Emmanuel Frénod

Michaël Gutnic

Sever A Hirstoaga

FIRST ORDER TWO-SCALE PARTICLE-IN-CELL NUMERICAL METHOD FOR VLASOV EQUATION

The aim of this work is to build an accurate numerical method for the simulation of the long time evolution of the Vlasov solution f ε with an electric field E ε = E0+εE1 for small ε. To this purpose, we use the Two-Scale Convergence theory to determine a first order approximation F + εF1 of f ε , then particle approximations to build an algorithm to obtain a numerical approximation of F + εF1.

Résumé. On cherche à construire une méthode numérique pour l'évolution en temps long de la solution f ε de l'équation de Vlasov avec un champ électrique E ε = E0 + εE1 pour ε petit. À cet effet, on utilise la théorie de la convergence deux échelles pour obtenir une approximation d'ordre un F +εF1 de f ε , puis une méthode particulaire pour construire l'algorithme d'approximation numérique de F + εF1.

Introduction

In this paper, we are interested in the solution f ε to the following Vlasov problem set in a bi-dimensional phase space

     ∂f ε ∂t + v ε ∂f ε ∂r + E ε - r ε ∂f ε ∂v = 0, f ε (t = 0, r, v) = f 0 (r, v).
(1.1)

where f ε : (t, r, v) → f ε (t, r, v), for t ∈ [0, T ), r ∈ R + and v ∈ R. In (1.1), let

E ε (t, r) = E 0 t, t ε , r + ε E 1 t, t ε , r , (1.2)
where E i=0,1 : (t, τ, r) → E i=0,1 (t, τ, r) are regular and 2π-periodic with respect to τ . Our purpose is to build a numerical method which is efficient for simulating the long time evolution of f ε in the limit ε → 0.

In Section 2, we use Two-Scale Convergence theory to determine a first-order approximation of f ε , that is f ε (t, r, v) ≈ F (t, t/ε, r, v) + ε F 1 (t, t/ε, r, v) where F : (t, τ, r, v) → F (t, τ, r, v) and F 1 : (t, τ, r, v) → F (t, τ, r, v) are 2π-periodic functions with respect to τ ∈ R.

We first show in Section 2.1 that, for the zero order term, there exists a function G : (t, q, u) → G(t, q, u) solution of an initial boundary condition partial differential problem such that F (t, τ, r, v) = G(t, R τ (r, v)), where R τ is the 2D-rotation of angle τ . In Section 2.2, we handle first order term and show that there exists two functions G 1 : (t, q, u) → G 1 (t, q, u) and W : (t, τ, q, u) → W (t, τ, q, u) such that F 1 (t, τ, r, v) = G 1 (t, R τ (r, v)) + W (t, τ, R τ (r, v)) where W can be computed from ∇ q,u G and G 1 is yet the solution of an initial boundary condition partial differential problem.

In Section 3, we introduce a PIC based approximation to build a numerical algorithm which will allow us to solve numerically the problems satisfied by G and G 1 as well as to compute a numerical approximation of W .

On Two-Scale Convergence statements

The concept of Two-Scale Convergence was introduced at the end of the 80's by Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF][START_REF] Nguetseng | Asymptotic analysis for a stiff variational problem arising in mechanics[END_REF]. In 1992, Allaire gave a very understanble proof of this result [START_REF] Allaire | Homogenization and Two-scale Convergence[END_REF]. Then several authors used this theory to build numerical methods called Two-Scale Numerical Methods (e.g. Frénod and Sonnendrücker [START_REF] Frénod | The Finite Larmor Radius Approximation[END_REF], Frénod, Salvarani and Sonnendrücker [START_REF] Frénod | Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method[END_REF], or Mouton [START_REF] Mouton | Two-scale semi-lagrangian simulation of a charged particles beam in a periodic focusing channel[END_REF]). For more details about Two-Scale Convergence theory, we refer to these authors and their references.

Let us here only recall the definition of the Two-Scale Convergence.

Definition 2.1. A function f ε : (t, r, v) ∈ [0, T ) × R 2 → f ε (t, r, v) Two-Scale converges as ε → 0 to a function F : (t, τ, r, v) ∈ [0, T ) × R × R 2 → F (t, τ, r, v) 2π-periodic in τ , if: T 0 R 2 f ε (t, r, v) [φ] ε (t, r, v) dr dv dt → T 0 2π 0 R 2 F (t, τ, r, v) φ(t, τ, r, v) dr dv dτ dt, (2.1)
for all φ ∈ C 0 where

C i := C i c [0, T ); C i # (R; C i c (R 2 )) for i = 0, 1.
Remark 2.1 (Notations). In the above definition and in the following of the paper, we introduce the following general notations:

(i) Subscript # in space definition stands for 2π-periodicity in τ . (ii) Bracket [•] ε stands for [φ] ε (t, r, v) := φ(t, t/ε, r, v) for any function φ depending of the four variables (t, τ, r, v) ∈ [0, T ) × R × R 2 .
Moreover, in the whole section we will use the 2D-rotation of angle τ of a two-dimensional vector.

Definition 2.2. For all τ ∈ [0, 2π] and all (r, v) ∈ R 2 , we define the 2D-rotation R τ of angle τ by:

R τ (r, v) = cos τ -sin τ sin τ cos τ r v = r cos τ -v sin τ r sin τ + v cos τ . (2.2)
Its inverse in then given by R -τ whose first component is denoted by R -τ r .

Remark 2.2. Thanks to the above definition, we have that :

R τ (r, v)• q u = r v • R -τ (q, u). (2.3)
Theorem 2.1. Let (f ε ) ε be a sequence of solutions to Vlasov problem (1.1) with the electric field E ε given by (1.2). Therefore there exists two functions

F : (t, τ, r, v) ∈ [0, T ) × R × R 2 → F (t, τ, r, v) and F 1 : (t, τ, r, v) ∈ [0, T ) × R × R 2 → F (t, τ, r, v) 2π-periodic with respect to τ ∈ R such that f ε (t, r, v) ≈ F t, t ε , r, v + ε F 1 t, t ε , r, v , (2.4)
Moreover, regarding the zero order term, there exists a function G : (t, q, u)

∈ [0, T ) × R 2 → G(t, q, u) such that F (t, τ, r, v) = G(t, R τ (r, v)), (2.5
) where R τ is the rotation defined in Definition 2.2 and G is solution to the problem

         ∂G ∂t (t, q, u) + 1 2π 2π 0 R τ 0, E 0 t, τ, R -τ r (q, u) dτ • ∇ q,u G(t, q, u) = 0, G(t = 0, q, u) = 1 2π f 0 (q, u).
(2.6)

Regarding the first order term, there exists two functions

G 1 : (t, q, u) ∈ [0, T ) × R 2 → G 1 (t, q, u) and W : (t, τ, q, u) ∈ [0, T ) × R × R 2 → W (t, τ, q, u) such that F 1 (t, τ, r, v) = G 1 (t, R τ (r, v)) + W (t, τ, R τ (r, v)), (2.7)
where on the one hand, W is given by

W (t, τ, q, u) = τ 2π 2π 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ - τ 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ • ∇ q,u G(t, q, u), (2.8)
and on the other hand, G 1 is solution to the problem

                                                         ∂G 1 ∂t (t, q, u) + 1 2π 2π 0 R τ 0, E 0 t, τ, R -τ r (q, u) dτ • ∇ q,u G 1 (t, q, u) = 1 2π 2π 0 τ 0 R σ 0, ∂E 0 ∂t t, σ, R -σ r (q, u) dσ - τ 2π 2π 0 R σ 0, ∂E 0 ∂t t, σ, R -σ r (q, u) dσ dτ • ∇ q,u G(t, q, u) + 1 4π 2π 0 ∇ q,u R σ 0, E 0 t, σ, R -σ r (q, u) dσ 2π 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ - 1 2π 2π 0 ∇ q,u R τ 0, E 0 t, τ, R -τ r (q, u) τ 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ dτ •∇ q,u G(t, q, u) dτ.
G 1 (t = 0, q, u) = 0.

(2.9) Remark 2.3. Note that the expression of W in (2.8) as well as problems (2.6) and (2.9) do not depend on 1/ε-frequency oscillations any more. The proof of Theorem 2.2 is given in the two following sections.

Zero order approximation

Here, we briefly recall arguments developped in [START_REF] Frénod | Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method[END_REF]. Let us consider the Vlasov equation for f ε in problem (1.1). First, we multiply by f ε and integrate over r ∈ R + and v ∈ R to obtain:

1 2 d f ε (t, •, •) 2 L 2 (R 2 ) dt = 0, (2.10)
which leads to the following estimate:

f ε L ∞ ([0,T );L 2 (R 2 )) ≤ C. (2.11)
Then it is known that, up to a subsequence, (f ε ) ε Two-Scale converges to some F : (t, τ, r, v) →

F (t, τ, r, v) with F ∈ L ∞ [0, T ); L ∞ # (R, L 2 (R 2 )
) . Now, we multiply Vlasov equation in (1.1) by [φ] ε with φ ∈ C 1 , integrate over t ∈ [0, T ) and (r, v) ∈ R 2 and then integrate by parts. Thus we show that F satisfies in a weak sense the following equation

∂F ∂τ + v -r • ∇ r,v F = 0, (2.12)
where we note that ε is not involved anymore. From (2.12), we deduce that F is constant along the characteristics associated to this advection equation, and deduce from Theorem 4.2 in [START_REF] Frénod | Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method[END_REF] that there exists a function G : (t, q, u)

∈ [0, T ) × R 2 → G(t, q, u) such that (2.5) is satisfied with G is solution to problem (2.6) of Theorem 2.1.
Remark 2.4. Let G : (t, q, u) ∈ [0, T ) × R 2 → G(t, q, u) be a given function and define the family

F ε ε>0 by [F ] ε (t, r, v) = G t, R t/ε (r, v) .
(2.13)

Then (i) ∇ r,v [F ] ε (t, r, v) = R -t/ε ∇ q,u G t, R t/ε (r, v) . (ii) ∂[F ] ε ∂t (t, r, v) + 1 ε v -r ∇ r,v [F ] ε (t, r, v) = ∂G ∂t t, R t/ε (r, v) .

