says that for each level of income y and for 0 q rq  we have ( , ) ( , ) qq u r y u r y    because the status quo is always preferred to the net loss of the public good  .

Let us set ( ) (

where s corresponds to WTA -. With () q zr being the supremum for ( , ) q u r y and () q zr being the supremum for ( , ) q u r y  is there y that gives 0 ( , ) > ( ) q u q y z r  or ( , ) > ( ) qq u r y z r s    ?

We know that 0 qq q r r     so for all y we have 0 ( ) ( ) q z q z r  and 0 ( ) ( )

q u r y z q s  . We conclude that 0 ( , ) < ( ) ( , ) qq u q y z r s u r y     .

Proof of Proposition 2. v is nonincreasing and concave. We have ( ) ( ) ( ) ( )

expression of the inequality is the tangent of v at q r . It gives ( ) ( ) ( ) ( ) q q q q v r v r v r r       when 0 q r    which is independent of q r . The value function is bounded.


Arnaud Z Dragicevic
email: arnaud.dragicevic@cirano.qc.ca.

Meunier Guy
email: guy.meunier@gis-larsen.org.

Guy Meunier

Isabelle Holland

Competitive Private Supply of Public Goods

Keywords: provision privée des biens publics, déculpabilisation, statut social, compétition, transfert de revenu, subventions public good private supply, guilt relieving, social status, competition, income transfer, subsidies A13, C7, H2, H41

come

Introduction

The voluntary offset market enables agents to pay for their negative externalities issued from carbon emissions by investing in projects that reduce emissions or sequester carbon, such as tree planting or renewable energy. The reduction of carbon emissions is a public good because, once provided, agents can enjoy the benefits devoid of rivalry, without excluding anyone from its consumption. Some people believe that the voluntary carbon offset market is inefficient. One of the arguments put forward is that offsetting validates polluting behavior. Likewise, offsetting is said to operate like charities: voluntary supplies never provide enough public good because of the free-rider incentive. And when private arrangements finance a public good, free-riding on other people's provisions is rational.

However, free-riding is limited to some extent because agents who purchase offsets may also derive private benefits. [START_REF] Olson | The Logic of Collective Action: Public Goods and the Theory of Groups[END_REF] advances the hypothesis that free-riding can be overcome through social incentives. According to him, agents do not privately supply a public good for its direct material benefit, but to achieve social objectives like prestige or respect; this would explain why individuals do less free-riding than what the economic theory suggests. Following this rationale, [START_REF] Hawkes | Why Hunter-Gatherers Work: An Ancient Version of the Problem of Public Goods [and Comments and Reply[END_REF] show that in ancient times hunters and gatherers tended to share their resources because the cost of exclusion from the groupwhere every agent prefers a supplier to a consumer as a neighborwas too high to risk, thus making resources a public good. This impure approach of pro-social behavior has been modeled by [START_REF] Andreoni | Impure Altruism and Donations to Public Goods: A Theory of Warm-Glow Giving?[END_REF] who justifies private provisions in terms of warm-glow or joy-of-giving. Our approach differs from Andreoni's and rejoins Olson's, for we consider social status gained by agents who privately supply a public good from its relative perspective. As a matter of fact, supplying to the public good can generate the private benefits of guilt reliefwhich we find more convincing than warm-glowand social status 1 . In the first case, agents want to feel better about themselves, because they want to recover self-esteem after producing a public bad. Indeed, guilt is the source of altruistic acts in subjects who cause harm [START_REF] Regan | Guilt, Perceived Injustice, and Altruistic BehaviorCompetitive altruism: From reciprocity to the handicap principle[END_REF]. If an agent feels guilty, because she believes she bears responsibility for carbon excesses, then guilt alleviation through carbon offsetting is a private benefit derived from the supply of the public good2 . The motivation for it is internal. It is thus an intrinsic incentive. Previous work has indicated that guilt can significantly increase an individual's likelihood of engaging in charitable behavior [START_REF] Strahilevitz | Donations to Charity as Purchase Incentives: How Well They Work May Depend on What You are Trying to Sell[END_REF]. A major element in generating charitable donations through the use of guilt appeals instills a sense of responsibility to help [START_REF] Basil | Guilt Appeals: The Effects of Responsibility and Altruistic Norms[END_REF]. Responsibility may stem from causing something to occur or from failing to avoid the arrival of some occurrence [START_REF] Miceli | How To Make Someone Feel Guilty: Strategies of Guilt Inducement and Their Goals[END_REF]). Since guilt is positively related to donation [START_REF] Hibbert | Guilt Appeals: Persuasion Knowledge and Charitable Giving[END_REF], guilt alleviation has a positive impact on the environmental awareness.

