WELFARE ORIENTED PRICING VS. PROFIT ORIENTED PRICING AND
WELFARE GAIN OF PPP
Alain Bonnafous*
Laboratoire d'Economie des Transports
(CNRS, Université Lyon 2, ENTPE)
9 March 2016
Ground address: LET, ISH, 14 av Berthelot, 69363 Lyon, France.
* Professor (CNRS-LET), email: alain.bonnafous@let.ish-lyon.cnrs.fr
ABSTRACT
This paper deals with the alternative between the socio-economic or the financial
point of view and is focussed on the issue of pricing for PPP infrastructures. With a
profit pricing there is a welfare loss for each new project but the need of subsidy is
lower and the same public budget could allow a more important investment
programme and thus a greater welfare gain than with a welfare pricing. This
conjecture is illustrated by empirical simulations. This paradoxical result means that
the optimal pricing theory
cannot be generalised from one single project to a
programme of several projects.
Bonnafous,
Welfare Oriented Pricing vs Profit Oriented Pricing and Welfare Gain of PPP
1. INTRODUCTION
This paper deals with a research programme from which a previous communication
presented during the 10th WCTR (Bonnafous and Jensen, 2005) already stemmed.
We need to start with a reminder of the framework and of the main results of this
paper because the present communication remains situated in the field of
consequences of some mathematical properties of the same relationships between the
need of subsidies and some economic parameters. One of the main issues of this
reminder, presented in the section 2, concerns the “tyranny” of the financial internal
rate of return (IRR) and suggests to consider the question of optimal pricing.
Indeed, in the previous communication, both the financial and the socio-economic
evaluation of each investment project were based on the same hypothesis of pricing.
From the moment the financial IRR appears as a better ranking criterion for a welfare
purpose than the socio-economic internal rate of return (ERR), we can wonder
whether the optimal pricing with respect to an optimised welfare for each project
remains the optimal pricing when we consider a programme of new projects planned
under a public financing constraint.
More precisely, this paper examines if, for this kind of programme, a profit oriented
pricing, with which public funding can be reduced, provides a better welfare level
than a welfare oriented pricing. The difference between these two sorts of pricing
will be emphasised in the section 3.
Some answers to our central question will be stated in section 4, these result being
based on numerical simulations.
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2. REMINDER ON THE OPTIMAL RANKING OF INVESTMENT
The previous paper concerned the choice of infrastructure investments and more
specifically the optimal ranking of the projects that are to be implemented. The
framework was that of Public Private Partnerships, or more generally, that of projects
financed both by subsidies and by theirs own revenues such as tolled highways.
The optimal investment programme (i.e. which provides the maximum of socioeconomic net present value or NPV) must be defined under a constraint of annual
subsidies. This previous communication demonstrated that the optimal ranking was
that of decreasing ratio NPV/subsidies. It also proved that the ranking of decreasing
socio-economic internal rate of return (IRR) does not provide a better welfare gain
than the ranking of decreasing financial IRR. To the contrary, when the financial IRR
is higher, the need of subsidy is lower and the same public budget allows for a more
important investment programme. The lower the level of this budget, the higher the
interest to follow the financial IRR ranking of the project.
To confirm these points, we used a set of 17 toll highway projects for which
homogeneous economical data were available (Cf. Appendix 1). These 17 French
projects were in competition in the early 90s. Most of them have been carried out
with a small contribution from public money owing to the former French financing
system, that of crossing subsidies between tolled highways (before an european
reform made that impossible)). In this exercise our working hypothesis is a
mechanism of PPP in which for each project a subsidy can help the operator to reach
a target IRR.
Subsidizing rates have been calculated from the relationships of our PPP financing
model (presented in the appendix 2), taking 8% as the target IRR for the operator.
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We assume that there exists a budget constraint F the first year, this constraint
increasing by 2,5 % yearly. To make clear the role of the budget constraint, we have
changed its value between 0<F> and 2<F>, <F> being the mean subsidy required by
all the projects (<F>= 472 Meuros). For a given order of the projects, the value of F
determines the rhythm of completion of projects, since each project needs to draw on
this budget an amount depending on available public subsidies each year.