First order approximation

We consider the zero order Two-Scale approximation

F of f ε introduced in section 2.1. Let us define [F ] ε as in Remark 2.1(ii). Then thanks to Remark 2.4(ii) ∂[F ] ε ∂t (t, r, v) + 1 ε v -r • ∇ r,v [F ] ε (t, r, v) = ∂G ∂t t, R t/ε (r, v) . (2.14)
For the sake of simplicity in computations below, we introduce the notation

E ε = E 0 (t, t/ε, r) + ε E 1 (t, t/ε,
∂[F ] ε ∂t + 1 ε v -r • ∇ r,v F ε + E ε ∂[F ] ε ∂v = = E ε ∂[F ] ε ∂v + ∂G ∂t t, R t/ε (r, v) , (2.15) = 0 E ε • ∇ r,v [F ] ε + ∂G ∂t t, R t/ε (r, v) , (2.16) = 0 E ε • R -t/ε ∇ q,u G t, R t/ε (r, v) + ∂G ∂t t, R t/ε (r, v) , (2.17) = R t/ε (0, E ε )• ∇ q,u G t, R t/ε (r, v) + ∂G ∂t t, R t/ε (r, v) . (2.18)
To handle the right-hand-side of (2.18), we introduce the function Υ ε such that

Υ ε t, τ, r, v := R τ (0, E ε )• ∇ q,u G t, R τ (r, v) + ∂G ∂t (t, R τ (r, v)) . (2.19)
Then defining [Υ ε ] ε as in Remark 2.1(ii), we can subtract (2.18) from (1.1) and multiply the result by 1/ε to obtain

∂ ∂t f ε -[F ] ε ε + 1 ε v -r • ∇ r,v f ε -[F ] ε ε + E ε ∂ ∂v f ε -[F ] ε ε = - 1 ε [Υ ε ] ε . (2.20) Now, let a function W ε : (t, τ, q, u) ∈ [0, T ) × R × R 2 → W ε (t, τ, q, u) be such that ∂W ε ∂τ (t, τ, R τ (r, v)) = -Υ ε (t, τ, r, v). (2.21)
On the one hand, if we substitute Υ ε by its expression in (2.19) and then consider equation (2.21) in variables (q, u) = R τ (r, v), we obtain

∂W ε ∂τ (t, τ, q, u) = -R τ 0, E ε t, τ, R -τ r (q, u) • ∇ q,u G(t, q, u) - ∂G ∂t (t, q, u). (2.22)
We substitute the time derivative of G in the right-hand-side of (2.22) by its expression in (2.6).

Integrating the result with respect to the second variable from 0 to τ gives

W ε (t, τ, q, u) = τ 2π 2π 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ - τ 0 R τ 0, E 0 t, τ, R -τ r (q, u) dσ • ∇ q,u G(t, q, u) (2.23) -ε τ 0 R τ 0, E 1 t, τ, R -τ r (q, u) dσ • ∇ q,u G(t, q, u).
When ε goes to 0 in (2.23), we obtain the limit W of W ε defined by (2.8) in Theorem 2.1.

On the other hand, let us introduce W ε define by

W ε (t, τ, r, v) := W ε (t, τ, R τ (r, v)). Therefore, (2.21) is equivalent to ∂ W ε ∂τ t, τ, r, v + v -r • ∇ r,v W ε t, τ, r, v = -Υ ε (t, τ, r, v). (2.24) Using Remark 2.1(ii), we note that ∂ t [ W ε ] ε = [∂ t W ε ] ε + [∂ τ W ε ] ε /ε and ∂ r,v [ W ε ] ε = [∂ r,v W ε ] ε so that considering (2.
24) in τ = t/ε and multiply the result by 1/ε leads to

∂[ W ε ] ε ∂t - ∂ W ε ∂t ε + 1 ε v -r • ∇ r,v [ W ε ] ε = - 1 ε [Υ ε ] ε . (2.25)
Now, we subtract (2.25) from (2.20) and subtract from the both sides of the result the term

E ε ∂ v [ W ε ] ε to obtain ∂ ∂t f ε -[F ] ε ε -[ W ε ] ε + 1 ε v -r • ∇ r,v f ε -[F ] ε ε -[ W ε ] ε + E ε ∂ ∂v f ε -[F ] ε ε -[ W ε ] ε = - ∂ W ε ∂t ε -E ε ∂[ W ε ] ε ∂v . (2.26)
We want to obtain an a priori estimate in order to prove convergence result. Therefore we multiply

(2.26) by f ε -[F ] ε ε -[ W ε
] ε and integrate by part. This leads to

1 2 d dt f ε -[F ] ε ε -[ W ε ] ε 2 L 2 (R 2 ) (t) ≤ ∂ W ε ∂t ε + 0 E ε • ∇ r,v [ W ε ] ε L 2 (R 2 ) (t) f ε -[F ] ε ε -[ W ε ] ε L 2 (R 2 ) (t).
(2.27)

If the first factor on the right-hand-side of (2.27) is bounded, by Gronwall Lemma, we obtain the following estimate

f ε -[F ] ε ε -[ W ε ] ε L ∞ ([0,T );L 2 (R 2 ))
≤ C.

(2.28)

From this estimate, we deduce the following Two-Scale convergence result

f ε -[F ] ε ε -[ W ε ] ε -→ F 1 -W , (2.29)
where W = W (t, τ, R τ (r, v)) and W given by (2.8). Now, we want to compute F 1 . Returning to (2.26), we multiply by φ ∈ C 1 (see Definition 2.1), integrate over t ∈ [0, T ) and (r, v) ∈ R 2 and then integrate by parts. We obtain

- T 0 R 2 f ε -[F ] ε ε -[ W ε ] ε ∂φ ∂t ε + E ε ∂φ ∂v ε dr dv dt - 1 ε T 0 R 2 f ε -[F ] ε ε -[ W ε ] ε ∂φ ∂τ ε + v -r • [∇ r,v φ] ε dr dv dt = - T 0 R 2 ∂ W ε ∂t ε - 0 E ε • ∇ r,v W ε ε [φ] ε dr dv dt.
(2.30) Multiplying (2.30) by ε and passing to the limit using the Two-Scale convergence (2.29), we obtain

∂ ∂τ F 1 -W + v -r • ∇ r,v F 1 -W = 0, (2.31)
in a weak sense. Thus, F 1 -W is constant along the characteristics so that there exists a function

G 1 : (t, q, u) ∈ [0, T ) × R 2 → G 1 (t, q, u) such that F 1 (t, τ, r, v) -W (t, τ, r, v) = G 1 (t, R τ (r, v)), (2.32)
and therefore, thanks to the definition of W , we obtain the decomposition (2.7) of F 1 in Theorem 2.1. Now, let γ : (t, q, u)

∈ [0, T ) × R 2 → γ(t, q, u) a function in C 1 c [0, T ); C 1 c (R 2 ) such that φ(t, τ, r, v) = γ(t, R τ (r, v)).
(2.33)

We consider (2.30) for this function φ. Then, the second term on the left hand side cancels and we obtain

T 0 R 2 f ε -[F ] ε ε -[ W ε ] ε ∂φ ∂t ε + E ε ∂φ ∂v ε dr dv dt = - T 0 R 2 ∂ W ε ∂t ε - 0 E ε • ∇ r,v W ε ε [φ] ε dr dv dt.
(2.34)

Thanks to the Two-Scale convergence, we pass to the limit as ε → 0 and integrate over τ ∈ [0, 2π] which leads to

T 0 2π 0 R 2 G 1 (t, R τ (r, v)) ∂φ ∂t + 0 E 0 • ∇ r,v φ dr dv dτ dt = T 0 2π 0 R 2 ∂ W ∂t - 0 E 0 • ∇ r,v W φ dr dv dτ dt. (2.35)
Then substituting φ by its expression in terms of γ and changing the variables

(r, v) in (q, u) = R τ (r, v) that is (r, v) = R -τ (q, u) gives T 0 2π 0 R 2 G 1 (t, q, u) ∂γ ∂t (t, q, u) + R τ 0, E 0 t, τ, R -τ r (q, u) • ∇ q,u γ(t, q, u) dq du dτ dt (2.36) = T 0 2π 0 R 2 ∂W ∂t (t, τ, q, u) + R τ 0, E 0 t, τ, R -τ r (q, u) • ∇ q,u
W (t, τ, q, u) γ(t, q, u) dq du dτ dt.

