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A Modelisation of Public Private Parternshipswith failure time

Caroline HILLAIRET and Monique PONTIER

Abstract A commissioning of public works (Maitrise d’ouvrage publique, namelyMOP in French) is a system where the community has commissioned equipment(hospital, prison ....) for its own needs and bear the cost, partly by self and partlyby a loan from a bank. On another hand, Public-Private Partnership (PPP) meansthat community agrees on a period (15-25 years) with the contractor, and is billedrent. More or less it means “leasing” purchase, covering three parts: deprecia-tion of equipment, maintenance costs, financial costs. This new formula is basedon an “ordonnance” of June 17, 2004, amended by the law of 28 July 2008 (seelegifrance.gouv.fr), justified by the emergency of requested equipment constructionor its complexity. Our aim is to study the advantages and disadvantages of the newPPP formula. Here is a particular case of a risk-neutral consortium. We discuss ofthe advantages of outsourcing (‘externalisation’) in terms of model parameters andprove that externality is interesting only in case of large enough noise when we ex-clude the risk of bankruptcy. Indeed, this risk does not seem covered under currentlegislation. Finally, we study what could happen in case of failure penalities to bepaid by the private consortium. In such a case, externality could be interesting insome context as high noise, high reference cost, short maturity, high enough penal-ity.

1 Introduction

In the classic formula of a public project, a commissioning of public works (Maitrised’ouvrage publique, namely MOP in French) the community realizes equipments(hospital, prison ....) for its own needs and bear the cost, partly by self and partly bya loan from a bank.

Caroline HILLAIRET, CMAP, Ecole Polytechnique, e-mail: [email protected]·Monique PONTIER, IMT, Universite de Toulouse. e-mail: [email protected]

1

2 Caroline HILLAIRET and Monique PONTIER

In the formula “ partnership agreement ”(or public-private partnership, namelyPPP) community agrees on a period (15 to 25 years) with the contractor, and ischarged a rent. Somehow, it is a lease purchase, covering three parts: deprecia-tion of equipment, maintenance costs, financial costs. This formula is based onan “ordonnance” of June 17, 2004, amended by the law of 28 July 2008 (seelegifrance.gouv.fr). The justification for this device is mainly based on the urgencyof requested equipment construction or its complexity. Here is studied the advan-tages and disadvantages of the PPP contract system. We have not included problemsof taxation, (as the “ordonnance” and the law do). However, it should be noted thatpart of VAT (value added tax) is recoverable in case of MOP, whereas in the frame-work of a PPP, not only it is not, but it is added at the VAT payable on the loan.This particularity could have an influence, but this problem is not addressed in thispaper. Here in particular, in the case of a risk-neutral consortium, we discuss thebenefits of outsourcing in terms of model parameters. We show that when includingthe risk of bankruptcy, the externality can be interesting when a penalty is imposedon the consortium in case of bankruptcy and in a certain context: for instance whenuncertainty is high enough, or the reference cost is important, or short maturity, orsufficient penalty. In fact, this corresponds to a risk transfer from public to private.

Section 2 sets the problem, following Iossa et al. model [2], and introduces thevarious parameters of the problem. In Section 3 we solve an optimization problemsimultaneously for the consortium and the public community. Then we study inSection 4 the effects of introducing a bankruptcy time whose risk does not seemcovered under the legislation above-named; this changes the model. If no penaltyis required, the result of the optimization yields to choose a minimal externality incontrast to the result in case of absence of bankruptcy (Section 3). Finally, in Section5, the consortium is obliged to pay penalties in case of bankruptcy: it is the only casediscussed here where in a particular configuration of the game settings, outsourcingcan be interesting for both parties. Section 6 gathers these results.

2 The problem setting

We follow here the framework of [2] by adding a stochastic view point.To modelize the randomness of the model, we introduce a filtered probability space(Ω ,F = (Ft)t∈[0,T],P).

Theoperational cost(Cs)s∈[0,T] of the infrastructure maintenance is a nonnega-tive F-adapted process.(Cs)s∈[0,T] is a rate (its unit is euros per unit time) and canbe written as

Cs = θ0−es−δa+ εs, s∈ [0,T]. (1)

where

• θ0 = benchmark cost of the maintenance,

A Modelisation of Public Private Parternships with failure time 3

• es = effort on the maintenance done at dates to reduce the cost, it is a nonnegativeF-adapted process,

• a = effort on the construction to improve the infrastructure quality, it is a param-eter inR+,

• (εs)s∈[0,T] is a centered boundedF-adapted process that modelizes the randomoperational risk of the activity. We will assume thatε ∈ [−m,M], dt⊗dPa.s.

• δ is the externality, it is a parameter inR+.

The externality represents the impact of the infrastructure on the maintenancecost. We assume that improving the infrastructure quality reduces the maintenanceoperational cost, thus the externalityδ is non negative.The maintenance cost is payable by the consortium until the maturityT or until apossible default of the consortium. We will assume the natural condition that thecosts are nonegative a.s. This condition leads to some constraints detailed in section3.2, using the expression of the optimal efforts.

The community pays to the consortium a rentt(c) which is a function of thecostc : this rent permits both to pay the consortium for its work and to cover themaintenance costs that are in its charge. We assume that the community chooses alinear expression for the rent:

t(c) = α−βc, with β ≥−1, andα such that a.s.t(Cs)≥Cs ∀s∈ [0,T].

t(c)−c is a decreasing function of the costsCs, and thus an increasing function ofthe effortses. The largerβ is, the greater is the incitement to the consortium to makeeffort on the infrastructure, but at the cost of a greater risk premiumα.

Remark 1 εs being in the interval[−m,M] ∀s∈ [0,T] (m> 0, M > 0) the condi-tion t(Cs)≥Cs ∀s∈ [0,T] is satisfied as soon asα ≥ (β +1)(θ0 +M).

The consortium aims to maximize its terminal utility, discounted at the rater ≥ 0,its optimisation problem can be formulated as follows

max(a,e)∈[0,+∞[×E

(E(∫ T

0e−rs (U(t(Cs)−Cs,s)−φ(es))ds

)−ψ(a)

)(2)

with E = (es)s∈[0,T] F adapted such that∀s∈ [0,T]es≥ 0a.s..The functionsφ andψ represent the effort cost, and following [2] we will chooseφ(a) = a2

2 andψ(e) = e2

2 . U is an utility function that modelizes the consortium riskaversion.