In figure 1, we compare the total socioeconomic NPV (NPVse) returned by the 17
highway projects for different rankings, as a function of F. More precisely, we plot
the % of gain obtained by choosing the specified ranking criterion compared to the
total NPV returned by using the ERR as the ranking criterion. We use two other
criteria: the financial IRR and the « output », defined as O(i) = NPV(i)/sub(i), where
sub(i) represents the amount of public subsidies required by this project to obtain the
targeted IRR (8%, as assumed above). Note that the curve « output » is very close to
the curve labeled "Optimum" (triangles) which was obtained by a numerical
algorithm, inspired from the Monte Carlo method and providing the maximum of
total NPV.
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Figure 1: Comparison of the total socioeconomic NPV
returned by the 17 highway projects for different rankings
(We plot the % increase of the program NVP compared to that of the ERR
ranking)
Nevertheless, what is important for the present paper is the result which confirms in
Figure 1 that the financial IRR is a better ranking criterion than the ERR, and this is
the truer the tighter the budget constraint. When this constraint is lower than some
value (close to one third of the average subsidy needed for a project), the NPV output
of the program obtained with the ERR ranking is frankly disastrous when compared
with the pure financial IRR order.
This paradoxical result is attributable to the financial constraint. Indeed, when the
public financing capacity is very low, an investment programme which takes little
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Welfare Oriented Pricing vs Profit Oriented Pricing and Welfare Gain of PPP
account of the financial profitability of the project will very quickly consume the
public funds available, resulting in a slower rate of entry into service. Thus, this
"budget effect" explains the paradox in terms of the length of network brought into
service according to the order of programme implementation.
The present paper deals with the same kind of alternative between the socioeconomic or the financial point of view but focussed on the issue of pricing. On a
tolled road, for instance, the pricing depends on whether its objective is that of
maximising the profit of the operator or that of maximising the welfare. On the basis
of the previous results on the optimal ranking, we can wonder what is the optimal
pricing for a programme of new projects.
With a profit oriented pricing there is a welfare loss for each new project but the
need of subsidy is lower. We have thus to consider the budget effect: the profit
pricing improves the financial IRR and, under the same public budget constraint,
could allow for a more important investment programme and thus a greater welfare
gain than under the welfare pricing. In order to explore this issue we need to
formalise the relationships between the selected price, on one side, and both the
welfare gain and the need for subsidies, on the other side.
3. WELFARE PRICING VS. PROFIT PRICING
The contrast between welfare and profit is a classical economic problem (for instance
Mills, 1995). The optimal charge of a transport infrastructure use is based on the
social marginal cost (Boiteux, 1956  1960). More precisely, the « price-relevant
marginal cost » equals the social marginal cost minus the private marginal cost. This
pricing is usually opposed to the optimal pricing for the operator revenue. Facing to
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these two facets, the Ramsey-Boiteux pricing reconciles each other. In the present
paper we do not use exactly the same distinction: to the profit oriented pricing we
will oppose a welfare oriented pricing based on the scarcity of public financing
capacities.
In order to facilitate the analysis, we consider a standard case in which the demand d
is a linear function of the toll p and takes the form:
d  d0    p
(1)
Were d0 and  are fixed parameters. The users surplus S is therefore given by:
S
 d0
(  p)2
2 
( 2)
And the gross revenue for the operator is given by:
R  d 0  p    p2
(3)
Thus the classical distinction proposed by Jules Dupuit (1844 and 1949)) between
two pricing strategies appears clearly in Figure 2:
- either the public authority decides to favour the interest of the users and
consequently to choose a free use of the infrastructure,
- or the public authority decides to use the road pricing in order to minimise the
public expenses.
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In the firs case the user surplus reaches its maximum value (d02/2). In the second,
the revenue reaches its maximum value (d02/4). Note that the optimal price for the
operator provides the internalisation of only half of the maximal user surplus because
we are in the case of a single price.