We finally deduce that G 1 satisfies, in a weak sense, the following problem

                   ∂G 1 ∂t (t, q, u) + 1 2π 2π 0 R τ 0, E 0 t, τ, R -τ r (q, u) dτ • ∇ q,u G 1 (t, q, u) = - 1 2π 2π 0 ∂W ∂t (t, τ, q, u) + R τ 0, E 0 t, τ, R -τ r (q, u) • ∇ q,u W (t, τ, q, u) dτ G 1 (t = 0, q, u) = 0.
(2.37)

To handle the right hand side of (2.37), we need to compute the partial derivatives of W . First from (2.8), we have that

∂W ∂t (t, τ, q, u) = τ 2π 2π 0 R σ 0, ∂E 0 ∂t t, σ, R -σ r (q, u) dσ - τ 0 R σ 0, ∂E 0 ∂t t, σ, R -σ r (q, u) dσ • ∇ q,u G(t, q, u) + τ 2π 2π 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ - τ 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ • ∇ q,u ∂G ∂t (t, q, u),
(2.38) so that we can then replace ∂ t G by its expression in (2.6) to get

∂W ∂t (t, τ, q, u) = τ 2π 2π 0 R σ 0, ∂E 0 ∂t t, σ, R -σ r (q, u) dσ - τ 0 R σ 0, ∂E 0 ∂t t, σ, R -σ r (q, u) dσ • ∇ q,u G(t, q, u) + τ 2π 2π 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ - τ 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ • • 1 2π 2π 0 ∇ q,u R σ 0, E 0 t, σ, R -σ r (q, u) dσ ∇ q,u G(t, q, u)
+H q,u G(t, q, u)

1 2π 2π 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ , (2.39)
where H q,u is the Hessian matrix with respect to variable q an u. Next, from (2.8), we deduce that

∇ q,u W (t, τ, q, u) = τ 2π 2π 0 ∇ q,u R σ 0, E 0 t, σ, R -σ r (q, u) dσ - τ 0 ∇ q,u R σ 0, E 0 t, σ, R -σ r (q, u) dσ • ∇ q,u G(t, q, u) + H q,u G(t, q, u) τ 2π 2π 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ - τ 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ .
(2.40)

We consider the scalar product of (2.40) by vector R τ 0, E 0 t, τ, R -τ r (q, u) , sum the result with (2.39) and then integrate over τ ∈ [0, 2π]. Using Fubini's Theorem for the terms involving the Hessian matrix of G which is symmetric yields

2π 0 ∂W ∂t (t, τ, q, u) + R τ 0, E 0 t, τ, R -τ r (q, u) • ∇ q,u W (t, τ, q, u) dτ = 2π 0 τ 2π 2π 0 R σ 0, ∂E 0 ∂t t, σ, R -σ r (q, u) dσ - τ 0 R σ 0, ∂E 0 ∂t t, σ, R -σ r (q, u) dσ • ∇ q,u G(t, q, u) dτ + 2π 0 τ 2π 2π 0 ∇ q,u R σ 0, E 0 t, σ, R -σ r (q, u) dσ - τ 0 ∇ q,u R σ 0, E 0 t, σ, R -σ r (q, u) dσ R τ 0, E 0 t, τ, R -τ r (q, u) - 1 2π 2π 0 ∇ q,u R σ 0, E 0 t, σ, R -σ r (q, u) dσ τ 2π 2π 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ - τ 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ • ∇ q,u G(t, q, u) dτ - 2π 0 H q,u G(t, q, u) τ 2π 2π 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ - τ 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ • • 1 2π 2π 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ -R τ 0, E 0 t, τ, R -τ r (q, u) dτ. (2.41)
For both last factors on the right hand side of (2.41), we remark that the second one is the exact derivative of the first one with respect to τ . Since the Hessian matrix is symmetric and does not depend on τ , we can take it out of the integral. Thus, the remaining integrand is the exact derivative of a quadratic form with respect to variable τ , so that integrating over a period in τ gives 0. Moreover, we can perform an integration by parts in the first part of the second term on the right hand side of (2.41). We obtain

2π 0 ∂W ∂t (t, τ, q, u) + R τ 0, E 0 t, τ, R -τ r (q, u) • ∇ q,u W (t, τ, q, u) dτ = 2π 0 τ 2π 2π 0 R σ 0, ∂E 0 ∂t t, σ, R -σ r (q, u) dσ - τ 0 R σ 0, ∂E 0 ∂t t, σ, R -σ r (q, u) dσ • ∇ q,u G(t, q, u) dτ - 2π 0 1 2π 2π 0 ∇ q,u R σ 0, E 0 t, σ, R -σ r (q, u) dσ -∇ q,u R τ 0, E 0 t, τ, R -τ r (q, u) τ 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ - 1 2π 2π 0 ∇ q,u R σ 0, E 0 t, σ, R -σ r (q, u) dσ τ 2π 2π 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ - τ 0 R σ 0, E 0 t, σ, R -σ r (q, u) dσ • ∇ q,u G(t, q, u) dτ.
(2.42)

In the second term on the right hand side of (2.42), we remark that first and last products cancel. Moreover, τ /2π is the only factor depending of τ in the third product, integration of which over [0, 2π] gives π. Substituting the right-hand side of equation in problem (2.37) by its expression in (2.42) finally leds to problem (2.9)

Numerical algorithm

To build our numerical algorithm, we use a Particle-in-Cell (PIC) method, that deals with macroparticles rather than directly with the distribution function.

Theorem 2.1 tells that, provided that problem (2.6) is solved and space derivatives of G can be computed, then we can compute W from (2.8) and source term in problem (2.9). Then it remains to solve problem (2.9) to obtain G 1 . From G, G 1 and W , we finally get the Two-Scale first order approximation of f ε thanks to (2.4), (2.5) and (2.7). Thus, the main steps of the algorithm are the following.

Mains steps of the algorithm

(i) Compute a particle approximation of G, that is push the macroparticles with respect to the advection operator associated to the partial differential equation in problem (2.6) as in [START_REF] Frénod | Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method[END_REF]. (ii) Compute an approximation of the gradient ∇ q,u G from the particle approximation of G which is compatible with the desired particle approximation of W. (iii) Compute a particle approximation of W from equation (2.8). (iv) Compute a particle approximation of the source term of the partial differential equation in problem (2.9), and then a particle approximation of G 1 by solving problem (2.9) with the same advection operator than in problem (2.6).

To deal with the first step, we introduce the following particle approximation of function G G(t, q, u)

= Np k=1 ω k δ(q = Q k (t)) δ(u = U k (t)), (3.1)
where δ is the Dirac mass, N p is the number of macroparticles and (Q k (t), U k (t)) is the position in phase space of macroparticle k which moves along a characteristic curve of the equation in (2.6).

Hence our problem is reduced to compute the location of macroparticles at the next time step from their positions at the previous time step, as the solution of the following dynamical system

d dt Q k U k = 1 2π 2π 0 R τ 0, E 0 t, τ, R -τ r (Q k , U k ) dτ. (3.2)
The main difficulty to solve coupled problem (3.2) lies in the fact that we need to approximate the integral term. A solution to this problem is described in section 5.1 of [START_REF] Frénod | Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method[END_REF].

Once the macroparticles are pushed along the characteristics, we can compute G at the next time step. Applying the ad hoc rotation, we then obtain a particle approximation of F from (2.5). Finally, evaluating this approximation in τ = t/ε gives the zero order approximation of f ε .

For the second step of the algorithm, we need to recover an approximation of the space derivatives of G thanks to the particle approximation of G. Therefore, a regularization of approximation (3.1) is needed. To this purpose, we introduce a regular function γ α with support included in the intervall [0, α], and such that the function γ

α k : (q, u) → γ α (q -Q k (t)) 2 + (u -U k (t))
2 has an integral over q and u with worth 1. Thus a regularization of (3.1) is given by

G(t, q, u) = Np k=1 ω k γ α (q -Q k (t)) 2 + (u -U k (t)) 2 , (3.3)
Thus we can compute numerical approximations of the space derivatives of G.

Now we can handle the third step of the algorithm. Indeed, in (2.8), we can compute an approximation of the right hand side which has the following shape

k=1 β k (t, τ, q, u) (γ α ) (q -Q k (t)) 2 + (u -U k (t)) 2 . ( 3.4)
In order to get W , we need to distribute the above approximation over the macroparticles. Let the particle approximation of W be given by

W (t, τ, q, u) = Np k=1 β k (t, τ ) δ(q = Q k (t)) δ(u = U k (t)), (3.5)
which can be regularized by

W (t, τ, q, u) = Np k=1 β k (t, τ ) γ α (q -Q k (t)) 2 + (u -U k (t)) 2 . ( 3.6)
Then we want that all the integrals of the sums in (3.4) and (3.6) over each subdomain of the phase space are almost equals. From the numerical point of view, we want to verify this proximity only for the support

C i of functions γ α i that is Ci Np k=1 β k (t, τ ) γ α k (q, u) dq du = Ci Np k=1
β k (t, τ, q, u) γ α k (q, u) dq du, (3.7) which writes again β k (t, τ, q, u) γ α k (q, u) dq du.

(3.8)

Therefore we can determine the coefficient β k of the particle approximation of W up to the solution the linear system (3.8).

Finally, we consider the last step of the algorithm. We want to build a particle approximation of G 1 from problem (2.9). We first note that in this problem, the advection operator is exactly the same than in problem (2.6). Therefore the macroparticles to approximate G 1 can be chosen as being localized exactly at the same phase space locations as the ones used to approximate G. Thus G 1 is approximated by

G 1 (t, q, u) = Np k=1 ω 1 k (t) δ(q = Q k (t)) δ(u = U k (t)), (3.9)
where the weights ω 1 k now depend on time t, which allows to take into account the effect of the source term on the right-hand side of the equation in (2.9). Once more, we can regularize this approximation by

G 1 (t, q, u) = Np k=1 ω 1 k (t) γ α (q -Q k (t)) 2 + (u -U k (t)) 2 .
(3.10)

Then we handle the right hand side of equation in problem (2.9) exactly as for W . Indeed, approximation of this term has the following shape η k (t, τ, q, u) γ α k (q, u) dq du, (3.12) which writes again η k (t, τ, q, u) γ α k (q, u) dq du.