In parallel, agents compete to be formally acknowledged as being the most concerned about the public good. This prosocial behavior can be due to social pressure and norms and corresponds to an extrinsic incentive. An agent who offsets receives a proof acknowledging her provision to the public good. She thus sends a signal to make other agents aware of her polluting abatement. Following this rationale, producers will also promote their offsets as part of their corporate social responsibility policy [START_REF] Kotchen | Offsetting Green Guilt[END_REF]. People have a preference for showing altruism in situations that facilitate broadcast opportunities, and the provision of a public good is certainly one such situation (Smith and Bliege Bird 2000). If high status brings with it high earnings, then status seeking behavior can be explained as a part of economic behavior [START_REF] Ball | The Economic Value of Status[END_REF]. According to competitive altruism, despite the dearness of being publicly generous, agents can promote their generosity as potential exchange partners, reaping the benefits later on [START_REF] Roberts | Competitive altruism: From reciprocity to the handicap principle[END_REF]. Agents also refuse transactions that are in their best economic interest when they feel they are an insult to their dignity [START_REF] Bénabou | Incentives and Prosocial Behavior[END_REF]. Experimental literature has confirmed the role of individual status as an incentive affecting market outcomes [START_REF] Ball | Status in Markets[END_REF]donors (Duffy andKornienko 2005). Because of the rivalry and excludability in social 1 Neither effect is exclusive, feelings being rarely disconnected. We just specify the domination of a motivation over another.

hierarchy, agents have to compete before attaining some desired social status: if an agent desires to be the first or among the first in some venture, she might have to make the most efforts to reach her goal. Making the most efforts means that she has knowledge of her challengers and of the efforts she has to invest. De facto, what type of incentives should be introduced to increase private provisions? How does competition influence an agent's supply of a public good? Do agents become more generous by guilt or by craving for social status? For example, competitive mechanisms such as contests have shown to increase the voluntary provision of a public good [START_REF] Kolmar | Contests and the Private Provision of Public Goods[END_REF].

This paper investigates how competition influences private provisions of the public good when agents are stirred by an intrinsic impulse, meaning that they mainly maximize utility from guilt relief, as opposed to when they are stirred by an extrinsic impulse, suggesting that they mainly maximize utility from social status. The paper answers the current call for understanding the behavior of individual consumers in the voluntary offset market (House of Commons 2007) and more generally how private agents behave toward public goods in a competitive setting. Our public goods game unveils several results: first, we find that when status seeking dominates guilt relief, private provisions become strategic complements: an attribute which increases the aggregate level of the public good. We prove sufficient conditions for existence and uniqueness of a Nash equilibrium. We then show that when agents behave according to their best-response functions, the aggregate level of the public good depends on the disparity between agents' incomes, whichdepending on the nature of the provisionsinduces a particular income transfer policy. At last, we prove that in case of see-through, subsidies have an ambiguous impact on the aggregate level of the public good.

We give a basic account of the social status function and present the public goods game in Section 2. In Section 3, we analyze the impact of a fiscal policy on the best response functions.

Section 4 discusses the social optimum characteristics. We provide a representative model with logarithmic preferences and describe explicit properties of equilibria in Section 5. Section 6 concludes.

The public goods game

Let us consider two agents i and j, with ji  . Let i w be agent i's endowment, let i x denote her consumption of the private good, let G be the aggregate level of public good and let i g account for her provision to the public good. The aggregate level of public good is the sum of the two agents' provisions ij G g g . The relief of an agent is not exclusory with regard to the other agent. Private contributions due to guilt are thus additive. On the other side, agent i's social status3 is determined by her relative contribution iij s g g . Indeed, as reported by [START_REF] Auriol | Status and Incentives[END_REF], social status is a scarce resource: increasing an agent's status requires that another agent's status decreases. Agents now have preferences represented by the following utility function ( , , )

i i i i u u x G s  (1)
Considering agent j's provision j g as exogenous, agent i maximizes her utility by solving the following program , max ( , , )

ii i i i xg u x G s subject to i i i
x g w  and 0

i g  (1')
Let us now determine the Nash equilibrium of the public goods game. Each agent's bestresponse function fully specifies her equilibrium strategy. This strategy involves choosing a level of private supply to the public good. We first analyze the best response functions of each agent.

We thus study the two motives for contributing to the public good: to relieve guilt and to acquire social status.

Assume the marginal utility from the provision to the public good to be ( , , )

i i i i i i ii u u u H x G s G s x          (2)
The first term denotes the marginal utility from the public good. The second term represents the marginal efficacy of social status. The last term is the marginal fall in the consumption of private goods. We then make three assumptions on H .

2 2 2 2 0 i i i i i i i i i H u u u x x G x s x               (A1) (A1)
0 i i i i i i i H u u u G G G s G x               (A2)
(A2) states that the marginal utility of the supply of the public good decreases with G. As a matter of fact, if the level of the public good increases independently of agent i's supply, there is no incentive to contribute to the public good. This is a formal foundation for the free-riding issue.

Considering negative externalities, it simply means that any agent can compensate for the damage caused, and all agents can profit from its reparation4 .

2 2 2 2 0 i i i i i i i i i H u u u s s G s s x               (A3)
(A3) implies that an increase in social status creates negative incentives: the agent tends to reduce her supply to the public good, because she no longer has to compete for social status.