Figure 2 : Stylised demand function, users surplus and revenue
d
S R
d0
Smax = d02/2
Rmax = d02/4
O
PRmax = d0/2
Pmax= d0/
P
d = d0 – .p
Users Surplus S
Operator benefit R
In order to simplify the calculations, we shall assume that variable costs are
negligible and that only the constant costs are to be covered by the operator. This
hypothesis means that the maximum of the gross revenue and the maximum of the
net revenue are procured by the same value of p (in our stylised case this value is
equal to d0/2). Thus, regarding any optimal pricing problem, we shall only consider
the variable elements of the objective functions. These functions are therefore R(p)
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Welfare Oriented Pricing vs Profit Oriented Pricing and Welfare Gain of PPP
for the operator and U=R(p)+S(p) for the public authority. We abstract from the
function U the existence of a constant of derivation as it is not relevant for the
subsequent analysis.
Nevertheless, when the projects are financed both by subsidies and by theirs users,
any additional unit of revenue is also one saved unit of public expense and
consequently one saved unit of tax. From the moment it is an accepted fact that the
tax system is not neutral and not optimal, one unit of saved public expense has a
value higher than one. Thus, as a measure of an element of the collective wealth, the
relevant amount of the revenue is not R but .R were  is a scarcity coefficient of
public funds. Thus, the welfare objective function becomes:
U  R S
( 4)
or alternatively:
U
d 02
1
 (  1)  d 0  p    p 2 (   )
2
2
(5)
In the very particular case in which the tax system is perfectly neutral (such that  = 1)
equation (5) becomes (6):
U
d 02  2
 p
2b 2
(6)
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Welfare Oriented Pricing vs Profit Oriented Pricing and Welfare Gain of PPP
When  = 1, it is quite natural for the function U to be a decreasing function of p
because for any additional value op p there is a loss of S from which only a part is
internalized under the form of additional R which the operator must receive. Thus, in
this particular case, the welfare function is maximized for p = 0.
In the more general case, the derivative dU/dp can be written as :
1
U '  (  1)  d 0  2   (   )  p
2
( 7)
Thus the welfare function (5) reaches its maximum for:
pU max 
 1
1

2

d0
2
(8)
It may be easily seen that this value of pUmax varies from 0 to pRmax, the value of p
for which the gross revenue for the operator is maximum. Equation (8) shows
furthermore that the higher the scarcity coefficient of public funds, the nearer of
pRmax.the value of pUmax. When  tends to infinity pUmax tends to pRmax i.e. to d0/2.
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Welfare Oriented Pricing vs Profit Oriented Pricing and Welfare Gain of PPP
Figure 3: The utility function in case of scarcity coefficient for public funds
S, R
U
O
PUmax
PRmax
P
Users Surplus (S)
U=S+.R ( = 1)
Gross Revenue (R)
U=S+.R ( > 1)
If we consider the linear demand function as an approximate figure of the real
function and if we retain for  the value used by the French Administration
(presently: 1.3), we obtain:
pUmax = 0,375.pRmax
These relative positions of the two optimal prices will be useful later on for our
exercise of simulation.
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4. THE OPTIMAL PRICING OF A PROJECT IS NOT ALWAYS OPTIMAL
FOR A PROGRAMME OF PROJECTS
Let us go back to the initial question, that of whether the optimal pricing in the sense
of the welfare maximisation for each project remains optimal once we consider the
pricing of a programme of new projects planned under the public financing
constraint.
In order to prove that this proposition is wrong we only need to find one case of the
programme for which a profit oriented pricing provides a higher overall socioeconomic NPV than a welfare oriented pricing.
For a perfect evidence, we should compare programmes with the genuine values of p
optimising respectively the welfare gain and the revenue for each project. Today we
do not know these values. Nevertheless, we know that for the set of 17 projects of
toll highway projects the level of p used for the ex ante evaluation was equal to the
mean toll on the French network during the early 90’. Thus, in order to select a
reasonable rough estimate of values for the two sort of pricing we can assume that
this mean toll was somewhere between pUmax and pRmax and even that it was on a
median position pm. In this case, from equations (3) and (7) it follows that, ift R was
boosted by 16 %, it would lead to a loss of U of 8 %. Symmetrically, a reduction of
tolls R of 16 %, would increase U of 8 %.