Definition 2. A functionU : (t,c)→U(t,c) is called utility function if(i) U : [0,T]×]0,+∞[→ R is continuous.(ii) ∀t ∈ [0,T], U(t, ·) is strictly increasing and strictly concave.(iii) The derivatives∂

∂ tU , ∂

∂cU , exist and are continuous on[0,T]×]0,+∞[.


This optimisation problem (2) will be reformulated in Section 4 in the case of apossible default of the consortium at a random timeτ.

On the other hand, the community aims to maximize the social welfare definedas the social value of the project minus the rent payed at the consortium. The com-munity optimization problem can be formulated as follows

max(α,β )∈A

SW: (α,β ) 7→(

E[B0 +

∫ T

0e−rsb(es)ds−

(∫ T

0e−rst(Cs)ds−C0

)])(3)

with B0 ≥ 0, b : R+ → R+ C1 and increasing, (for exampleb(x) = bx, b > 0), brepresents the community utility.A = (α,β ),α ≥ 0,β ≥−1 such thatt(Cs)≥Cs ∀s∈ [0,T] in order that the con-sortium is refund of the maintenance costs ;C0 is the initial cost payable by theconsortium,B0 is the initial social value of the project.

3 Solution of the problem without default

3.1 Maximization of the consortium utility

Proposition 1. The parameters of the rent(α,β ) being fixed, there exists an uniquesolution(a, es) at the optimization problem (1), given by

es = (β +1)U ′ (α− (β +1)(θ0−es−δa+ εs))a = δE(

∫ T0 e−rsesds).

Furthermore,a 7→ es(a)

is decreasing : the more effort the consortium makes for the construction, the lesseffort it has to do for the maintenance.

Proof: We have to optimize the function

(a,e) 7→= E(∫ T

0e−rs

(U(α− (β +1)(θ0−es−δa+ εs))−

12

e2s))

ds

)− 1

2a2,

which is concave ina and ines and thus which is maximum when its gradient iszero. This leads to the couple (a,es) solution of system in Proposition 1. We claimthat for alla, there exists an unique nonnegative solutiones(a). Indeed, sinceU isstrictly concave and increasing, we have

U ′ (α− (β +1)(θ0−es−δa+ εs)) > 0,


andg : x 7→ (β +1)U ′ (α− (β +1)(θ0−x−δa+ εs)) is decreasing. Thuses(a) isthe abscissa of the intersection of the bisector and the functiong graph, and it issolution of the implicit equation

F(x,a) = x− (β +1)U ′ (α− (β +1)(θ0−x−δa+ εs)) = 0.

The relation betweenes anda is reflected by the derivative

deda

=−∂aF∂xF

=(β +1)2δU”(α− (β +1)(θ0−x−δa+ εs))

1− (β +1)2U”(α− (β +1)(θ0−x−δa+ εs)).

SinceU” < 0, deda < 0, a 7→ es(a) is decreasing. We do the same for the function

h : a 7→ δE[∫ T

0 e−rses(a)ds] and we conclude by the existence of an unique optimala solution of equationa = h(a). •

Notation: We introduce the following notation, useful for the rest of the paper

At :=∫ t

0e−rsds.

Example 1 : linear utility U(x) = η + γx.In this case, the consortium is risk neutral. The rent rule being fixed, the optimalefforts for the consortium are given by

es = γ(β +1) ∀s∈ [0,T]a = δ es(

∫ T0 e−rsds) = δγ(β +1)AT .

Example 2 : quadratic utilityU(x) = x− γ

2x2 with γ > 0 such thatt(Cs) < 1γ∀s∈ [0,T].

In this case, the risk aversion of the consortiumγ1−γx is an increasing function of hiswealth. The rent rule being fixed, the optimal efforts for the consortium are givenby a = δ

(β+1)(1−γ(α−(β+1)θ0))+AT1+γ(β+1)2(1+δ 2AT )

es = (β+1)1+γ(β+1)2 (1− γ(α− (β +1)θ0 +(β +1)δ a− (β +1)εs)) ∀s∈ [0,T]

The noise(εs) being centered,E(es) does not depend ons. Furthermore,εs takingvalues in[−m,M] ∀s∈ [0,T] (m> 0, M > 0), the conditiont(Cs) < 1

γ∀s∈ [0,T]

is satisfied as soon asα − (β + 1)(θ0−m) + (β+1)2

1+γ(β+1)2 ((1+ δ 2)(1− γ(α − (β +

1)θ0))+ +δ (β +1)M) < 1γ.

Remark 3 Comparison between these two examples: in the case of a quadraticutility function,E(es) ≤ (β + 1) and is a decreasing function of the risk aversion,whereas for a linear utility with slopγ > 1, E(es) = es > (β + 1). Thus the more


risk averse the consortium is, the less effort it will do both for the construction andfor the maintenance of the infrastructure.

Finding an explicit solution of the community optimization problem being te-dious in a general setting, we will from now on focus on the framework of linearutility functions both for the community and the consortium

U(x) = γx, γ > 0 ; b(x) = b.x,b > 0.

This leads to the following constraints on the externality.

3.2 Constraints on the externality

It seems natural to assume the cost being nonnegative a.s. This leads to some con-straints on the parameters that we will explicit, using the expression of the optimalefforts in the framework of a linear utility. Moreover, in practice, the communitycan not outsource more than a given levelδmax. We fix δmax such thatCs≥ 0 almostsurely:

Cs≥ 0⇐⇒ θ0 ≥ es+δa− εs.

Using the expressions ofa ande (see example 1) :

Cs≥ 0⇐⇒ θ0−m≥ γ(β +1)(1+δ2AT) (4)

This is a constraint linkingδ andβ . Note that this inducesθ0 ≥msinceβ ≥−1.

3.3 Maximization of the community social welfare

Our aim is to find explicit solutions in order to quantify the advantages of outsourc-ing, with the linear utilitiesU(x) = γx, γ > 0 et b(x) = b.x,b > 0. The rent rulebeing fixed, the consortium optimal efforts are given by

es = γ(β +1) ∀s∈ [0,T]a = δγ(β +1)AT .