Given the three assumptions and following the work by [START_REF] Andreoni | Impure Altruism and Donations to Public Goods: A Theory of Warm-Glow Giving?[END_REF], we now consider that individuals obtain guilt relief and social status from their private supply of the public good. Following the first order condition, agent i's best response, that is,  

, i i i r w g , is to have i g such as ( , , )
0 i i i i j i j r H w g g g g g      (3)
A Nash equilibrium of the public goods game is a couple of strategies

i i i j g r w g  with ji  (4)
Let us now look at the second order condition to see whether contributing to the public good does in fact maximize an agent's function. The second order condition is satisfied for

0 i i i i i i i i dH H H H dg G s x           (5)
The sign of the differential implies a diminishing marginal utility of the public good as the agent supplies the public good. Negativity depends on three terms. The first term measures the outcome of any provision to the public good on the marginal utility of the public good. This is our indicator of free-riding. The second term values the outcome of a shift in the social status on the marginal utility of the supply of the public good. It allows us to study the interactions between the aggregate level of the public good and the social status in the utility function. The third term assesses the impact of a decrease in private goods' consumption on the marginal utility of the public good.

The effect of agent j's supply on the marginal utility of agent i's supply is

i i i j i i dH H H dg G s    (6)
This effect is ambiguous, for the first term is negative while the second one is positive. The first term denotes a typical free-riding issue: an increase of agent j's provision reduces agent i's incentive to contribute; except that the second term denotes status seeking, thus an opposite effect, as social status decreases with agent j's supply. Indeed, agent i suffers from the reduction in the level of public good, thus any private provision that increases the public good also increases agent i's utility. Provided that any supply removes her feelings of guilt, she can freeride on others' provisions and allocate all her endowment to the private goods instead. This is a counter-incentive to supply the public good. In parallel, agent i suffers from status loss in social status pursuit every time others supply the public good. Therefore j g is also an incentive to contribute in order to maintain the level of social status.

The sign of the best-response function slope of agent i is

/ / ij i j i i dH dg r g dH dg    (7)
The sign depends on which effect prevails: guilt relieving or status seeking. According to the terms of [START_REF] Bulow | Multimarket Oligopoly: Strategic Substitutes and Complements[END_REF], if free-riding dominates social status pursuit or / < 0 ij rg  , we are in presence of strategic substitutes, and strategic complements vice versa. Despite the fact that in the standard public goods games the only effect at stake is free-riding (even in the presence of an impure public good) and public good provisions are always strategic substitutes: injecting competition for social status converts the provisions into strategic complements in some cases.

A Nash equilibrium is a set of provisions that satisfies the aggregation of supplies. Let us prove its existence and uniqueness. For a Nash equilibrium to exist, one must verify

// , ( 1,1) // i j j i i i j j dH dg dH dg dH dg dH dg   (8)
The slopes of the best-response functions are bounds within the interval ( 1,1)  . The binding conditions are sufficient for the existence of a unique Nash equilibrium.

Proposition 1: If (8) is satisfied, there exists a unique Nash equilibrium.

Let us now see what happens when the policy of income transfer is instituted. Consider the ratio which confronts the two motives involved in the public good's supply. The expression returns to an intrinsic impulse coefficient such as

/ ( / ) (2 / ) ii i i i i i Hx H x H s         (9)
The numerator measures the marginal utility of the public good and stands for the intrinsic (contrite) impulse of guilt relief to supply the public good. It depends on agent i's income and thus on her opportunity loss when she does not purchase the private goods. Here, agent i is indifferent between consuming her own supply or benefiting from agent j's supply of the public good. In Andreoni's terminology, this phenomenon means pure altruism or selflessness of agent i. We consider the numerator as a measure of free-riding on others' provisions instead.

The denominator represents the influence of social status on the marginal utility of the supply of the public good and stands for the extrinsic (social) impulse of status seeking to supply the public good. Just as with the numerator, it depends on agent i's income, but it depends on social status above all, that is, marginal utility of the public good derived from her own provision (analogue to impure altruism). Given that status is acquired by relative provisions, the effect of social status counts twice. First, consuming more of the x's decreases agent i's provision to the public good and thus her social status; second, more of j g implies lower social status for agent i, all else being equal. For those reasons, the intrinsic impulse coefficient is inversely proportional to status seeking.

Proposition 2: An income transfer from agent j to agent i , such that

0 ij dw dw    increases G if and only if > ij  .
Agents are unwilling to perfectly substitute their provisions to offset a transfer. If > ij  then agent i is less status seeking than agent j. Hence, the policy of income transfer will increase (decrease or not change) the aggregate level of the public good if and only if the income gainer is less status seeking than (more status seeking than or equally status seeking than) the income loser, because the former has lower incentives to supply than the latter. This proposition is comparable to that of Andreoni, but our interpretation is dissimilar. In fact, since competition for social status encourages agents to supply the public good, only an increase in income will motivate the lower income agent to supply more5 , for it enables her to compete for social status.

In equilibrium, she will allocate the increase in income in both private and public goods. Without transfer, her position discourages her to race for social status and she can only relieve her guilt.