(3.13) Therefore we can determine the coefficient η k up to the solution the linear system (3.13). Since G 1 is solution to problem (2.9), we finally have that

dω 1 k dt = η k , (3.14)
that needs to be numerically solved to get the full approximation of G 1 .

Np

η

  k (t, τ, q, u) (γ α ) (q -Q k (t)) 2 + (u -U k (t)) 2 .(3.11)Therefore in order to distribute this approximation on the macroparticles, we want to determineη k such that Ci Np k=1 η k (t, τ ) γ α k (q, u) dq du = Ci Np k=1

  r) and omit the dependency of [F ] ε and its partial derivatives in (t, r, v). First, we add the term E ε ∂[F ] ε /∂v on both sides of equation (2.14) to get (2.15). Then we rewrite the righthand-side to show up the term ∇ r,v [F ] ε and get (2.16). Using Remark 2.4(i) then Remark 2.2, we obtain respectively (2.17) then (2.18).


Magnus Söderberg
email: magnus.soderberg@mines-paristech.fr

Makoto Tanaka
email: mtanaka@grips.ac.jp

Spatial price homogeneity as a mechanism to reduce the threat of regulatory intervention in locally monopolistic sectors

Keywords: regulatory threat, district heating, Sweden JEL Classification: L12, L21, L51, L97

We claim that a reason for why unregulated investor-owned local monopolies do not always charge the monopoly price is that they are threatened by customer complaints that may lead to retaliations from local elected officials. When investor-owned monopolies are exposed to this threat they will mimic the price(s) of their neighbour(s); the stronger the threat, the higher the spatial price correlation. The threat increases when elected officials have pro-consumer preferences and neighbours are geographically close. The empirical analysis, based on a complete cross-sectional data set from the Swedish district heating sector in 2007, confirms the theoretical predictions.

INTRODUCTION

It has been noted that local monopolies owned by private investors do not always charge prices that are significantly higher than those charged by publicly owned monopolies, and/or that the introduction of formal price regulations does not always reduce prices [START_REF] Edwards | The effects of public ownership and regulatory independence on regulatory outcomes[END_REF][START_REF] Stigler | What can regulators regulate? The case of electricity[END_REF][START_REF] Wallsten | An econometric analysis of the telecom competition, privatization, and regulation in Africa and Latin America[END_REF]. One may ask, therefore, why local monopolies owned by private investors (IOMs) do not always set their prices at a higher level. Some commentators have suggested that the explanation may be that local monopolies owned by public bodies (POMs) are not relevant benchmarks since they might not maximise social welfare, 1 and others have pointed at the difficulty of designing regulatory models that appropriately incentivise price reductions (e.g. [START_REF] Joskow | Incentive regulation and its applications to electricity networks[END_REF]. These explanations assume that the price-setting incentives are primarily homogenous within sectors, whereas empirical studies have pointed at substantial pricesetting heterogeneity within monopolistic (network) sectors [START_REF] Altissimo | Inflation persistence and price-setting behaviour in the Euro area. A summary of the IPN evidence[END_REF][START_REF] White | Power struggles: explaining deregulatory reforms in electricity markets[END_REF]. In the present paper, we propose a new explanation to why IOMs depart from the textbook behaviour, an explanation that is based on heterogeneous threat of regulatory intervention.

Our claim is that locally, unregulated IOMs voluntarily hold back their prices when they are threatened by complaints from customers that are likely to result in retaliations from local elected officials. The probability of a customer complaint is positively correlated with the heterogeneity in pricing behaviour that the customer observes when comparing the price in her/his jurisdiction with the price(s) in neighbouring jurisdiction(s). Vote-maximising elected officials typically have an arsenal of different responses to choose from in this situation, ranging from relatively soft interventions such as more restrictive building/location permit processes and use of fuel types to more extreme interventions such as receivership (i.e. that the local public administration takes over the responsibility of the utility's operative work) and expropriation (i.e. that ownership is legally transferred to the local council). 2 However, elected officials are either pro-consumer or pro-firm, which affects their probability to intervene. IOMs respond to this threat by mimicking the prices set by their neighbours, with the level of threat being positively correlated with the spatial price correlation.

Our claim is related to the literature on regulatory threat, which says that firms self-regulate their present profits to gain higher profits in the future [START_REF] Brunekreeft | Regulatory threat in vertically related markets: the case of German electricity[END_REF][START_REF] Leidy | Rent dissipation through self-regulation: the social cost of monopoly under threat of reform[END_REF][START_REF] Block | The spillover effect of antitrust enforcement[END_REF]. Firms' incentive to react to threat has been justified from their desire to reduce the 1 Two established arguments are [START_REF] Niskanen | Nonmarket decision making: the peculiar economics of bureaucracy[END_REF] budget-maximising principle and [START_REF] Stigler | The theory of economic regulation[END_REF] suggestion about industry capture.

2 Both receivership and expropriation have been used in the locally monopolized electricity distribution sector in Sweden, and re-municipalisation (not renewal of concession contracts) has occurred in the German electricity distribution sector and the French water sector in recent years in response to high consumer prices. overall degree of scrutiny from outside agents, transfer scrutiny to other firms [START_REF] Decker | Implications of regulatory responsiveness to corporate environmental compliance strategies[END_REF] and/or prevent more stringent regulatory activity in the future [START_REF] Lutz | Strategic quality choice with minimum quality standards[END_REF]. In previous studies it has been taken as given that a firm's exposure to threat is positively correlated with its price level, or price cost margin [START_REF] Glazer | Pricing by the Firm Under Regulatory Threat[END_REF][START_REF] Brunekreeft | Regulatory threat in vertically related markets: the case of German electricity[END_REF][START_REF] Bawa | Dynamic behaviour of a firm subject to stochastic regulatory review[END_REF]. 3 The validity of this claim hinges on the assumption that an isolated price level is informative to policy makers.

However, even if one imposes the weaker assumption that policy makers can react to price changes, this can be an unrealistically strong assumption when outside agents suffer from a lack of cost information. 4 As an alternative, it has been suggested that monitoring agents can compare conditions set simultaneously by (many) firms and that this can explain firms' behaviour relatively well [START_REF] Gilpatric | Regulatory enforcement with competitive endogenous audit mechanisms[END_REF]. 5 In an unregulated market where customers cannot rely on policy makers to effectively monitor monopolies, they can themselves form expectations about the monopolies' degree of 'abusiveness'. If customers compare conditions set simultaneously by different monopolies, they can compare their own conditions with those set in neighbouring jurisdictions. The preference for closer neighbours follows from [START_REF] Dixit | Trade expansion and contract enforcement[END_REF] random matching model, which postulates that individuals are more likely to 'match' the closer (according to physical distance or some socioeconomic criteria) they are located. The presence of strategic interaction between neighbouring jurisdictions has been confirmed empirically in studies on a broad range of local policy setting (e.g. [START_REF] Bordignon | In search of yardstick competition: a spatial analysis of Italian municipality property tax setting[END_REF][START_REF] Buettner | Local business taxation and competition for capital: the choice of the tax rate[END_REF][START_REF] Brueckner | Testing for strategic interaction among local governments: the case of growth controls[END_REF]. Customers' tendency to use conditions in neighbouring jurisdictions as a basis for why they complain about monopolistic conditions was also observed by [START_REF] Söderberg | Uncertainty and regulatory outcome in the Swedish electricity distribution sector[END_REF] in his review of a large number of customer complaints in the regulated Swedish electricity distribution sector.

Several studies in the marketing literature suggest that when a price is perceived as unfair, it provokes anger and increases the likelihood of customer responses, e.g. switching of supplier [START_REF] Antón | The mediating effect of satisfaction on consumers' switching intention[END_REF][START_REF] Athanassopoulos | Customer satisfaction cues to support market segmentation and explain switching behaviour[END_REF][START_REF] Campbell | Perceptions of price unfairness: antecents and consequences[END_REF]. 6 Similar arguments have been used to build theoretical models in the economic literature (Di Tella and Durba, 2009; Rotemberg, forthcoming, 3 More recent theoretical studies have extended the earlier models of regulatory threat in different directions, e.g. the pricing decision under the threat of potential divestiture of firms [START_REF] Tanaka | The Effects of Uncertain Divestiture as Regulatory Threat[END_REF] and the pricing of a utility with an expanding network facing a threat of regulatory intervention [START_REF] Chisari | Pricing dynamics of network utilities in developing countries[END_REF]Kessides, 2009). 2005). In [START_REF] Rotemberg | Customer anger at price increases, changes in the frequency of price adjustment and monetary policy[END_REF] model, firms internalise that customers react negatively when they become convinced that a price is unfair. Di Tella and Dubra (2009) reach a similar conclusion and show that equilibria exist where customers are not angry. 7 Rotemberg (forthcoming) introduces the notion of a reference price that the current price is compared with to form an opinion about whether a price is fair or not. He suggests that this reference price can be a price charged previously by the firm.

As pointed out above, the present paper takes an alternative position as it uses prices set simultaneously by neighbours as reference price(s).

Lastly, we also draw on the literature that focuses on the influence of local political ideology on regulatory decisions. These studies have suggested that left-wing governments tend to be relatively more pro-consumer [START_REF] Holburn | Interest group representation in administrative institutions: The impact of consumer advocates and elected commissioners on regulatory policy in the United States. POWER working paper 002[END_REF]Holburn andVanden Bergh, 2006, Besley and[START_REF] Besley | Elected versus appointed regulators: theory and evidence[END_REF][START_REF] Cambini | Regulatory independence and political interference: evidence from EU mixed-ownership utilities' investment and debt[END_REF]. Similar to [START_REF] Biggar | The effect of future price expectations on customers' willingness to make sunk investments in reliance on a monopoly service[END_REF], we highlight the influence of local political ideology on the pricing strategies of district heating utilities, even in the absence of any formal regulatory mechanism.