Proposition 2. We recall the social welfare (3) :

SW(α,β ) =(

E[B0 +

∫ T

0e−rsb(es)ds−

(∫ T

0e−rst(Cs)ds−C0

)])with Cs = θ0− es−δ a+ εs and t(Cs) = α−βCs. We assume that

0≤ δ2 ≤ δ

2max=

θ0−2m− γ(b+1)γAT

. (5)


Then the community optimal policy is given by

β =γb+θ0− γ(1+δ 2AT)

2γ(1+δ 2AT)(6)

α =γb+θ0 + γ(1+δ 2AT)

2γ(1+δ 2AT)(M−bγ− γ(1+δ

2AT)). (7)

Remark here that assumption (5) implies that the benchmark costθ0 is boundedfrom below, otherwise negative costs can occur.

Proof: Sincees = γ(β +1) is constant :

SW(α,β )−B0−C0

AT= E[bes−α +βCs] = bes−α +β (θ0−es(1+δ

2AT))

= (b−β (1+δ2AT))es−α +βθ0

= (b−β (1+δ2AT))γ(β +1)−α +βθ0.

SW is a polynomial function of degree 2 inβ :

SW(α,β )−B0−C0

AT=−β

2γ(1+δ

2AT)+β (γb+θ0− γ(1+δ2AT))−α +bγ

The dominating coefficient is negative, thus there exists an unique maximum achievedfor

β =γb+θ0− γ(1+δ 2AT)

2γ(1+δ 2(AT))(8)

that can be also written as

γ(1+δ2AT) =

γb+θ0

2β +1, (9)

as soon as the constraint (4) is satisfied, that is as soon as

θ0−m− 12(bγ +θ0 + γ(1+δ

2AT))≥ 0,

which is indeed satisfied since the externalityδ is bounded from above byδ 2max =

θ0−2m−γ(b+1)γAT

.

The choice ofα must satisfy the constraintt(Cs)≥Cs, that is

α ≥ (β +1)(θ0 + εs− es−δ a) ds⊗dP a.s.

The constraint must be satisfied in the linear case wherees = γ(β +1), a= δγ(β +1)(AT), andεs≤M. We chooseα saturating this constraint,α is given in terms of


β and using relation (9)

α = (β +1)(θ0 +M− γ(β +1)(1+δ2AT))

= (β +1)(θ0 +M− (β +1)γb+θ0

2β +1)

α =β +1

2β +1[β (θ0 +2M−bγ)+M−bγ].

Sinceα ≥ (1+ β )Cs and since the cost is nonnegative via (4), we necessarily haveα ≥ 0. •

Remark that (9) implies thatβ is a decreasing function of the externalityδ , thatsatisfies

bγ +mθ0−2m−bγ

≤ β ≤ bγ +θ0− γ

2γ:

The upper bound corresponds toδ = 0 (no outsourcing), the lower bound corre-sponds toδ maximum (5). Thus the study of the impact of the externalityδ onthe social welfare can be done through the study of the functionβ 7→ SW(α(β ),β )where we replaceδ by its function ofβ using (9).

Proposition 3. We assume (5). If bγ − γ2

bγ+θ0≤ M – that is if the noise level is

high enough – the social welfare is optimal for the maximal externalityδ =√θ0−2m−γ(b+1)

γAT. Otherwise, is the noise level is lower, the social welfare is opti-

mal for δ = δmax or for δ = 0 (depending on whether SW(βmax) < SW(βmin) ornot).In conclusion, if we exclude the default risk, the externality is attractive only if thenoise level is high enough.

Proof: We want to optimize the following function

β 7→ SW(α,β )−B0−C0

AT= β

2 γb+θ0

2β +1−α +bγ,

that is, replacingα by its optimal value function ofβ

β 7→ 12β +1

[β 22(bγ−M)−β (3M +θ0−4bγ)−M +2bγ].

We recall the constraint onβ

βmin =γb+m

θ0−2m−bγ≤ β ≤ βmax=

γb+θ0− γ

2γ.

More precisely, we study on the interval[βmin,βmax] the function


f (β ) =1

2β +1[β 22(bγ−M)−β (3M +θ0−4bγ)−M +2bγ]. (10)

Differentiating with respect toβ , we get

2β +14(bγ−M)

f ′(β ) = β2 +β − θ0 +M

4(bγ−M)

The discriminant of this polynomial function of degree 2 is∆ = 1+ θ0+Mbγ−M = θ0+bγ

bγ−M .

If ∆ > 0, the positive root isβr =−1+

√θ0+bγ

bγ−M2 . Remark that

βr < βmax⇐⇒M < bγ− γ2

bγ +θ0.

This can only happen ifbγ − γ2

bγ+θ0> 0, that is (by solving the second degree in-

equation(bγ)2 +bγθ0− γ2 > 0) if bγ > θ02 (√

1+ 4γ2

θ20−1).

• First case :bγ ≤M (i.e. high level of noise),f (and thusSW) is a strictly decreasing function ofβ (and thus strictly increasingin δ ). The social welfare is optimal forδ = δmax (maximal externality) for a highlevel of noise.

• Second case :bγ− γ2

bγ+θ0≤M ≤ bγ (i.e. medium level of noise).

Sinceβr > βmax, f (and thusSW) is again a strictly decreasing function ofβ (andthus strictly increasing inδ ). The social welfare is optimal forδ = δmax.

• Third case :bγ − γ2

bγ+θ0≥ M (i.e. low level of noise andbγ large enough). The

optimal externality depends on whether or notβr is greater thanβmin:

– If βr > βmin, thenSW is a strictly decreasing function ofβ on [βmin,βr ], andstrictly increasing on[βr ,βmax]. Thus, the social welfare is optimal forδ =δmax or for δ = 0 (whetherSW(βmax) < SW(βmin) or not).

– If βr ≤ βmin, thenSW is a strictly increasing function ofβ (and a strictlydecreasing function ofδ ). Thus, the social welfare is optimal when there is nooutsourcing (δ = 0).

To summarize this third case, the derivativef ′(β ) being successively negativethen positive, the optimum is achevied at one of the interval bound and is equalto SW(βmax)∨SW(βmin).

•In conclusion, when we exclude the default risk of the consortium, outsourc-

ing becomes attractive only if the noise level is high enough. This correspondsto a risk transfer from the community to the consortium.