The direct consequence is free-riding on other agents' provisions. Another way of understanding the proposition is: since the higher income agent proveswith a higher level of supply which reflects higher incometo be more extrinsically impulsed, she does not have to contribute more to the public good. She is in no doubt to hold the social status ex ante.

Our model is a way-out to impure altruism and warm-glow giving. We identify pure altruism as guilt relief and thus free-riding, while impure altruism corresponds to the willingness to compete for social status, which is observable via any non-anonymous donation. The model is thus an alternative and a more realistic way to explain prosocial behavior, especially when agents compete to publicly offset a public bad.

Fiscal policy

Assume that the government subsidizes private supplies at a rate  , and pays for this subsidy by levying a lump-sum tax  . All net fiscal receipts fund the public good.

i i i i u u x Y s  (10)
The social status of agent i is determined by her private supply

1 i i y g      thus 1 ij i yy s     (11)
Subsidizing works as a multiplying effect and reinforces the incentive to privately supply.

Moreover, subsidies have a direct effect on the social status. With higher subsidies and fixed total contribution, agents' private supplies as well as the difference between social statutes rise.

Agent i's budget constraint is subsequently rewritten

i i i
x y w . Her best response ( , )

i i j r w y with ji  is to have i y such as 1 , , ( ) 0 1 i i i i j i j r F w r r y r y           (12)
A Nash equilibrium of the public goods game is a couple of strategies ** , ij yy such that each strategy is the best response to the other agent's strategy ** ( , )

i i i j y r w y  with ji  11 , 11 i i i j i F F F F F F F y Y s x y Y s                       (13)
Let us now take a further look at the effect of the subsidy on the reaction function; the partial derivative yields the following

2 1 1 1 (1 ) 1 1 i i i i i dF u u u u d s s Y s x                         (14)
The effect of subsidizing private supplies is ambiguous, for the first term is positive while the second one is negative. The first term represents the multiplying factor from the subsidy: by means of Y, an increase in subsidizing reinforces the incentive to supply and provides more social status to the agent. On the contrary, due to (A3), the agent tends to reduce her supply to the public good. This is captured by the second term.

Proposition 3: Given preferences of the form ( , , )

i i i u
x Y s and an interior equilibrium, (i) an increase in the lump-sum tax  leaves the aggregate level of the public good unchanged: the crowding-out is complete; (ii) an increase in subsidies  gives an ambiguous outcome on the aggregate level of the public good: the crowding-out is unknown.

First, similarly to the pure public good scenario without see-through, and contrary to Andreoni's Proposition 3(a), our results show no influence of the lump-sum tax, which translates the inconsequence of the level of taxation. Indeed, raising taxes makes people contribute less by the amount of fiscal receipts. The crowding-out is complete and public and private supplies are totally substitutable; a dollar spent on taxes provides the same stimulus to private supplies than a dollar of direct grants.

Second, contrary to the pure public good scenario with see-through, and similarly to Andreoni's Proposition 3(b), our results show high influence of the subsidy, that is, we are in presence of a multiplying factor of subsidies which was previously termed as greater stimulus to charity. Imparting subsidizing incites agents to contribute more than by the amount of the government's expenditures. At this point of analysis, the crowding-out is incomplete and public and private supplies are imperfect substitutes; a dollar spent on subsidies provides a greater stimulus to private supplies than a dollar of direct grants. However, additional social status discourages agents to privately supply. In conclusion, the crowding-out is unknown.

Let us now study the influence of subsidizing on the intrinsic impulse coefficient

  / ( / ) 2 /(1 ) ii i i i i i Fx F x F s            (15) As 1   , 0 i   
, which implies that subsidizing reinforces the pursuit of social status, for it works as a multiplying factor on the extrinsic motivation. Yet, the higher the subsidizing, the higher the contributing, as subsidizing provides social status for free. We also know that higher social status creates negative incentives to privately supply the public good. Therefore, subsidizing has an ambiguous impact on private supplies, because agents can obtain social status without increasing their provisions to the public good. Due to see-through, agents are aware of the governmental gifting that replaces its direct granting to the public good.

Social optimum

A social Pareto optimum is a repartition of contributions that maximizes a weighted

welfare function ij W u u   (16)
where  represents the weight of agent j's utility. With fixed revenues i w , j w first order conditions are

0 j i ii du du dg dg   (17)
and subsequently

i i i i i i i i du u u du F dg G s dx       (18)
The effect from the contribution of agent i on agent j's utility is

j j j j ij du u u F dg G s    (19)
The first term is the public good dimension of agent i's supply and represents a positive externality of her contribution on agent j's utility; it possibly indicates agent i's suboptimal contribution. The second term is the private good dimension of agent i's supply and represents a negative externality on the social status of agent j. Which term dominates determines the impact of agent i's actions on agent j.