The claim that spatial price heterogeneity is a source of regulatory threat in local monopolistic sectors

has not yet been modelled theoretically, and it has only received sporadic empirical attention. The purpose of this paper is to provide a theoretical foundation for this mechanism and to evaluate its existence empirically based on the Swedish district heating sector. This sector consists of a mixture of unregulated IOMs and POMs that are confined by municipal borders. The Swedish district heating sector provides a unique setting for studying monopolies' price incentives since, contrary to most network sectors, the IOMs are not subject to formal price control mechanisms. Also, the Swedish electoral system can be characterised as bi-partisan where the ruling party/coalition is either left-or right-wing. There is empirical evidence that left-wing local councils in Sweden generally intervene more in markets [START_REF] Pettersson-Lidbom | Do parties matter for economic outcomes? A regression-discontinuity approach[END_REF], and [START_REF] Biggar | The effect of future price expectations on customers' willingness to make sunk investments in reliance on a monopoly service[END_REF] find that the Swedish district heating utilities adopt a more pro-consumer pricing behaviour under left-wing local councils.

We utilise a complete cross-sectional data set from 2007 that has not been used before in published work.

Our theoretical model, which can be viewed as a new variant of the Bertrand model, complements existing theoretical explanations for regulatory threat by showing that local monopolists, under certain conditions, have incentives to mimic the prices set by their neighbours. Similar 'neighbouring' effects have been observed previously in the context of regulatory threat. For example, [START_REF] Block | The spillover effect of antitrust enforcement[END_REF] show that the cost of highway construction is reduced after antitrust enforcement in neighbouring jurisdictions, and [START_REF] Eckert | The geographical distribution of environmental inspections[END_REF] find that firms are more likely to comply with environmental regulations when neighbours have recently been found in violation.

Several empirical studies have looked at firms' pricing behaviour in response to policy makers' (likely) actions [START_REF] Olmstead | Rationing without Government: The West Coast Gas Famine of 1920[END_REF][START_REF] Erfle | Media, Political Pressure, and the Firm: The Case of Petroleum Pricing in the Late 1970s[END_REF][START_REF] Driffield | Effectiveness and Effects of Attempts to Regulate the UK Petrol Industry[END_REF]. The study that is most similar to ours is [START_REF] Ellison | Coordinating on Lower Prices: Pharmaceutical Pricing Under Political Pressure[END_REF] investigation of pharmaceutical prices in the U.S. during a period of a relatively intense scrutiny from policy makers.

Similar to our study, they suggest that firms voluntarily hold back prices to reduce the threat of regulatory intervention and that firms' responses are positively correlated with their degree of regulatory vulnerability. However, our study differs from [START_REF] Ellison | Coordinating on Lower Prices: Pharmaceutical Pricing Under Political Pressure[END_REF] in terms of both market structure and source of threat since they consider a market subject to competition and when the level of threat is related to the firms' own price levels.

The paper proceeds with a section that lays the foundation of our theoretical model. Our propositions are presented in Section 3. A background on the Swedish district heating sector is given in Section 4, and the empirical analysis is presented in Section 5. Section 6 concludes.

THE MODEL

Consider a set of unregulated locally monopolistic firms i where each firm faces its own demand for heat, ) ( i i p q , and i p denotes the price. We assume that 0

< ′ i q and 0 ) ( lim = ∞ → i i p p q i . The cost of providing the heat is denoted )) ( ( i i i p q c
. If the firm is an IOM, its profit is expressed as

)) ( ( ) ( ) ( i i i i i i i i p q c p q p p - = π ,
∫ ∞ ′ ′ = i p i i i i i p d p q p cs ) ( ) ( , where 0 ) ( ) ( < - = ′ i i i i p q p s c and 0 ) ( ) ( > ′ - = ′ ′ i i i i p q p s c
for the relevant price range. . Lastly, we mention some relevant facts that we will come back to later:

m i i i p p p < < , i i i p π π > ) ( , i i i s c p cs < ) ( and i i i w s p sw < ) ( hold, since 0 > ′ i π , 0 < ′ i s c and 0 < ′ i w s
, respectively.

Structure of firms

A fully privately owned firm is only concerned with its own profit. A public firm, on the other hand, is concerned with consumer surplus as well as its own profit. Considering the private and public structures of firms, we define the surplus of firm i as follows:

) ( ) ( ) ( i i i i i i i p cs p p S α π + = , (1)
where

[ ) 1 , 0 ∈ i α . 8 When 0 = i α
, the firm is fully private and only seeks to maximise its profit. When

i α < 0
, the firm is a public firm that maximises the weighted sum of its profit and consumer surplus.

A larger i α implies that the firm has a more public character, putting more emphasis on consumer surplus. 9 When i α approaches 1, the firm becomes the (local) council itself and seeks to maximise full social surplus. We will discuss this extreme case later.

Noting that 0 ) ( > ′ ′ i i p s c and [ ) 1 , 0 ∈ i α , ) ( ) ( i i i i i p s c p s c ′ ′ < ′ ′ α holds. After adding ) ( i i p π on both sides, we obtain ) ( ) ( ) ( i i i i i i i p s c p p S ′ ′ + ′ ′ = ′ ′ α π ) ( ) ( i i i p s c p ′ ′ + ′ ′ < π , 0 < (2)
where the last inequality comes from the strict concavity of )

( i i p sw . Thus, ) ( i i p S is strictly concave. The maximiser of the firm's surplus is denoted ) ( max arg i i s i p S p = . s i p is unique since ) ( i i p S is
s i i p p < , i.e. 0 ) ( ) ( ) ( > ′ + ′ = ′ i i i i i i i p s c p p S α π (3) for s i i p p < . Furthermore, i i i S p S > ) ( holds for s i i i p p p < < since 0 ) ( > ′ i i p S
for this price range.

Expected surplus of firms

Next we introduce a functional relationship between the probability of regulatory intervention and the prices set by the firms. The probability of regulatory intervention is assumed to be a function of the price difference between two neighbouring firms, i.e. ) (

i i i p p - - θ
. The regulatory authority observes prices as well as introduces price regulation in district i with probability ) (

i i i p p - - θ where ) (⋅ i θ is increasing, i.e. 0 > ′ i θ . i i p p - -
can be both positive and negative. We consider the range of probability, 1 0 < < i θ , for the relevant price range. 10 It should be noted that the probability of intervention is positive even when firm i charges a price below the neighbour's price. Furthermore,

we assume that ) (⋅ i θ is strictly convex, i.e. 0 > ′ ′ i θ for the relevant range.
Once the regulatory authority decides to intervene in district i, price regulation is introduced. We consider that the price will be regulated at i p . At this level, the firm obtains a constant surplus,

i i i i i i i i i i i s c p cs p p S S α π α π + ≡ + = ≡ ) ( ) ( ) (
. Considering a risk-neutral firm, we can now express its expected surplus as

i i i i i i i i i i i i S p p p S p p p p ES ) ( ) ( )) ( 1 ( ) , ( - - - - + - - = θ θ ) ) ( )( ( ) ( i i i i i i i i S p S p p p S - - - = - θ . ( 4
)
The second term on the second line of (4) can be interpreted as a form of expected penalty. When firm i chooses some price level i p given i p -, its surplus ) ( i i p S is reduced by

i i i S p S - ) ( with probability ) ( i i i p p - - θ
. This highlights that the expected surplus of firm i depends not only on its own price i p , but also on i p -in the neighbouring district -i. Using the terms of profit and consumer surplus, expected surplus can be written as )) ( (

i i i i i i i i i i s c cs cs ES α π α π θ α π + - + - + = .
(5)

Firm i maximises its expected surplus given the price of neighbouring firm -i. The first-order condition for firm i can be written as

0 ) ( ) 1 ( = - ′ - ′ - = ∂ ∂ i i i i i i i S S S p ES θ θ . ( 6
)
Recall that

i i i S p S > ) (
for

s i i i p p p < < , 0 > ′ i S for s i i p p < , and 0 < ′ ′ S , as discussed in subsection 2.1. Moreover, note that 1 0 < < i θ , 0 > ′ i θ , and 0 > ′ ′ i θ , as discussed here. We then have 0 ) ( 2 ) 1 ( 2 2 < - ′ ′ - ′ ′ - ′ ′ - = ∂ ∂ ≡ i i i i i i i i i i S S S S p ES θ θ θ φ (7) for s i i i p p p < <
regarding the second-order condition.

The council

Before turning to the analysis of regulatory threat, we explore the decision of the council in relation to the model of the firm. As discussed in the introduction of this section, the council wants to set the price at i p to maximise the social surplus ) ( i i p sw in district i. This coincides with the decision of the firm, which wants to maximise its expected surplus ) , (

i i i p p ES - when i α approaches 1. When i α approaches 1, we have )) ( ( ) ( i i i i i i i i i i i i s c cs cs w s sw sw ES + - + - + = - - = π π θ π θ and ) ( ) 1 ( i i i i i p ES w s sw w s i i - ′ - ′ - = ∂ ∂ θ θ . Noting that 0 ) ( = ′ i i p
0 ) ( 2 ) 1 ( < - ′ ′ - ′ ′ - ′ ′ - i i i i i i i w s sw w s w s θ θ θ . ( 8
)

EFFECTS OF REGULATORY THREAT

We investigate the effects of regulatory threat in detail. As discussed in the previous section, the expected surplus of the firm is affected by the neighbouring price through the probability of regulatory intervention. Thus, even though the firm is a local monopolist, each firm strategically chooses its price in each jurisdiction, taking into account the price(s) of neighbouring jurisdiction(s).