We conclude this section with a toy numerical example in order to quantify numer-ically this benchmark noise level under which outsourcing is not attractive.

3.4 Numerical example

In this example we takeθ0 = 100 euros per unit time. The noise represents therandomness of the cost around this value, that isΘ0 = θ0 + ε is a random variablewith values in[θ0−M,θ0 +M] (here we takem= M). In the case of a linear utilityU(x) = γx, we letγ = 25 euros per unit time andb = 1.

θ0 = 100 ; γ = 25 ; b = 1. (11)

Proposition 4. For θ0 = 100 ; γ = 25 ; b = 1, outsourcing is attractive if and onlyif noise level M is greater than50

3 (that is around 16,7% of the benchmark costθ0).

Proof: First, the levelbγ − γ2

bγ+θ0given in Proposition 3 is equal to 20 in this

example, thus ifM ≥ 20 the maximal externality is optimal. Remark that in thiscase

βmax= 2, ∀M ; f (βmax) = 50−3M,

where f (which has the same behavior asSW) was defined in (10):

f (β ) =1

2β +1[β 22(bγ−M)−β (3M +θ0−4bγ)−M +2bγ].

Now we study the case whereM < 20. Proposition 3 says that the optimum dependson the position off (βmin) with respect to this value 50−3M. We have

βmin =25+M75−2M

.

We setM = 5µ, then we obtain

f (βmin) =2500−20×50µ +95µ2

5(15−2µ),

to be compared tof (βmax) = 5(10−3µ). Then, f (βmin) < f (βmax) if and only ifM < 50

3 ∼ 16.7, leadingβ optimal equal toβmax, andδ = 0. •

4 Introduction of a default time, without penalty

We extend here the previous model in a more dynamic point of view and by intro-ducing a default time. We still consider linear utilities (U(x) = γx andb(x) = bx)and we consider the operational cost as a semimartingale :


dCs = (θ0−es−δa)ds+σdWs.

The community chooses then the following expression for the rent

dt(Cs) = αds−βdCs.

We define the default timeτ as the first time as the consortium can not refund itsdebt anymore. In a first step, we assume that no penalty is imposed to the consortiumin case of default (the case of penalty will be studied in Section 5).

4.1 Utility maximisation for the consortium

The consortium refund the debt at a rateD (dDs = De−rsds) that is deducted fromits profit. Its aim is to optimize (withU(x) = γx being its utility function)

(e,a,τ) 7→ E(∫

τ∧T

0e−rs(γ[dt(Cs)−dCs−dDs]−

12

e2s))− 1

2a2

= E(∫

τ∧T

0e−rs[γ(α−D)− γ(β +1)(θ0−es−δa)− 1

2e2

s]ds

)− 1

2a2

sinceE(∫

τ∧T0 e−rsdWs

)= 0.

Proposition 5. We assume that the initial effort does not depend on the default time(which is unknown at date 0). Then the optimal policy of a risk neutral consortiumis given by

es = γ(β +1)11[0,τ∧T](s)a = δγ(β +1)AT .

Proof: Since the default timeτ is unknown at the initial date, we do the opti-mization only in(e,a) with the fact that the efforte is done on the interval[0,τ],thus

es = γ(β +1)11[0,τ∧T](s)a = δγ(β +1)E(

∫τ∧T0 e−rsds)

But rather than taking an initial efforta depending onτ, it is more relevant to takethe optimal initial effort as in the case with no default (thus we may overevaluate it):

a = δγ(β +1)E(∫ T

0e−rsds) = δγ(β +1)AT . •


4.2 Definition of the default time and of the communityoptimization problem

We introduce the initial fund financing the project :DAT =∫ T

0 e−rsDds.The consortium must refund its debt,dt⊗dP a.s.:

DAT + t(Ct)−Ct −Dt ≥ 0,

that is

Yt = DAT +∫ t

0e−rs(α−D−(β +1)(θ0−γ(β +1)(1+δ

2AT))ds−∫ t

0e−rs(β +1)σdWs≥0.

Thus the default occurs when this constraint is not satisfied anymore.

Definition 4. The default timeτ is defined as

τ = inft : Yt < 0.

If r = 0 (thenAt = t) :

τ = inft :∫ t

0e−rs(β +1)σdWs> DAT +(α−D−(β +1)(θ0−γ(β +1)(1+δ

2AT))At.

If r > 0 :

τ = inft :∫ t

0e−rsdWs > Ar −Bre

−rt,

where

Ar =rDAT +(α−D− (β +1)(θ0− γ(β +1)(1+δ 2AT))

r(β +1)σ,

Br =α−D− (β +1)(θ0− γ(β +1)(1+δ 2AT))

r(β +1)σ.

We remark here that this defaut timeτ is increasing inα: the greater the communityrent is, the longer the consortium avoids the default (and this∀r). Considering that itis optimal for the community to postpone the default as longer as possible, we willchoose the maximumα satisfying the constraints detailed in the following.

We adapt the definition ofSWbecause in case of default, the community shouldtake over from the consortium to refund the debt

SW(α,β )−B0−C0 = E(∫ T

0e−rsbesds−

∫τ

0e−rsdt(Cs)−

∫ T

τ

e−rsDds

)

= E(∫

τ∧T

0e−rs[bγ(β +1)−α +β (θ0− γ(β +1)(1+δ

2AT))]ds−D∫ T

τ

e−rsds

)


= [D+bγ(β +1)−α +β (θ0−γ(β +1)(1+δ2AT))]E[Aτ∧T ]−DAT .

Introducing

H := D+bγ(β +1)+β (θ0− γ(β +1)(1+δ2AT)),

SW(α,β )−B0−C0 +DAT = (H−α)E[Aτ∧T ],

which is the product of a decreasing and an increasing function inα. If α ≥H, thenα 7→ SW(α,β )−B0−C0 +DAT is decreasing thus an optimalα must be less thanH and we get

SW(α,β )−B0−C0 +DAT = (H−α)E[Aτ∧T ]≥ 0.

Therefore, the optimum exists in the interval[0,H] and we will study the followingfunction of the parametersα,β ,δ

E[Aτ∧T ] = E[Aτ I τ<T ]+ATP(τ > T).