Simulations were made of investment programmes depending on what we call the
welfare oriented pricing (pm -  pm) or what we call the profit oriented pricing (pm +
 pm). We used the set of projects already well-tested. For each pricing hypothesis,
subsidy rates and socio-economic NPV’s were calculated owing to equations of our
PPP financing model (Cf. Appendix 2).
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Thus, for each pricing alternative we know the ratio NPV/Subsidy of each project
and it is easy to establish the optimal ranking based on this ratio. It was assumed that
the projects were subject to a budget constraint under which the annual public
financing was restricted respectively to 150, 300, 450 and 600 Meuro during the first
year of the programme. Subsequently the financing was assumed to increase by
2.5 per cent a year. Each of these programmes was assumed to run for 15 years. The
results are given in the table 1 below.
The paradoxical facet of these results is that the use of a profit oriented pricing
improve significantly the welfare gain of the programme even though each project
provides a lower NPV than with a welfare oriented pricing. That is of course a
consequence of the contents of the two programmes, the most effective being
composed of a greater number of projects.
Our intuition is thus confirmed: we find the same kind of results than those which
were reminded in section 2. There is also the same kind of explanation in the sense
that the rate of subsidy of a PPP project is strictly depending of the financial IRR
which is itself depending of the pricing system.
Nevertheless, the more radical explanation derives of the properties of the function
between the need for subsidy and the additional IRR which the operator must receive
(Cf. Appendix 2): the concavity of this function means that the first differences
between the targeted IRR and the IRR of the operation can be extremely costly,
particularly in the case of projects with a low intrinsic IRR. Thus any improvement
of this intrinsic IRR by a profit oriented pricing decreases the need of subsidy and
increases the number of implemented projects.
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Welfare Oriented Pricing vs Profit Oriented Pricing and Welfare Gain of PPP
Table 1: Comparison of the total NPV returned by the optimal programme
regarding two different pricing systems
Budget
Subsidies and
Welfare oriented
Profit oriented
constraint
NPV
pricing
pricing
150
Total programme
20227
34093
15,0
22,0
26199
39168
8,5
14,2
27865
43184
5,9
8,7
30829
45832
4,5
6,9
NPV
Total
NPV / Subsidy
300
Total programme
NPV
Total
NPV / Subsidy
450
Total programme
NPV
Total
NPV / Subsidy
600
Total programme
NPV
Total
NPV / Subsidy
There is nevertheless a slight difference with the previous results. We found that the
pure financial IRR is a better ranking criterion than the ERR and that this is the truer
the tighter the budget constraint. We do not observe on the results represented in
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Welfare Oriented Pricing vs Profit Oriented Pricing and Welfare Gain of PPP
Figure 4 the same role of the tightness of the constraint: the pricing effect becomes
significant as soon as the budget constraint is active.
Figure 4:Comparison of the total socio-economic NPV
returned by highways for alternative pricing systems
25,0
20,0
15,0
NPV/Subsidy
10,0
5,0
0,0
150
300
450
600
Budget constraint
Welfare oriented pricing
Profit oriented pricing
Several points still remain to be examined. For instance, because our results concern
only a particular case, it would be useful to specify the general conditions under
which an optimal programme based on the profit oriented pricing could provide a
more important welfare gain than an optimal programme based on welfare pricing.
5. A POLICY ORIENTED CONCLUSION
Intuitively, the optimal welfare pricing of each project would be the one that
provides the highest welfare gain when the project is integrated in a larger
programme, that of several projects. From the moment there is at least one case in
which this rule does not work, this case is an evidence that the intuitive rule is
wrong. From an academic point of view, it is clear that the results above open some
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Welfare Oriented Pricing vs Profit Oriented Pricing and Welfare Gain of PPP
new questions for both the optimal pricing theory and the PPP’s programmes
assessment.
From a policy oriented point of view we can summarise our result in a few words: In
a programme of PPPs under the public budget constraint, it could be of the general
interest to allow the operator to determine the infrastructure pricing.
REFERENCES
Boiteux, M. (1956), Sur la gestion des monopoles publics astreints à l’équilibre
budgétaire, Econometrica, Vol. 24.
Boiteux, M. (1960), Peak-Load Pricing, Journal of Business, April, Vol. 33, n°2, pp.
157-179.