This can be regarded as a new variation of the Bertrand model where spatial price competition is driven by regulatory threat.

Technically, we can obtain the Nash equilibrium by solving the first-order conditions ( 6

s i i i p p p < < * . Proof. Noting that 0 ) ( = ′ s i i p S , i s i i S p S > ) ( and 0 > ′ i θ , we have 0 ) ) ( ( * < - ′ - = ∂ ∂ - -= = i s i i i p p p p i i S p S p ES i i s i i θ .
Thus, firm i can make more profit by choosing a price less than s i p for any given *

i p -. Next, noting that 1 0 < < i θ and 0 ) ( > ′ i i p S , we have 0 ) ( ) 1 ( * > ′ - = ∂ ∂ - - - = = i i i p p p p i i p S p ES i i i i θ .
This implies that firm i can make more profit by choosing a price greater than i p for any given In what follows, our analysis is concerned with the relevant price range of

s i i i p p p < < unless otherwise stated.
We further examine the strategic reaction of the firm under regulatory threat. The first-order condition (6) for firm i yields the best response function )

( i i i p BR p - =
. When the decisions of the firms are 'strategic complements', the slope of the best response curve is positive. In contrast, when the decisions of the firms are 'strategic substitutes', the slope of the best response curve is negative. 11 In the next proposition we show that i p and i pare strategic complements.

Proposition 2. Prices are strategic complements, i.e. 0

2

> ∂ ∂ ∂ - i i i p p ES . Proof. Noting that i i i S p S > ) ( , 0 > ′ i S , 0 > ′ i θ and 0 > ′ ′ i θ , we have 0 ) ( 2 > - ′ ′ + ′ ′ = ∂ ∂ ∂ ≡ - i i i i i i i i i S S S p p ES θ θ ψ . (9)

■

This proposition implies that the slope of the best response curve is positive:

0 ) ( 2 2 2 > - = - = ′ ∂ ∂ ∂ ∂ ∂ - - i i i i i p ES p p ES i i i i p R B φ ψ . ( 10
)
The current model is different from the standard Bertrand model. Each firm is a local monopolist that faces its own demand. Despite this fact, regulatory threat induces each local monopolist to react strategically, taking into account the price(s) of neighbouring jurisdiction(s). As a result, regulatory threat leads to a positive price correlation among jurisdictions. An interesting question is how the structure of the firm affects its strategic reaction. Thus, we next examine the effects of changing i α on the slope of the best response curve, )

( i i p R B - ′
. We obtain the following results:

Proposition 3. When i α increases, the slope of the best response curve decreases, i.e.

0 < ∂ ′ ∂ i i R B α .
11 See e.g. [START_REF] Tirole | The Theory of Industrial Organization[END_REF].

Proof. See Appendix 1. ■ If a firm is fully private with 0 = i α , the slope of the best response curve is the steepest. If a firm gains a more public character with a larger i α , the slope of the best response curve becomes flatter.

These results follow from the publicly owned firm's incentive to restrain its price, taking account of the consumer surplus in its district. Therefore, when i α increases, the slope of the best response curve decreases, and hence the positive correlation of prices among districts becomes weaker.

As discussed in Section 2, when i α approaches 1, the firm becomes the council itself, and the price it chooses approaches i p . i p is determined irrespective of the neighbouring price since i p is the maximiser of full social surplus )

( i i p sw . Consequently, i i R B α ∂ ′ ∂ approaches 0 when i α approaches 1.
Lastly, we extend and refine our model by introducing a proxy parameter (0,1)

i μ ∈
that represents the strength of regulatory threat. We assume that a higher value of i μ corresponds to a situation where the regulatory threat is stronger. Specifically, two situations are worth noting, based on descriptive statistics presented in Section 4. First, utilities operating under pro-consumer councils will face more severe downward pressure on prices than those operating under pro-firm councils. Thus, it is assumed that i μ takes a higher value when elected officials have more consumer-friendly preferences. Second, neighbours located relatively close to each other are subject to more serious price comparison than distant neighbours. Hence, it is assumed that i μ takes a higher value when neighbouring jurisdictions are geographically closer.

Given some observed level of price difference between two districts, the probability of regulatory intervention should be higher when i μ is larger. Thus, it is straightforward to rewrite the probability of intervention as ( )

i i i i p p μ θ - -
. 12 The expected surplus of firm i can now be rewritten as

( , ) ( ) ( )( ( ) ) i i i i i i i i i i i i ES p p S p p p S p S μ θ - - = - - - . ( 11
)
12 We can also assume multiple parameters for the strength of the regulatory threat: for example, (0,1)

THE SWEDISH DISTRICT HEATING SECTOR

The district heating technology builds on the principle of centralised heat production where the heat is distributed under high pressure to customers' properties through a network of underground pipelines carrying hot water or steam. At the customer's property, a heat exchanger extracts heat energy and the cooler water is returned to the heat centre to be re-heated and re-distributed. District heating meets approximately 50% (or 47 TWh) of the total heat demand in Sweden and it is the most common heating alternative for multi-dwelling houses in 234 of the total 290 municipalities (SCA, 2009;SCB, 2009). Networks only exist in densely populated areas over a certain size and it is rare that networks are connected over municipal borders.

In 2007, 30 % of the utilities were IOM and 36% operated in municipalities with a right-wing dominated council. 13 While ideological composition of the local council must be treated as endogenous in the subsequent analysis, we will argue that IOMs and POMs are structurally similar and can be considered as random samples. This argument is based on the high degree of similarity 13 In our empirical investigation, an investor-owned district heating utility is defined as a utility where private investors control any share > 0 of the utility. [START_REF] Söderberg | The role of model specification in finding the influence of ownership and regulatory regime on utility cost: the case of Swedish electricity distribution[END_REF] shows that private investors tend to determine the economic behaviour of Swedish energy utilities irrespective of whether they are minority or majority owners.

between the two sub-samples in terms of climate conditions, population characteristics, input price levels and network characteristics (detailed statistics are provided in Table A2.1 in Appendix 2).

Municipal-level data from 2007 shows that the average price per MWh for a multi-dwelling property that consumes 193 MWh of heat is 653 SEK and 663 SEK when the utility is a POM and operating under left-and right-wing councils, respectively. 14 When the utility is IOM, the price is 688 SEK when the council is left-wing and 710 SEK when the council is right-wing. Although these prices are not statistically different, they do indicate that utilities under left-wing councils charge lower prices than those under right-wing councils, and that POMs charge lower prices than IOMs.

Pair-wise correlations between prices in municipality i and its neighbours i -, where i is IOM and

{ } 1 ..., , 2 , 1 - = - i i
is ordered in distance between i and i -, show that the correlation is 0.44 between i and 1 = i and 0.28 between i and 2 = i (the results are similar when i operates under left-and right-wing councils). The correlations for higher values of i are insignificant. Similar calculations for when i is POM show that the correlation is 0.31 when 1 = i and the council is left-wing.

Correlations for higher values of i -, and when the local council is right-wing, are insignificantly different from 0. The picture that emerges from these descriptive figures is that (i) neighbouring prices are significantly correlated, yet the correlation is generally confined to the closest neighbour, (ii) IOMs are substantially more affected by neighbours' prices than are their POM counterparts, and

(iii) utilities operating under left-wing (rather than right-wing) councils attach more weight to the price of their closest neighbour.

There is no statistics available on complaints about district heating prices, but a recent study on the characteristics of district heating coverage in local and national Swedish newspapers shows that prices are the most frequently covered topic and that the tone is more critical when the utility is an IOM [START_REF] Palm | Medias rapportering om fjärrvärme[END_REF]. This shows that customers have an effective channel to reach out with their complaints to elected officials. Anecdotal evidence from utility representatives and media reporting also indicates that municipalities can (and do) make decisions that have a substantial impact on utilities' financial performance.

14 These prices represent the price paid by a multi-dwelling property that is considered to be a 'national average property'. See Section 5.1 for further details.

EMPIRICAL EVALUATION

Based on the insights from Section 4, we want to empirically investigate whether i p is influenced by 1 = -i p , i.e. the price charged by i's closest neighbour. Propositions 3 and 4 postulate that the relationship between i p and 1 = -i p is stronger when i is an IOM and when i faces a stronger regulatory threat. This translates into a functional form that can be written as

β x W i i i p p + = = -1 γ ,
where γ is the parameter denoting the spatial association between i p and

1 = -i p
. W is a weight matrix consisting of predetermined proxies for the level of threat between i and all i -.