4.3 Solution in the caser = 0

If r = 0,τ = inft : Wt > A−Bt,

where

A =DAT

(β +1)σ,

B =D+(β +1)(θ0− γ(β +1)(1+δ 2T))−α

(β +1)σ.

We defineK := D+(β +1)(θ0− γ(β +1)(1+δ

2AT)). (12)

4.3.1 The law of the default time, caser = 0

Using 3.2.3. page 148 [3], the law ofτ is given by

Proposition 6. If r = 0,, the default time is defined by

τ = inft : Wt >DAT

(β +1)σ+

α−K(β +1)σ

t

where K is defined in (12). Then the density ofτ onR+ is


t 7→ DAT

(β +1)σ√

2πt3exp[− 1

2t

(DAT−(α−K)t

(β +1)σ

)2

].

If r = 0, Pτ < ∞= exp(A(K−α)−|A(K−α)|) (cf. [4] page 197). Thus,E[τ] isfinite if and only if α < K. In order to postpone the default, we takeα ≥ K.

Corollary 5 If r = 0, we chooseα = K, and the default time is defined asτ =inft : Wt > DAT

(β+1)σ , the density ofτ onR+ is

t 7→ DAT

(β +1)σ√

2πt3exp[− 1

2t

(DAT

(β +1)σ

)2

].

4.3.2 Constraints on the parameters, caser = 0.

A raisonnable constraint is to take a nonnegative instantaneous operational cost (inexpectation) :

E[Csds] = (θ0− γ(β +1)(1+δ2T))ds≥ 0.

that isθ0 ≥ γ(β +1)(1+δ

2T).

We have previously justified choosingα ≥ K to move the default back, such thatE[τ] = +∞.

Proposition 7. The expected instantaneaous cost being nonnegative, andα ≥ K(such that E[τ] = +∞) induce the following constraints

θ0−bγ(β +1)≤ γ(β +1)(1+δ2T)≤ θ0. (13)

Furthermore0≤ K ≤H and this prove the existence of an optimalα in the interval[K,H].

Proof: The expected instantaneaous cost being nonnegative is equivalent to

E[Csds] = (θ0− γ(β +1)(1+δ2T))ds≥ 0,

thus we get the right hand side inequality

θ0 ≥ γ(β +1)(1+δ2T).

This implies

K = D+(β +1)(θ0− γ(β +1)(1+δ2T))≥ D≥ 0.

The left hand size inequality follows from

H−K = (β +1)bγ−θ0 + γ(β +1)(1+δ2T)≥ 0. •


We now chooseα = K, that maximizes the first factorSW−B0−C0 + DAT . Weremark that in this case, in expectation, the instantaneous rent is positive :

E[t(Cs)−Cs]ds= α− (β +1)(θ0− γ(β +1)(1+δ2T)) = D > 0.

4.3.3 Study of the social welfare, function ofβ ,δ , caser = 0.

If r = 0, the law ofτ is explicit, furthermore (Corollaire 5)we chooseα = K =D+(β +1)(θ0− γ(β +1)(1+δ 2T)) and thus the factor of the expectation is

H−K = (β +1)bγ−θ0 + γ(β +1)(1+δ2T).

Corollary 6 If r = 0, we chooseα = K < H, and the functionSW(α,β )−B0−C0 +DAT = (H−K)E[τ ∧T]

= [(β +1)bγ−θ0+γ(β +1)(1+δ2T)]

[∫ ∞

0(t ∧T)

DAT

(β +1)σ√

2πt3exp[− 1

2t

(DAT

(β +1)σ

)2

]dt

].

With the choiceα = K, the default time is the hitting time ofDAT(β+1)σ by a Brownian

motion, thus the density ofτ is DAT

(β+1)σ√

2πt3exp[− 1

2t

(DAT

(β+1)σ

)2]. Assuming that

β ,δ satisfy (13), Corollary 6 gives the function we want to optimize :

(β ,δ ) 7→ [(β +1)bγ−θ0+γ(β +1)(1+δ2T)]

∫R+

t∧TDAT

σ(β +1)√

2πt3exp[− 1

2t

(DAT

(β +1)σ

)2

]dt.

Proposition 8. Let r = 0. We assume that the default time is postponed as longer aspossible and that, the consortium optimal policy(es, a) being established, the PPPcontract requires nonnegative (in expectation) operational cost and rent. Then theoptimal rent rule and the optimal externality are

α∗ = D,

−1 < β∗ =

θ0

γ−1, (14)

δ∗ = 0.

Outsourcing is not optimal in this case.

Proof: The function

δ 7→ [(β +1)bγ−θ0+γ(β +1)(1+δ2T)]

∫R+

t∧TDAT

σ(β +1)√

2πt3exp[− 1

2t

(DAT

(β +1)σ

)2

]dt

is increasing inδ and using (13),(1+δ 2T)∗ = θ0γ(β+1) . This optimum is greater than

1 thus we get the constraint forβ :


γ(β +1)≤ θ0.

Replacing 1+δ 2T by its optimal value, we want to optimize the function

β 7→ bγ

∫R+

t ∧TDAT

σ√

2πt3exp[− 1

2t

(DAT

(β +1)σ

)2

]dt.

This function is increasing, thusβ ∗ = θ0γ−1, and using the expression of(1+δ 2T)∗

we get thatδ ∗ = 0. Finally,

α∗ = D+(β ∗+1)(θ0− γ(β ∗+1)) = D. •

The interpretation is the following: if there is no penalty in case of a default, thecommunity optimal policy is to outsource the less possible (and MOP are better andmore secure than PPP). Furthermore, we remark that at the optimum,E(Cs) = 0,and the rent ist(Cs)−Cs = D− θ0

γCs, E(t(Cs)−Cs) = D. Thus, the rent coincides,

in expectation, to the refund of the consortium debt.

4.4 Solution of the problem in the caser > 0

If r > 0 :

τ = inft :∫ t

0e−rsdWs > Ar −Bre

−rt,

where

Ar =rDAT +(α−D− (β +1)(θ0− γ(β +1)(1+δ 2AT))

r(β +1)σ,

Br =α−D− (β +1)(θ0− γ(β +1)(1+δ 2AT))

r(β +1)σ.