At the social optimum, we have 1 0, 0

j i ij ij du du FF dg dg       (20)
The two equations determine a Pareto optimal couple of contributions , ij gg. The ambiguity of the sign of the externality implies an unclear outcome on the social optimum at interior equilibrium: it is unclear whether an agent undercontributes or overcontributes as regards the social optimum. In case of undercontribution, we face the issue of the weak version of the free-riding hypothesis, where the public good is provided, but its allocation is suboptimal [START_REF] Brubaker | Free Ride, Free Revelation, or Golden Rule?[END_REF].

Let us now study the social optimum in presence of a fiscal policy. Let us denote ( ), ( ) ij yy  the Nash equilibrium with fiscal policy ( , , ) ( , , )

i i i i j j j j W u w y Y s u w y Y s     (21)
The effect of a change in subsidy amounts to

j i ij y y W W W yy               (
                                       (23)
The first two terms represent the effect of the subsidy on the contributions to the public good. The third term encapsulates the multiplying effect of the subsidy on social status. By the envelope theorem, the precedent becomes 2

(1 )

               (24)
In the end, we obtain 2 (1 )

j j j i j i i i i j i j u u du y y u u du G s G s ds ds                                   (25)
The expression divulges a compound ambiguous effect. The two first terms represent the effect of an agent's supply on the marginal utility of the other agent's supply. We learn from Section 1 that this effect is ambiguous, and depends on the predominance of social status. The third term illustrates the impact of subsidies on social status. Once again, the effect is ambiguous, for both factors have an ambiguous sign. However, note that if ij yy  , it is likely that the agent i's marginal reward from social status, that is / ii us  , is lower than the marginal reward of agent j, or It is straightforward to see that an increase in subsidies expands the disparities between social statuses. Those who already held social statuses by providing higher private supplies obtain additional status for free, enlarging the fore social distance. This result implies that high levels of subsidizing can worsen the distance to social optimum of the public good.

Concerning Paretian optimality, agents can always choose to contribute less than what is socially optimal while gaining social status due to see-through. It means that in presence of subsidies, corner equilibria do not arise, because agents are always incited to contribute a positive amount. Nevertheless, we can allude that if social status strongly dominates guilt relief, there is an inevitable breaking point from which subsidies no longer increase private supplies up to the social optimum, due to the gratuitousness of the additional social status that they provide.

The logarithmic model

The explicit program

Following the model by [START_REF] Kumru | The Effect of Status on Voluntary Provision[END_REF] x . The second term corresponds to the utility that agent i obtains from her supply of the public good, which is nondecreasing in i g . We assume that individuals originate guilt relief from their private supply to the public good. Agent i's preferences when she provides the public good by The expression denotes the utility that agent i gets from the aggregate level of the public good scaled by a specific index 0 i   : some willingness to alleviate guilt is expressed by either agent6 .

Agent i gets utility from social status when she provides the public good by i g . Her status is given by the distance between her provision and that of agent j's such as ()

i i j gg   for ji  ( 28
)
Agent i enhances her status in the social hierarchy if her provision outdistances agent j's;

otherwise, her social status deteriorates. The status is scaled by a specific index i  , with 0 i   , which measures agent i's willingness to pursue social status. When agents provide identical provisions, the term vanishes. In the equilibrium, agent i knows whether she acquires social status through her private supply of the public good ( ij gg  ).

We thus see that agent j's provision is both a strategic substitute and a strategic complement of agent i's utility. As a strategic substitute, two obvious interpretations come out.

First, agent i suffers from the public bad, thus any private provision that increases the public good also increases agent i's utility. Second, since any provision can remove her feelings of guilt, she can free-ride on others' provisions and allocate all her endowment to the consumption of the private goods. To consider agent j's provision as a strategic complement is to consider that agent i suffers from status loss in social hierarchy every time agent j provides the public good. In this case, agent j's private provision decreases agent i's utility.

Reaction functions

Now suppose both agents decide to submit their provisions to the public good. Given j g , differentiating () u  with respect to i g gives i r , agent i's best-response function 1 ( , ) 22

i i i j i j A r w g w g  if max , ii j ii ww g AA     (29)
. where ( ) /( ) Her provision is always the half of her income, but she has no incentive to contribute more than that. This is the behavior of an autonomous agent who disregards the provisions of the opponent. We could think of an agent who invariably contributes to the public good in order to alleviate her guiltbecause some moral obligation incites her to do so -, but who does not discredit the positive spillover on her social rank, even if she is not centered upon the social ranking matter. This agent denies the possibility of acting as a free-rider.

i i i i i A        . Whether

At last, when <

ii  or

0 i A  , i r holds if / j i i g w A 
, otherwise ii rw  and agent i allocates her full income to the supply of the public good. Provisions are then strategic complements: every time agent j increases her supply, agent i has an incentive to increase her supply to stay in the race for the social status up to the point where her full income is spent.

According to the foregoing results, Fig. 1. illustrates the best-response functions which meet at the bisection line, observed from symmetric cases ij     and ij     . Each best-response function -initiated from the reference point which is the opponent's null provision is v-shaped, i.e. separated into two segments following opposite slopes.