To allow public and private utilities to have different values of γ, and for private utilities to respond differently to left-and right-wing dominated local councils, it is necessary to add two additional spatial terms where the elements in the second and third weight matrixes are multiplied with the type of ownership and ownership/political majority, respectively. The measures of ownership and political orientation can be continuous, as in the theoretical model, yet here we use two binary variables where b takes the value 1 if the utility is classified as an IOM and r takes the value 1 if the local council has a right-wing majority. 15 To increase flexibility, we allow the price variables to be transformed by λ,

where

[ ] 1 , 1 - = λ
and where λ=0 represents the natural logarithm. Hence, the general model that we want to estimate is formulated as

i i i i i i p p p p ε γ γ γ λ λ λ λ + + + + = = - = - = - β x W W W 1 3 3 1 2 2 1 1 1 , ( 12
)
where the elements in W 1 , W 2 , and W 3 are calculated as

⎪ ⎩ ⎪ ⎨ ⎧ = - = - - - otherwise 0 1 for 1 , , , 1 i d w i i i i , ⎪ ⎩ ⎪ ⎨ ⎧ = - = - - - otherwise 0 1 for 1 , , , 2 i d b w i i i i i and ⎪ ⎩ ⎪ ⎨ ⎧ = - = - - - otherwise 0 1 for 1 , , , 3 i d b r w i i i i i i . i i d - ,
is the Euclidian distance between the centres of the largest urban communities in i and -i. The inversed distance reduces the weight for more distant neighbours. Similar weighting structures are common in the applied spatial literature (e.g. [START_REF] Lambert | A two-step estimator for a spatial lag model of counts: theory, small sample performance and an application[END_REF][START_REF] Zhou | Predicting the distribution of households and employment: a seemingly unrelated regression model with two spatial processes[END_REF]Lee and Yu, 2009). 16 However, when only -i=1 is included on the RHS, it is possible that distance is already internalised in the model, and hence the value of including the distance will be investigated more closely.

The x vector consists of b, r and cost characteristics. Cost is assumed to be influenced by the unit costs of fuel ( f i p ) and labour ( l i p ). 17 In addition, firms that rely on physical networks typically have a cost structure that depends on network characteristics. We include the amount of energy produced (q i ) to account for economies of scale in the production, and total network length ( i ) to account for scale effects in the distribution. Also, two strategic behaviours can influence the pricing behaviour when networks are subject to expansion. First, [START_REF] Chisari | Pricing dynamics of network utilities in developing countries[END_REF] point out that a utility can have incentives to keep its price low to attract customers during the initial phase of the expansion, and then raise the price as the network approaches its optimum size. This suggests that price and market share are positively related. However, non-adopters of district heating would prefer a high price to increase public revenue, yet this group will only dominate as long as district heating has a market share below 50% in the heating market. Once the market share exceeds 50%, adopters will be in majority and will lobby for lower prices. This line of argument suggests that the relation between market share and price is negative. To control for these potential effects, we add a variable m i for the district heating's market share in the local residential energy market. Utilities' mix of customer types is accounted for by including share of detached houses, s i . Nine firm dummies are also included to control for instances when a particular utility is responsible for the district heating operation in more than one municipality.

Data

Data is collected from the Swedish Energy Markets Inspectorate, The Swedish District Heating Association, Statistics Sweden, the 'Nils Holgersson's annual price comparisons', and the utilities/municipalities directly. The data set is cross-sectional for the year 2007 and represents all 242 municipalities where district heating is a significant source of heating in the largest urban area, excluding Gotland, which is an island. Descriptive statistics and further details about the variables in (12) are provided in Table A2.2 in Appendix 2.

The price i p is the average price (fixed plus variable) paid per MWh for a multi-dwelling property that consumes 193 MWh of heat and requires 3 860 m 3 of water pass-through per year. The correlations with prices for other standardised customer types (80MWh, 500 MWh and 1 000 MWh) are high (between 0.94 and 0.96), which means that i p can be viewed as the average price for all multi-dwelling properties.

17 Cost of capital is not considered since it is not subject to geographical heterogeneity.

Fuel price is total costs of fuels divided by total amount of kWh produced. This price can be negative since utilities are paid to dispose of public waste, which sometimes is a substantial fuel type. Labour price is average municipal salary (net local taxes) in the public sector. This is preferred to figures based on utilities' own accounting statements since it eliminates the risk of including rents captured by strong unions and self-rewards by executives.

Estimation

The derivation of an estimable spatial autoregressive model is not covered here since it has been thoroughly dealt with elsewhere (e.g. [START_REF] Mobley | Estimating hospital market pricing: an equilibrium approach using spatial econometrics[END_REF][START_REF] Revelli | Performance rating and yardstick competition in social service provision[END_REF][START_REF] Zhou | Predicting the distribution of households and employment: a seemingly unrelated regression model with two spatial processes[END_REF].

Estimating ( 12) with the non-linear least square estimator, i.e. assuming strict exogeneity, shows that the specification suffers from over-parameterisation. A grid search approach starting with 1 -= λ and increasing in steps of 0.1 until λ=1 is therefore applied, showing that the residual sum of squared (RSS) for λ=1 is not significantly higher than the RSS for any other value of λ. This confirms previous findings that linear price models perform well in comparison with unconstrained transformations when firm heterogeneity is unobserved [START_REF] Cropper | On the choice of functional form for hedonic price functions[END_REF].

As we proceed, two problems can be noted. First, the three terms containing i p -are endogenous by construction (LeSage and Pace, 2009) and, second, the residuals ε i are likely to be heteroscedastic. 18Instrumental variable approaches with heteroscedasticity-consistent standard errors are offered by both 2SLS and two-stage, or 'optimal', GMM (OGMM),19 yet Lin and Lee (2010) note that OGMM is more efficient and potentially more robust. [START_REF] Kelejian | A generalized spatial two-stage least squares procedure for estimating a spatial autoregressive model with autoregressive disturbances[END_REF] argue that a subset of neighbours' covariates can be used as instruments to arrive at consistent spatial parameters. We chose this strategy and use neighbours' fuel and labour prices as instruments for p -i . Further endogeneity problems can occur if q i is correlated with ε i . Such correlation can occur from unobserved property characteristics. We use share of energy delivered to industrial customers (ind i ) and population in urban areas (pop i ) to control for this potential endogeneity. Although utilities determine the size of the network themselves, i is treated as exogenous since decisions to extend the network are typically made several years prior to the actual construction. In Section 4 we showed that public and private utilities are similar when comparing fundamental supply and demand conditions. b i is therefore treated as exogenous. Ideological composition of the local council can be endogenous to local government tariffs. 20 The indicator variable for right-wing majority (r i ) is instrumented with r i,t-2 , which refers to the situation in the previous electoral period.

An OLS estimation is displayed in Column (a) in Table 1. Column (b) shows the output for when p -i , q i and r i are treated as endogenous, using the OGMM estimator. The results are consistent in terms of signs and significance levels, but γ 2 (γ 3 ) increases (decreases) in magnitude as the exogeneity assumptions are relaxed. Column (c) shows ( 12) when ) is apparently strong enough to statistically eliminate the threat of intervention. 20 A strategically behaving council can increase the price when income goes up, and [START_REF] Leigh | Economic voting and electoral behaviour: how do individuals, local, and national factors affect the partisan choice?[END_REF] shows that income (among several other socio-economics factors) influences voting behaviour.

f i i i p d 1 1 1 , = - -= - , f i i i i p d b 1 1 1 , = - -= - , f i i i i t i p d b r 1 1 1 , 2 , = - -= - - , l i i i p d 1 1 1 , = - -= - , l i i i i p d b 1 1 1 , = - -= - , l i i i i t i p d b r 1 1 1 , 2 , = - -= - - , ind i , pop i, i t i b r 2 , - .
Other coefficients reveal that i p increases for higher q i and decreases for higher i which is consistent with the view held by industry representatives and the results by [START_REF] Biggar | The effect of future price expectations on customers' willingness to make sunk investments in reliance on a monopoly service[END_REF]. Input prices are positively correlated with p i . The tests of Hansen's J show that instruments are correctly excluded from the main equation and uncorrelated with ε i in both Columns (b) and (c).

CONCLUSIONS

In this paper we claim that a plausible explanation for why unregulated investor-owned local monopolies do not always charge the monopoly price is that doing so may lead to customer complaints that in turn may lead to retaliations from local elected officials. Customers compare their prices with those in neighbouring jurisdictions, and the probability of a complaint is positively correlated with the degree of spatial price heterogeneity. Elected officials have incentives to intervene when customers (i.e. voters) complain, yet the probability of intervention is reduced when elected officials have pro-firm preferences. We show that when utilities are subject to regulatory threat that originates from spatial price comparisons, the equilibrium price is located between the efficient and monopoly prices. Also, prices charged by neighbouring utilities are strategic complements, meaning that their prices are positively correlated. The spatial price correlation between neighbours increases when IOMs are exposed to relatively strong threat, i.e. when elected officials have pro-consumer preferences and neighbours are geographically close.

The empirical analysis is based on a complete cross-sectional data set from the Swedish district heating sector in 2007. This sector is unique in that it consists of local monopolies that are owned by both private investors (IOM) and municipalities (POM), yet the monopolies are not subject to any formal price regulation. The analysis supports the theoretical predictions by showing that 1) POMs tend to set lower prices than IOMs; 2) IOMs that operate under left-wing councils (i.e. councils that are relatively pro-consumer) have lower prices than IOMs under right-wing councils; 3) IOMs under left-wing councils internalise as much as 50% of their neighbours' prices; and 4) when the threat of intervention is lower, i.e. when IOMs operate under right-wing councils, there is no sign that they take their neighbours' prices into consideration.

Policy-wise one can conclude that the threat of regulatory intervention can effectively reduce IOMs' incentive to use their marker power, yet a favourable situation for consumers is fragile since a change in the council's ideological preferences can reduce the threat, which may lead to a substantial price increase.