4.4.1 Constraints on the parameters

By continuity, we have almost surely∫τ

0e−rs(β +1)σdWs = DAT +(α−D− (β +1)(θ0− γ(β +1)(1+δ

2AT))Aτ

that implies in the caser > 0

0= E[∫

τ

0e−rs(β +1)dWs] = DAT +(α−D−(β +1)(θ0−γ(β +1)(1+δ

2AT))E[Aτ ].

Thus


E[Aτ ] =DAT

D−α +(β +1)(θ0− γ(β +1)(1+δ 2AT)), (15)

this implies the constraint on the parameters (since 0≤ Aτ ≤ 1/r) :

0≤ E[Aτ ] =DAT

D−α +(β +1)(θ0− γ(β +1)(1+δ 2AT))≤ 1/r. (16)

We recallK = D+(β +1)(θ0− γ(β +1)(1+δ

2AT)),

thus we have the condition onα :

rDAT ≤D−α +(β +1)(θ0−γ(β +1)(1+δ2AT)) = K−α, α ≤K− rDAT . (17)

Furthermore, the instantaneous cost being nonnegative (in expectation) :

E[Csds] = (θ0− γ(β +1)(1+δ2T))ds≥ 0.

implies thatθ0 ≥ γ(β +1)(1+δ

2T).

4.4.2 Study of the social welfare, function ofβ ,δ ; caser 6= 0.

Proposition 9. Let r > 0. We assume that the default time is postponed as longer aspossible and that, the consortium optimal policy(es, a) being established, the PPPcontract requires nonnegative (in expectation) operational cost and rent. Then theoptimal rent rule and the optimal externality are

α = De−rT ,

−1 < β =θ0

γ−1, (18)

δ = 0.

Proof: We summarize the constraints : the cost rate is nonnegative (in expecta-tion) :

θ0 ≥ γ(β +1)(1+δ2AT)

as for the rent :

α ≥ (β +1)(θ0− γ(β +1)(1+δ2AT)) = (K−D).

The optimal parameters must satisfy

α ≤ H ∧ (K− rDAT).


As in the caser = 0, it seems to be relevant to chooseα such that to postponethe default as longer as possible, that isα = K− rDAT (that satisfies the constraintα ≥ K−D sincerAT ≤ 1). We get

H−α = (β +1)bγ−θ0 + γ(β +1)(1+δ2AT)+ rDAT

that is increasing inδ and must be nonnegative. This implies a constraint linkingβ

andδ :bγ(β +1)−θ0 + γ(β +1)(1+δ

2AT)+ rDAT ≥ 0.

Furthermore, with this choice ofα, we get

Ar = 0, Br =−DAT

(β +1)σ.

Thusτ = inft/∫ t

0 e−rsdWs > DAT(β+1)σ e−rt does not dependent onδ .

For continuity reason, ∫τ

0e−rsdWs =

DAT

(β +1)σe−rτ

thusE[e−rτ ] = 0, that isτ = +∞ a.s. andτ ∧T = T, Aτ∧T = AT . Therefore, for thischoice ofα,

SW(α,β ,δ ) = [(β +1)bγ−θ0 + γ(β +1)(1+δ2AT)+ rDAT ]AT .

SW is increasing inδ and the optimalδ is given by

(1+δ 2AT) =θ0

γ(β +1)

with the constraint θ0γ(β+1) ≥ 1, that isβ ≤ θ0

γ−1. Finally,

α = De−rT

and we easily check thatH−α = rDAT +(β +1)bγ is positive.The last step is to find the optimumβ +1 for the function

β 7→ f (β +1) = (rDAT +(β +1)bγ)AT .

This function is increasing, the optimalβ is given such as in the caser = 0 :

β =θ0

γ−1

andδ = 0. The community optimal policy is the same as in the caser = 0. •

In conclusion, whatever the interest rate is, outsourcing is NEVER optimalif we consider the possibility that the consortium defaults and if no penalty is


administered in case of default.

Given the maturity of PPP contract, it is obvious that we have to take into accountthe possibility of default. We will now focus on finding some cases where outsourc-ing is attractive if a penalty is administered in case of default.

5 Penalty in case of default withr = 0

Here we add in Section 4 model a penaltyρV(T− t)+ that the consortium shouldpay in case of default, whereas the community receives the compensationρ ′V(T−t)+. We assume the natural constraintρV ≤ D and we denoteρ = ρ ′+ ε whereεVis used to pay the liquidation cost. We summarize the constraints

ρV ≤ D ; ρ = ρ′+ ε, ε > 0. (19)

We consider the rentdt(Cs) = αds−βdCs, thus the consortium optimal policy re-mains the following.

Proposition 10.Considering the rent dynamic dt(Cs) = αds−βdCs and the opera-tional cost dynamic dCs = (θ0−es−δa)ds+σdWs, the consortium optimal policyis

et = γ(β +1)11[0,τ](t), a = γ(β +1)δT.

The default time is now defined as

τ = inft : (β +1)σWt > DT +(α−K)t−ρV(T− t)+∧T

whereK is defined in (12). Thusτ = τ ∧T with

τ := inft : (β +1)σWt > DT +(α−K +ρV)t−ρVT.

5.1 Constraints on the parameters

As in the previous section 4, we choose to postponed the default as longer as possi-ble, for both the consortium and the community interest :

α ≥ K−ρV = D+(β +1)(θ0− γ(β +1)(1+δ2T))−ρV.

Using the fact that the operational cost and the rent are nonnegative, we precise theconstraints on the parameters.

Proposition 11.We assume that the operational cost and the rent are nonnegative(in expectation) and we choose the bigger externality satisfying this assumption.


Then the optimal parametersα,β ,δ satisfy the following constraints :

γ(β +1)(1+δ2T) = θ0, (20)

γ(β +1) ≤ θ0, (21)

0≤ D−ρV ≤ α < D+bγ(β +1)−ρ′V, (22)

This last interval is not empty sinceρ > ρ ′ andγ(β +1)≥ 0.

Corollary 7 With the choice of a maximal externality, the consortium optimal ef-fort can be written with respect to(θ0,δ ,T) :

et =θ0

1+δ 2T11[0,τ](t), a =

θ0

1+δ 2TδT.

In this case dCs = σdWs on [0,τ].