Fig. 1. Agents i's and j's best-response functions

The black straight lines depict agent i's best-response functions. The grey straight lines depict agent j's best-response functions. We have three cases: (i) when the intrinsic impulse dominates >0 A , their best-response functions decrease in their opponents' provisions and the public good is weakly provided, for respective provisions are less than / 2, / 2 ij ww; (ii) when the extrinsic impulse dominates <0 A their best-response functions increase in their opponents' provisions and the public good is highly provided, equilibrium provisions' exceed / 2, / 2 ij ww;

(iii) when impulses are identical 0 A  , equilibrium provisions are / 2, / 2 ij ww.

The equilibrium

0 45° / jj wA / ii wA /2 j w /2 i w >0 A 0 A  <0 A g i g j
At a Nash equilibrium ** ( , ) ij gg each agent's provision is her best response to the other's.

We first consider an interior equilibrium where both agents' provisions are strictly positive but inferior to their incomes:

i i j j j i ij i j j i w A w w A w gg A A A A    (30)
Where ( ) /( )

i i i i i A        and ( ) /( ) i j j j j A       
. In this case, the aggregate level of the public good in equilibrium, that is, *

* * ij G g g , amounts to * 1 (2 ) (2 ) 4 j i i j ij G A w A w AA        (31)
As one can detect, when agents apply their best-response functions, the aggregate level of the public good depends on the relative distance between the social status and guilt relief indices. A policy of income transfer from agent j to agent i such that

Corollary 2: When

1 ij dw dw    impacts the aggregate quantity of public good jj ii ij i j i j gg gg dG dg dg w w w w                             (32) that is 20 1 4 ij ij dG A A AA     (32')
where 2( ) /( )( )

i j i j j i i i j j AA              .
Corollary 3: At interior equilibrium, an income transfer from agent j to agent i increases (decreases) the aggregate level of the public good if and only if > ( ) 0

i j j i      .
In equilibrium, when 0 ij   and each agent is indifferent between her supply and the other's supply, we have 0 dG  which is the standard result of neutrality obtained by [START_REF] Warr | The Private Provision of a Public Good is Independent of the Distribution of Income[END_REF].

Let us now consider corner solutions with either null or full-income provisions. In corner equilibria, in case of strategic substitutes, one of the agents provides a null supply. In case of strategic complements, one of the agents allocates her full income to the public good supply. If we analyze income transfers at the corner equilibria in the symmetric case, Figs. 2. and 3. depict provisions with respect to the income inequality. The x-axes denote agents' shares of total income: /( )

i i j w w w  , /( ) j i j
w w w  . The y-axis represents the aggregate level of the public good.

The total income is fixed. The broken black curve represents the provision of agent i while the broken grey curve represents the provision of agent j. The broken grey curve decreases while the broken black curve increases as the transfer between agents j and i occurs. The equality arises at 0.5. The solid black curve illustrates the sum of provisions, i.e. the aggregate level of the public good.

Fig. 2. Income transfer with strategic substitutes

In the case of strategic substitutes (the standard scenario in public goods games), the aggregate level of the public good decreases as the incomes' disparity shrinks. Indeed, at a corner solution, the lower income agent invariably free-rides on the supply of the higher income agent.

If the income is transferred from the lower income agent to the higher income agent, the latter should allocate the extra income into the public good supply 7 and the aggregate quantity should increase. This is a similar result to Theorem 5 from [START_REF] Bergstrom | On the Private Provision of Public Goods[END_REF] who show that equalizing income by transferring income from contributors to non-contributors will decrease the equilibrium supply of the public good, in the case of a pure public good ( 0 i   in our case).

7 For example, this can suggest that cutting taxes on the higher income agent and raising taxes on the lower income agent may increase private supply. In the case of strategic complements (the novel scenario in public goods games), the aggregate level of the public good decreases as the agents' income disparity grows. This time, the lower income agent allocates her full income to the supply of the public good in order to gain social status, thus saturating her supply capacity, whereas the higher income agent contributes less than her full income. An income transfer from the higher income agent to the lower income agent should increase the quantity of public good, because the lower income agent should allocate the money transfer to the provision of the public good.

At last, Fig. 4. shows the aggregate provisions to the public good in view of i A and j A , in both interior and corner equilibria. The kinks in the slope correspond to corner equilibria. When

i i j i i j i i i u w g g g g g x g w g                                    (33)
which we can rewrite as

1 ln( ) ln ( ) ( ) 1 i i i i j i i j w y y y y y            ( 33
i i i i i B            and /(1 ) / /(1 ) j j j j j B                    .
We see that the extrinsic motivation increases in  , which illustrates the multiplying effect. When the social status pursuit dominates guilt relief, that is ji yw  , any nonnegative amount enables agent i to gain social status. In this case, the joint supply of the public good in equilibrium, that is

i j i j i i i j j i g g g g w g g g g g                             (37)
Corollary 6: The difference between  and  determines whether an agent's supply has a positive or a negative externality on the utility of the other agent.