APPENDIX 1

Proof of Proposition

3 Note that 0 > ′ i π , 0 < ′ ′ i π , 0 < ′ i s c and 0 > ′ ′ i s c ,
i i i i s c cs - - < - < π π . ( 14
)
Next, rearranging (8) in subsection 2.3 yields

i i i i i i i i i w s w s w s sw w s ′ ′ - - ′ ′ < - ′ ′ - ′ ′ - < ) 1 ( θ θ θ ′ ′ - - ′ ′ < ′ ′ - ′ ′ - < ) 1 ( ) 1 ( 0 (16)
Then, it follows from ( 14) and ( 16) that:

) ) 1 ( )( ( ) ) 1 )(( (

i i i i i i i i i i i i s c cs s c s c π θ π θ θ θ π π ′ ′ - - ′ ′ - - < ′ ′ - ′ ′ - - . ( 17
)
We now check the sign of

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ + - = ∂ ∂ ∂ ∂ ∂ ′ ∂ i i i i i i i i i R B α φ α ψ φ α ψ φ 2 1 . The sign of i i R B α ∂ ′ ∂
depends on the sign inside the bracket. Noting that ( 13) and ( 17) hold, we obtain:

) )( 1 ( i i i i i i i i i i i i s c s c ′ ′ ′ - ′ ′ ′ - ′ = ∂ ∂ + ∂ ∂ - π π θ θ α φ ψ φ α ψ [ ) ) 1 )(( ( i i i i i i i s c s c ′ ′ - ′ ′ - - ′ ′ + θ θ π π θ ] ) ) 1 ( )( ( i i i i i i s c cs π θ π θ ′ ′ - + ′ ′ - + 0 < . ( 18
i i i i i i i i i i i i i BR S ψ φ φ ψ μ μ μ φ ψ μ φ ′′ ⎛ ⎞ ′ ∂ ∂ ∂ ⎜ ⎟ = - + ⎜ ⎟ ∂ ∂ ∂ ⎝ ⎠ ∂ = - ∂ > (23)
■

  proximity of neighbouring districts. Then the probability of intervention can be expressed as ( ) this alteration does not change the main results of the model. In the following, the double bars are used to express all corresponding terms that include i μ . We then obtain the following results regarding the effects of changing i μ on the slope of the best This proposition implies that the response of a firm to the neighbouring price is positively related to the strength of the regulatory threat. Firms are exposed to stronger regulatory threat when elected officials are more pro-consumer and neighbouring jurisdictions are geographically closer. When i μ increases, the slope of the best response curve increases and hence the positive correlation of prices among districts becomes stronger.

  can also observe that (c) is not as well-behaved as (a) and (b), which shows that it is important to incorporate the distance between neighbours in order to arrive at reasonable estimates. This finding supports Proposition 4.Based on estimates in Column (b), the average predicted price charged by POMs (669.2 SEK/MWh) is lower than the price set by IOM under left-wing councils (670.4 SEK/MWh), which in turn is lower than the price set by IOMs under right-wing councils (713.4 SEK/MWh). Although these prices are not statistically different, they point in the same direction as Proposition 1. As expected, there is no evidence that POMs take the prices of their closest neighbours into consideration. IOMs operating under left-wing councils ( 2 1 γ γ + ), i.e. when the councils are relatively pro-consumer, internalise 50% of the price adjustments made by their closest neighbours. The positive sign supports Proposition 4. Further support for Proposition 4 is provided by the e. that pro-consumer preferences of elected officials increase the threat. The combination of private ownership and the pro-firm preferences of right-wing councils (

Table 1 . Estimation output.

 1

													OLS	OGMM a	OGMM
													(a)	(b)	(c)
	Variable	Mean	SE	Mean	SE	Mean	SE
	d	1 -= , -i i	1	p	-	i	=	1	0.0245	0.0239	0.0161	0.0235
	i b	d	1 -= 1 , -i i	p	-	i	1 =	0.3275 **	0.1450	0.4843 **	0.2059
	i r	i b	d	1 -= 1 , -i i	p	-	i	1 =	-0.0227	0.2672	-0.6252	0.4779
	p	-i	=	1		0.2707	0.2502
	i p b	-i	1 =	0.6741	0.5182
	i i p b r	-i	1 =	-0.9566 *	0.4921
	b i												12.011	16.161	1.1954	21.248	-453.87	354.52
	i i b r								14.697	19.346	50.794	34.914	687.66 **	341.47
	q i												0.0464 **	0.0194	0.0576 ***	0.0203	0.0647 ***	0.0218
	i										-0.2552 **	0.0886	-0.2763 ***	0.0876	-0.2863 ***	0.0901
	p	i	f								0.1065 **	0.0528	0.0994 **	0.0479	0.1073 *	0.0556
	p	l i								0.0114 *	0.0067	0.0121 *	0.0063	0.0053	0.0080
	m i										-49.450	30.854	-34.654	30.775	-31.138	29.391
	s i												-207.85 ***	70.907	-219.44 ***	68.247	-185.68 **	79.268
	Constant	437.38 ***	142.44	421.10 ***	133.01	382.11 ***	166.92
	Ownership	Yes	Yes	Yes
	dummies
	Hansen's J	8.245	5.522
	Hansen's P>J	0.221	0.238
	R 2								0.453	0.446	0.375
	No. obs.	242	242	242
	Notes: * p < 0.10, ** p < 0.05, *** p < 0.01. SE are robust to arbitrary heteroscedasticity.

a Instruments:

  as stated in the beginning of Section 2. It follows from

	s	′ i w	<	0		and	s	′ ′ i w	<	0	in the beginning of Section 2 that	0	< π	i ′	<	-	i c ′ s	, and	0	<	s c	i ′ ′	<	i π ′ ′ -	. Thus, we
	obtain
			π	i ′	s c	i ′ ′	<	π	i ′ c s ′	i ′	.										(13)
	Note that	i π > , and i π	cs	i	<	c	s	i ′	, as stated in the beginning of Section 2. It follows from
	i sw	=		π	i	+	cs	i	<	π	i	+	c	s	i	=	s	i w	, that
			0														(							)

Table A2 .2 Descriptive statistics.

 A2

	Variable	Mean	Std.dev.	Min	Max
	i p													668.85	75.730	405.25	815.00
	d	1 -= 1 , -i i		p	-	i	1 =	57.047	155.13	6.7036	1 712.6
	i b	d	1 -= 1 , -i i	p	-	i	1 =	14.607	29.755	0	165.55
	i r		i b	d	1 -= 1 , -i i	p	-	i	1 =	6.9246	21.805	0	164.53
	b i																		0.6983	0.4599	0	1
	i i b r												0.1364	0.3439	0	1
	q i																		195.01	513.53	3.5	6 464
			i															81.914	138.97	3.2	1 200
	p	i	f													170.45	98.271	-126.22	711.98
	p	l i													21 512	651.42	20 100	24 100
	m i															0.3164	0.1734	0.0155	0.9329
	s i																		0.1083	0.0682	0	0.3437
	Instruments
	d	i 1 -= i 1 , -	f -p	i	1 =	21.068	121.08	-7.4644	1 807.6
	i b	d	i 1 -= i 1 , -	p	f -	i	1 =	16.997	121.13	-7.4644	1 807.6
	i r	t ,	-	2	i b	d	i 1 -= i 1 , -	p	f -	i	1 =	1.1314	4.2282	0	34.615
	d	i 1 -= i 1 , -	l -p	i	1 =	1 866.8	5 364.2	265.25	59 916
	i b	d	i 1 -= i 1 , -	p	l -	i	1 =	1 408.4	5 402.5	0	59 916
	i r	,		t	-	2	i b	d	i 1 -= i , -	1	p	l -	i	=	1	128.52	473.47	0	4 349
	t r 2 i , -	i b						0.0909	0.2881	0	1
	pop i										29 482	64 013	987	770 889
	ind i												0.1116	0 1041	0	0 6546
	Sources: p i (Nils Holgersson's annual price comparisons, www.nilsholgersson.nu);	i	1 , = -i

d

(based on coordinates recorded at www.eniro.se); b (annual reports, utilities/municipalities directly); q, (the Swedish District Heating Association (www.svenskfjarrvarme.se); f i p (Statistics Sweden and utilities/municipalities directly); l i p , m i , s i , pop i , ind i , r i (Statistics Sweden, www.scb.se).

For example, it has been claimed that the lack of relevant (cost) information is the primary complication involved in regulating locally monopolistic utility sectors[START_REF] Joskow | Regulation and Deregulation after 25 Years: Lessons Learned for Research in Industrial Organization[END_REF].

[START_REF] Gilpatric | Regulatory enforcement with competitive endogenous audit mechanisms[END_REF] compare two regulatory evaluation mechanisms -one where firms' present conditions are evaluated based on their previous conditions and one where several firms are compared with each other based on conditions set simultaneously.

In the context of local monopoly services, it has been demonstrated that demand is positively related to regulatory compliance[START_REF] Stafford | Can consumers enforce environmental regulations? The role of the market in hazardous waste compliance[END_REF].

This is confirmed empirically by[START_REF] Arora | Why do firms volunteer to exceed environmental regulations? Understanding particulation in EPA's 33/50 program[END_REF] as they show that firms are more inclined to voluntarily comply with regulatory conditions when they have relatively more customer contacts.

We rule out the extreme cases when the regulatory authority intervenes with 100 percent certainty (θ i =1), and when there is no risk of intervention (θ i =0).

Note that α used in Sections 2 and 3 is related to b as b=1-α.

[START_REF] Lee | Some recent developments in spatial panel data models[END_REF] use a piecewise function that is decreasing in the distance between i and -i.

For example, heteroscedasticity can occur from unobserved variation in network age since some recently established utilities might want to use revenues to finance network expansions. Utilities operating mature networks can therefore be expected to have lower price variability.

[START_REF] Kelejian | Specification and estimation of spatial autoregressive models with autoregressive and heteroscedastic disturbances[END_REF] report that the maximum likelihood estimator is significantly biased in the presence of heteroscedasticity.