Proof: The expectation of the instantaneous cost being nonnegative

E[Csds] = (θ0− γ(β +1)(1+δ2T))ds≥ 0.

that isθ0 ≥ γ(β +1)(1+δ 2T).Our goal here is to find situations where outsourcing is attractive, thus we ”a priori”chooseδ maximum 1+δ 2T =

θ0

γ(β +1).

This leads to the following constraint onβ (since 1+δ 2T ≥ 1)

β +1≤ θ0

γ.

With this choice of externality, the decision of postponing the default as longer aspossible leads to the constraint onα

α ≥ K−ρV = D−ρV. (23)

Furthermore, the instantaneous rent is nonnegative (in expectation)

E[t(Cs)−Cs] = α− (β +1)(θ0− γ(β +1)(1+δ2T))≥ 0,

thus, with the choice ofδ maximum,α ≥ 0. We compute the social welfare, withδmaximum and taking into account the compensation received in case of default :

SW(α,β )−B0−C0 = [D+bγ(β +1)−α]E[τ ∧T]−DT +ρ′VE[(T− τ)+],

that is, using(T− τ)+ = T− τ ∧T,

SW(α,β )−B0−C0 +(D−ρ′V)T = [D+bγ(β +1)−α−ρ

′V]E[τ ∧T]. (24)


This expression ofSWrequires the following constraint

D+bγ(β +1)−α−ρ′V > 0 i.e.α < D+bγ(β +1)−ρ

′V.

Furthermore, the constraints onα (α ≥ 0 and (23)) lead to:

(D−ρV)+ < α < D+bγ(β +1)−ρ′V.

Using assumption (19),(D−ρV)+ = D−ρV and

0≤ D−ρV ≤ α < D+bγ(β +1)−ρ′V.

This interval is not empty sinceρ > ρ ′ andbγ(β +1)≥ 0. •

5.2 Maximisation of the social welfare

To emphasize the dependency onβ , we now denotesτ by

τβ := inft : (β +1)σWt > (D−ρV)T +(α−D+ρV)t.

We remark thatβ 7→ τβ is decreasing. Using (24), we express the social welfareSWas a function ofβ .

Lemma 8 Up to an additive constant, the social welfare is the sum of two functionsof β +1 :

f (β +1)= bγ(β +1)E[τ∧T] = bγ

∫ ∞

0t∧T

(D−ρV)T

σ√

2πt3exp[− 1

2t

((D−ρV)T− (α−D+ρV)t

(β +1)σ

)2

]dt

andg(β +1) = [D−α−ρ

′V]E[τβ ∧T].

The following proposition gives the community optimal policy.

Proposition 12.We assume (19). We assume that we postpone the default as longeras possible and that, the consortium optimal policy(es, a) being established, thePPP contract requires nonnegative (in expectation) operational cost and rent. Thenthe optimal rent rule and the optimal externality are

(i) if D −α −ρ ′V ≤ 0, β = θ0γ

and the same conclusions as in the case with nopenalty hold (14).

(ii) if D −α−ρ ′V > 0, we chooseα = D−ρV (this does not contradict (ii) since

ρ ′ < ρ) and we denote A= γ(D−ρV)

√T

θ0σ. Then the sign( f +g)′( θ0

γ) is the one of the

following expression

(bθ0+2(ρ−ρ′)V)A(1−Φ(A))+bθ0A−1

(Φ(A)− 1

2−Aφ(A)

)−[2(ρ−ρ

′)V]φ(A).


For a ”small” A, ( f +g)′( θ0γ

) < 0, and there exists an optimalβ strictly less thanθ0γ

, thus the optimal externalityδ is strictly positive.

Proof: On the one hand, the functionf is increasing fromf (0) = 0 to

f (∞) = bγ

∫R+

t ∧T(D−ρV)T

σ√

2πt3dt =

4bγ(D−ρV)T√

T

σ√

2π.

On the other hand, concerning the functiong, two cases may occur :(i) if D−α −ρ ′V ≤ 0, g is also increasing, the optimalβ is θ0

γand the same con-

clusions as in the case without penalty hold (14).(ii) if D−α −ρ ′V > 0, g is decreasing and it is necessary to go into detail, usingthe constraints (21) and (22):

γ(β +1) ≤ θ0,

0≤ D−ρV ≤ α < D−ρ′V.

We will study the functionsf andg in the interval]0, θ0γ

]. To do this, and in orderto simplify the computations, we chooseα = D−ρV (thusτβ is a.s. finite with aninfinite expectation). We do the change of parameter :

ζ =(D−ρV)T(β +1)σ

, ζ ≥ γ(D−ρV)Tθ0σ

.

Thus

f (ζ ) = bγ(D−ρV)T

σ

∫R+

t ∧T1√2πt3

exp[− 12t

ζ2]dt, (25)

g(ζ ) = [(ρ−ρ′)V]ζ

∫R+

t ∧T1√2πt3

exp[− 12t

(ζ )2]dt.

We will use the following technical lemma :

Lemma 9 Let φ be the density of a standard centered Gaussian random variable,andΦ its cumulative function. Then for all positive A :∫ A

0u2

φ(u)du=−Aφ(A)+Φ(A)− 12

;∫ ∞

Au−2

φ(u)du= A−1φ(A)−1+Φ(A).

We now compute the derivative function of ˜g

Lemma 10

g′(ζ ) = [(ρ−ρ′)V]4ζ [

√T

ζφ(

ζ√T

)−1+Φ(ζ√T

)].

Proof: Before computing the derivative, we do the following change of variable

in g : u2 = ζ 2

t , t = ζ 2

u2 , dt =−2ζ 2

u3 du, andφ(u) = 1√2π

exp[−u2

2 ] :


g(ζ ) = [(ρ−ρ′)V]ζ

∫R+

ζ 2

u2 ∧Tu3

ζ 3

2ζ 2

u3 φ(u)du

= [(ρ−ρ′)V]2

∫R+

ζ 2

u2 ∧Tφ(u)du

= 2[(ρ−ρ′)V]

(∫ ζ√T

0Tφ(u)du+

∫ ∞

ζ√T

ζ 2

u2 φ(u)du

).

The previous lemma leads to

g(ζ ) = 2[(ρ−ρ′)V]

(T(Φ(

ζ√T

)− 12)+ζ

2(√

Tζ

φ(ζ√T

)−1+Φ(ζ√T

))

).