The sign of the effect, that is, the second term of the left hand side expression, is solely determined by the sign of   . At an interior equilibrium, if 0 A  , we fall on the standard case where contributions act like strategic substitutes. Free-riding then implies social suboptimal provision of the public good. Likewise, if 0 A  , we observe social superoptimal provision of the public good. When supplies are strategic substitutes, the aggregate level of the public good is suboptimal8. However, it is unclear whether an agent makes either a larger or a smaller contribution than the socially optimal one. This ambiguity is linked to the substitutability aspect.

Let us consider the scenario where agent j weekly free-rides, i.e. she undersupplies as regards the social optimum. Agent i does not internalize the positive externality of her supply and provides less than what agent j contributes in Nash equilibrium. In addition, her underprovision is either larger or lower than the social optimum. Graphically, we obtain the Nash and Pareto optima as follows.

Fig. 5. Agents i's and j's socially optimal response functions

Concluding remarks

When agents privately provide a public good, agents profit from donations to alleviate their guilt. Because guilt relief entails opportunity costs, agents refuse to pay for others' guilt or simply profit from efforts of the others, and this leads to free-riding and weak provision of the public good: a phenomenon amply covered by the economic literature.

The private provision of the public good is stimulated by the private benefit of the public good such as obtaining some social status, which ex post seems intuitive. In the case where provisions become strategic complements, a policy of income transfers from the higher income agent to the lower income agent should increase the aggregate level of the public good. Its purpose would be to activate the competition for social status, which increases the public good's level and thus social welfare. Therefore, competition can be an efficacious anti-free-riding shield 9 .

Yet, contrary to a common thought, a policy of subsidizing has an ambiguous effect on private contributions, because it provides social status for free whenever agents are endowed with 9 This is an opposite result to [START_REF] Holländer | A Social Exchange Approach to Voluntary Cooperation[END_REF] who finds that opening a market for the collective good lowers its provision. In any case, our model can stand for the wanting theoretical background which explains why agents under-react to the income transfer, that is, why lower income agents over-contribute and higher income agents under-contribute, in both experiments from [START_REF] Chan | The Voluntary Provision of Public Goods under Varying Income Distributions[END_REF] and [START_REF] Maurice | Income Redistribution and Public Good Provision: an Experiment[END_REF]. Indeed, neither has considered contributions as strategic complements.

Besides, [START_REF] Chan | The Voluntary Provision of Public Goods under Varying Income Distributions[END_REF] themselves conclude that the explanation for experimental results might be in a model where agents react to their competitors. Their intuition meets up with ours.

In general, our results could explain the institution of charity auctions, honor rolls of donors and the construction of socially responsible finance indices. More generally, it could relate to why institutions make use of agents' willingness to demonstrate their generosity if not their apparent selflessness. To some extent, our model could be an illustration of the theory of crowding out of intrinsic motivations by extrinsic incentives.

Appendix

  a positive sign. When social statuses are equal and agents are symmetric, the term vanishes.Proposition 4: Subsidies increase the difference in social status between agents.

  i allocates a part of her income to the supply of the public good. For sufficiently high values of j g , agent i can supply either nothing or her full income. Whichever occurs depends on the sign of ii If agent j surpasses this threshold, agent i has fairly no incentive to make positive provisions. In point of fact, even a quasi-null level of i g (nonnegative by definition) enables agent i to maximize her utility by allocating her income to more private goods while alleviating her guilt through agent j's provisions. We could think of an agent who pays to the collective high efforts in providing the public good while ending up self-pleased by giving agent i is equally concerned by guilt alleviation and social hierarchy. This time,

  i) the equilibrium provision of agent i decreases; (ii) the aggregate equilibrium quantity of public good decreases; (iii) and the equilibrium provision of agent j increases (decreases) if (

Fig. 3 .

 3 Fig. 3. Income transfer with strategic complements

  Fig. 4. The aggregate level of provisions

  through capacity; this policy can yield an outcome opposite to the aim of the government when it sets up subsidies.

  At interior equilibrium, the governmental subsidizing of private supplies increases the aggregate level of the public good.

		(	)	(				)	1	(	)	(	)	0
	*	* , amounts to *
				*		1		1	(1  ) (1 )    	(35)
	ij y y supply is a strategic substitute, if Y Y Corollary 4: In detail, the equilibrium contribution of agent i decreases with subsidizing, when the j i i j ij B w B w BB 2 2   (36) i j B ww B Corollary 5: An increase in governmental subsidizing (i) increases the agents' private supplies, 2 > [(2 ) / ]  5.5. Social optimum Following the separable nonlinear utility function, first order Pareto conditions with 1   when contributions are strategic complements; (ii) decreases agent i's private supply, when contributions are strategic substitutes, if and only if ij w w B B . and identical agents, that is, ij   and ij   , yield

Proof of Proposition 1

  Consider the ratio which measures the relative incentives to contribute to the public good Because of (11), the first factor of the left hand side is positive thus This equation holds if the right hand side is between 0 and