Bonnafous, A. (2002), Les infrastructures de transport et la logique financière du
partenariat public - privés : quelques paradoxes Revue Française d’Economie, vol.
17, n°1.
Bonnafous, A., Jensen, P. (2005), Ranking Transport Projects by their
Socioeconomic Value or Financial Interest rate of return ?, Transport policy. Vol.12,
Issue2. pp. 131-136 (Award of the best communication of the 10th WCTR, Istambul,
July.2004).
Dupuit, J. (1844), De la mesure de l’utilité des travaux publics, Annales des Ponts et
Chaussées, n° 116.
Dupuit, J. (1849). De l’influence des péages sur l’utilité des voies de communication,
Annales des Ponts et Chaussées, n°207.
Hayashi Y. & Morisugi H. (2000), « International comparison of background concept and
methodology of transportation project appraisal », Transport Policy, 7(1), pp.73-88.
Mills, G. Welfare and Profit Divergence for a Tolled Link in a Road Network,
Journal of Transport Economics and Policy, 1995, vol.29:2, pp. 137-46.
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Welfare Oriented Pricing vs Profit Oriented Pricing and Welfare Gain of PPP
Pons , D., Johenson, F. (2006), “L’influence du tarif des infrastructures sur la
rentabilité sociale d’un programme”, document de travail, available on:
http://halshs.archives-ouvertes.fr/halshs-00117036
APPENDIX 1: MAIN CHARACTERISTICS OF THE 17 PROJECTS
Figure 5 : ERR and IRR for 17 projects used for
the simulations
12,00%
10,00%
TRI
8,00%
6,00%
4,00%
2,00%
0,00%
0%
10%
20%
30%
40%
50%
60%
70%
80%
TRE
APPENDIX 2: THE PPP’s FINANCING MODEL
Suppose a standard case (Bonnafous, 2002) in which the capital cost C is incurred at
an annual rate c = C/d between the dates –d and 0. When the project comes into use
at time 0, the annual profit rate (revenues less operating costs) takes the form (a +
bt). We now introduce this further notation:


is a discount rate which may be used to calculate the NPV,
0
is that value of  which makes the NPV (of the unsubsidised project) equal
to zero – in other words,  0 is the internal rate of return,
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Bonnafous,
Welfare Oriented Pricing vs Profit Oriented Pricing and Welfare Gain of PPP

is the subsidy rate, expressed as a percentage of c (and hence also of C),

is the increase in the IRR that results from the subsidy rate  – that is to say,
for a subsidy rate  , the IRR becomes  0 +  .
If we recognise all benefits and costs up to a terminal date T, then:
0
T
NPV   - c.e .dt   (a  b.t).e αt .dt
-αt
d
( 9)
0
To simplify the analysis – with little empirical effect – now set the terminal date to
infinity. Then (9) becomes
NPV 
1
b
c(1 - ed )  a  



(10)
The IRR  0 of the unsubsidised project is therefore given by:
c(1 - e 0 d )  a 
b
0
0
(11)
When the subsidy  is applied, equation (3) becomes
(1 -  )c(1 - e ( 0   ) d )  a 
b
0
0  
(12)
If we think of the situation as one in which we want to find the subsidy rate  that
yields a specified IRR of (  0 +  ), then (12) may be written as
τ 1
a( 0   )  b
c( 0   )(e ( 0  ) d  1)
(13)
Here,  is expressed as a function of six variables. However, these are not all
independent, because  0 depends on a, b, c and d. Furthermore, equation (12)
implies that if  = 0, then  = 0.
The function (13), as a function of  0 (or  ), is concave. This property is illustrated
by the family of curves shown in Figure 6, which is drawn for the numerical case
given by c = 100, b = 1, d = 5 together with alternative values of a chosen so that as
a increases from one value to the next,  0 increases by 0.4 percentage points. Each
curve corresponds to a particular value for a, and shows how the required subsidy τ
increases as we increase the IRR from  0 to the target IRR, (  0 +  ).
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Welfare Oriented Pricing vs Profit Oriented Pricing and Welfare Gain of PPP
Figure 6 : The relationship between the subsidy rate and the IRR
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