Up to the multiplicative constant 2(ρ − ρ ′)V, the derivative of the first term is√Tφ( ζ√

T) and the second term is

√Tζ φ(

ζ√T

)−ζ2 +ζ

2Φ(

ζ√T

)

whose derivative is (φ ′(u) =−uφ(u)) :

√Tφ(

ζ√T

)− ζ 2√

Tφ(

ζ√T

)−2ζ (1−Φ(ζ√T

))+ζ 2√

Tφ(

ζ√T

)

and reduction leads to the result. •

The derivative off is

Lemma 11

f ′(ζ ) =−2bγ(D−ρV)T

σ

(1−Φ(

ζ√T

)+Tζ−2[Φ(

ζ√T

)− 12− ζ√

Tφ(

ζ√T

)])

.

Proof: We deduce from (25):

f ′(ζ ) =−bγ(D−ρV)T

σζ

∫R+

t ∧T1

t√

2πt3exp[− 1

2tζ

2]dt.

Doing a change of variable inf ′ :

f ′(ζ ) =−2bγ(D−ρV)T

σζ

∫R+

(ζ 2

u2 )∧T1

ζ 2

u2

√2π( ζ 2

u2 )3exp[−u2

2]ζ 2

u3 du=

=−2bγ(D−ρV)T

σ

∫R+

(ζ 2

u2 )∧T1√2π

u2

ζ 2 exp[−u2

2]du=


−2bγ(D−ρV)T

σ

(∫ ζ√T

0T

u2

ζ 2 φ(u)du+∫ ∞

ζ√T

φ(u)du

).

Lemma 9 yields :

f ′(ζ )=−2bγ(D−ρV)T

σ

(Tζ

−2[− ζ√T

φ(ζ√T

)+Φ(ζ√T

)− 12]+1−Φ(

ζ√T

))

. •

Proof: of the proposition 12, case (ii): We are looking at the sign of( f +g)′( θ0γ

)

which is the sign of−( f + g)′(ζ ) (in ζ = γ(D−ρV)Tθ0σ

). Using the two lemmas,

−( f + g)′(ζ ) =

2bγ(D−ρV)T

σ

(1−Φ(

ζ√T

)+Tζ 2 [Φ(

ζ√T

)− 12− ζ√

Tφ(

ζ√T

)])

−4ζ (ρ−ρ′)V[

√T

ζφ(

ζ√T

)−1+Φ(ζ√T

)].

Thus the sign of( f +g)′ is the one of

bγ(D−ρV)

√T

σ

(1−Φ(

ζ√T

)+Tζ−2[Φ(

ζ√T

)− 12− ζ√

Tφ(

ζ√T

)])

−2ζ√T

[(ρ−ρ′)V][

√T

ζφ(

ζ√T

)−1+Φ(ζ√T

)].

Forζ = γ(D−ρV)

θ0σ, we setA := γ

(D−ρV)√

Tθ0σ

, and the sign of( f +g)′( θ0γ

) is the one of

bθ0A

(1−Φ(A)+A−2[Φ(A)− 1

2−Aφ(A)]

)−2A[(ρ−ρ

′)V][A−1φ(A)−1+Φ(A)]

which is the expected expression of Proposition 12 (ii)

(bθ0+2(ρ−ρ′)V)A(1−Φ(A))+bθ0A−1

(Φ(A)− 1

2−Aφ(A)

)−[2(ρ−ρ

′)V]φ(A).

The asymptotics around zero of the two first terms are :

[2(ρ−ρ′)V +bθ0]A[1−Φ(A)] ∼ [2(ρ−ρ

′)V +bθ0]A2

,

bθ0A−1(

[Φ(A)− 12−Aφ(A)]

)∼ bθ0

5A2

6√

2π,

and the third term is equal forA = 0 to−2(ρ −ρ ′)Vφ(0) = −2(ρ−ρ ′)V√2π

< 0. Thus,

for A small enough,( f +g)′( θ0γ

) < 0. •


Remark 12 This condition “A= γ(D−ρV)

√T

θ0σsmall enough” is satisfied if

- the noise level is high (largeσ ),- the benchmark costθ0 is high,- the maturity T is short,- D−ρV is small, that is the penaltyρ is large enough.

In this section that modelizes the better the reality, we show that outsourcingis attractive for the community in case of high uncertainty or high noise, shortmaturity, high benchmark operational cost or a sufficently high penalty in caseof default.

6 Conclusion

Three models of PPP contracts have been studied in this paper :- The first one assumes that there is no default risk and that the contract does notend before maturity.- The second one introduces the default risk of the consortium, without any com-pensation for the community in case of an unreciprocated contract breaking-off.- The third one also considers the default risk of the consortium, and the consortiumhas to pay penalty in case of default, the community receiving a part of this penaltyas a compensation.

In the second model, whatever is the discount rate (positive or zero), the commu-nity optimal policy is to give up for outsourcing. In the first model, outsourcing isoptimal if the noise level around the maintenance benchmark cost is higher than athreshold: this corresponds to a risk transfer from the community to the consortium.Remark that this threshold is an increasing function of the benchmark cost and ofthe coefficient of the consortium utility. Similarly, in the third model with penaltyin case of default, outsourcing is optimal is the randomness is high enough, or if thecontract maturity is short, if the benchmark cost or the penalty are high enough.

Acknowledgements We thank Jerome POUYET who introduced us the economic bases of thissystem of Public Private Parternships. We also thank the colleagues who heared to us and askedsome questions which allowed us to improve our paper.

References

1. DOROBANTU D., MANCINO M.E., PONTIER M.: Optimal strategies in a risky debt con-text. In: Proceedings of 28th conference on quantum probability and related topics, Ham-mamet, November 2007, Stochastics,81:3, 269-277 (2009).


2. IOSSA E., MARTIMORT D., POUYET J.: Partenariats Public-Priv, quelques rflexions.Revue conomique,59 (3)(2008).

3. JEANBLANC M., YOR M., CHESNEY M.: Mathematical Methods for Financial Markets.Springer, Berlin, Heidelberg, New York (2009).

4. KARATZAS I. and SHREVE S.: Brownian Motion and Stochastic Calculus. Springer,Berlin, Heidelberg, New York (1988).

5. PATIE P.: On some first passage times problems motivated by financial applications. PhDThesis, ETH Zurich (2004).

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