	A similar expression holds for * j dg and summing both expressions gives * ( / ) ( / ) 1 ( 1 ( / ) 1 ( / ) i j j i i i j j i j j i r g r g dG dw dw r g     i    where , ( 1,1) jj ii ij i i j j AA             .                   r g The aggregation of provisions amounts to	j	)	i dw
	First, x r r x w w ** ( , ) ij gg is a Nash equilibrium if and only if * i g is a fixed point of the function: ( ( , ), ) i j j i  and ** () j i j r g g  . Second, if ( / ) ( 1,1) ij rg     and ( / ) ( 1,1) ji rg     , ij dG      1 2 22 11 22 44 i i i i j j j j j i i j i j g A w w A w g A w w A w A A A A                            
	( ( , ), ) x r r x w w 	has a unique fixed point and
	i j and the total provision is j i
	Proof of Corollary 1	// , // j j j i i i j dH dg i r dH dg dH dg g dH dg i    i j r g     * 1 (2 ) (2 ) 4 j i i G A w A w j AA   j      
					ij
	Given	j g , differentiating () u  with respect to	i g gives best-response * i g . At an interior solution
	Proof of Proposition 2 the first order condition is satisfied When 0   and agents are exclusively intrinsically impulsed
		ij
				G	 *	1 w r i i  22 ( i i r g  ) ii j i i ( r g       1 () 0 ) j  4 1 3 i i j j ij w w w w w w       	
	 //   ) ( / ) ( / ) ( / ) ( i i i i i i i i i dH dx j H x / ) ( i dH dg / i dH dg H G H s H G              H s / ) ( i i    H x / ) i i  Therefore, ( ( i   )( ) ( ) ( ) i i i i i i j i i j w r r g r g           , and Proof of Corollary 4	
	First of all, it is worth writing ( / ) (2 / i i i i H x i H x H s / ) ( i i      () / i H x / ) ( i i j H g H g / ) 1 ( / / i i i i r w i r g     1 1 1 1 ( , ) 2 2 ( ) 2 2 ii i j i i j i i j ii r g w w g w A g         1 ln( ) ln ( ) ( ) 1 i i i i j i i j w y y y y y                          The income transfer corresponds to 0 ij dw dw    . At the unique equilibrium ** ( , ) i j i i r g w g  and differentiation of this relation gives ** ii i j i rr dg dg dw gw   Proof of Corollary 2 ii  i w which is the case if ) j when 0 j i i g  when 0 ( /1 ) 1 0 ( ) ( /1 ) ii w y y y y y                i i i i j i i j w A (½) ii for any 0   ( ) ( /1 ) ( ) ( /1 ) i i i i i i j i i j w y y y y y               rw  * (1/1 ) (1/1 ) ii i i j y w y         	i r g w A j i i  )= 0   ii	,
					ji
	At an interior equilibrium, the two following equations are satisfied
	Since * j dg dG dg *  * 1 ( / * i , we have * i dg / ) i j i i j r g dg dG r g       At an interior equilibrium, the two following equations are satisfied * * ( / )( ) ( / i j i i r g dG dg r w dw ) i i        * ( / ) ( / ) 1 ( / ) 1 ( / ) i j ii i i j i j r g rw dw dG r g r g            ** ** 2 2 i i j i j i j j g A g w A g g w     ** ** i i j i j i j j y B y w B y y w          where (1/1 ) (1/1 ) , (1/1 ) ) jj ii ij BB                    . * . That is i i dw  
					i	i	j	j

i  according to the partial derivative of the reaction function * ( , ) ij G w w , agent i 's provision satisfies * 0 i A  (i.e. ii   ) and i A  (i.e. ii   ). When ( j g  . The same reasoning applies to agent j. 

[START_REF] Gilbert | Group Wrongs and Guilt Feelings[END_REF] speaks about membership guilt over group wrongs. This collective guilt will be shared by members of the collective in question in their capacity as group members.

A status-based model of market competition has previously been introduced by[START_REF] Podolny | A Status-based Model of Market Competition[END_REF].

According to[START_REF] Gilbert | Group Wrongs and Guilt Feelings[END_REF] since feeling guilt is unpleasant, it is liable to move one who feels it to act. And this will not necessarily be the personal undertaking of reparative action.

For example, OECD (2007) suggests monetary transfers in benefit of low income households when imposing environmental taxes.

Social comparison theory suggests that individuals have a need to compare themselves to individuals whom they deem are similar to them[START_REF] Goethals | Social Comparison Theory: Psychology from the Lost and Found[END_REF]).

In case of strategic complements, there is no such ambiguity and each agent socially overcontributes: the public good is at a level higher than the social optimum.

CIRANO is a private non-profit organization incorporated under the Québec Companies Act. Its infrastructure and research activities are funded through fees paid by member organizations, an infrastructure grant from the Ministère du Développement économique et régional et de la Recherche, and grants and research mandates obtained by its research teams.

The aggregation of joint supplies amounts to 1 2 1 11 1 11

and the total provision is * 1

(1 ) (1 ) 1

Subsidizing decreases

From which we obtain

The right hand side is negative if and only if >0 B and