Properties of Pareto-Efficient Contracts and Regulations for Road Franchising Hai Yang Chair...
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Transcript of Properties of Pareto-Efficient Contracts and Regulations for Road Franchising Hai Yang Chair...
Properties of Pareto-Efficient Contracts and Regulations for
Road Franchising
Hai Yang
Chair Professor
Department of Civil and Environmental Engineering
The Hong Kong University of Science and Technology
Outline
1. Introduction of BOT schemes
2. Theoretical analysis of a BOT toll road project
3. Conclusions
4. Future research
What is a BOT scheme?
BUILD
OPERATE
TRANSFER
Private firm constructs infrastructure facility
TollConcession period
Transfer to government
BOT is a form of project financing
To maximize total social welfare during the whole road life
Aims of public sector:
Aims of private sector:
To maximize net profit during the concession period
Why do we need BOT schemes?
Private sector: more efficient than the public sector, and therefore builds and operates facilities at less cost;
Public sector: facing taxpayer resistance, unable to finance facilities; short of funding
Private sector willing and able to undertake for a profit
Users who find it worthwhile to patronize this new road and pay charges
Users who do not use these new roads benefit from reduced congestion on the old ones
All may benefit whenever the charges cover all costs (including congestion and environmental costs)!
Aims of the research
Private investors:
Public sector:
to identify how, and under what circumstances, a highway BOT project is feasible and profitable, to identify project risks,
understanding how a proposed project will benefit the private investor, road users and the whole of society.
Aim of research: to establish a BOT contract acceptable to both parties
Model Formulation
City A City B
A single highway
Three fundamental decision variablesof a BOT project
Total social welfare (whole road life )
Profit(concession period T )
BOT
Contract
Concession period: TToll charge: pRoad capacity: y
Public sector
Private sector
A highway project
T̂
The Demand-Supply Equilibrium Condition:
Toll charge can be viewed as the function of travel demand q and road capacity y.
Model Formulation
,B q p t q y
Link travel time function
Value-of-Time
Inverse demand function
Toll charge
,p B q t q y
Homogeneous users with identical VOT;
The problem of the public sector:
ˆ, , , W T q y T S q y T T S y I y
0
, d ,q
S q y B w w qt q y
11 1max , : 0
qS y S q y q
Total consumer surplusduring concession period
Total consumers’ surplusduring post-concession period
Construction cost
Social welfare
Model Formulation
The problem of the private sector:
Total toll revenueduring concession period
Construction costProfit
, , , P T q y Tqp I y T q B q t q y I y
Model Formulation
Bi-objective programming for the BOT problem:
, ,
, ,max
, ,T q y
W T q y
P T q y
ˆ, , : 0 , 0, 0T q y T T q y where
Note: Perfect information for both the public sector and the private sector
Model Formulation
* * *, ,T q y
* * *, , , ,W T q y W T q y
* * *, , , ,P T q y P T q y
Definition (Pareto-Efficient Contract):
is said to be a Pareto-efficient
such that
and
with at least one strict inequality.
A BOT triple
, ,T q y contract if there is no other BOT triple
Model Formulation
(a) and (b) , and ;(c) .
Assumptions
Assumption 1
Assumption 2
Assumption 3
0B is a strictly concave function; qB q0t q 2 2 0t q 0t y
0I
Homogeneous of degree zero link travel time function
Constant return to scale in road construction
, , ,t q y t q y t
1yIE
I y ky(Elasticity of the investment cost in output capacity)
(k: the unit capacity cost)or
is volume-capacity ratio.
is a Pareto-efficient contract, then .
Properties of Pareto-Efficient Contracts
* * *, ,T q y
* ˆT T
Proposition 1: Under Assumption 1, if a triple
.Note:
1) Any BOT contract with concession period less than road life is wasteful, namely, renegotiating the contract can make at least one party better off.
2) This ‘‘lifetime concession period” result seems to be realistic because several BOT contracts around the world have been awarded for 99 years, including Highway 407 in Toronto, the Chicago Skyway and the Pocahontas Parkway (Virginia Route 495) in Richmond, Virginia.
Properties of Pareto-Efficient Contracts
Pareto-efficient contract, , solves * *ˆ, ,T q y
Proposition 2: Under Assumptions 1- 3, the v/c ratio for any
Thus, it is constant along the Pareto-optimal frontier and equals
2* *T̂t k
*
the socially optimal v/c ratio, .
Pareto Efficient Frontiers and Constant Volume/Capacity Ratio
Monopoly Optimum
y
y
00
Social Optimum
Pareto optimalsolution set
Demand
CapacitySocial welfare
Profit
Pareto optimalfrontier
00
Proposition 3: Under Assumptions 1, 2 and 3, for any
Pareto-efficient contract , the average social
cost per user is constant, namely,
Properties of Pareto-Efficient Contracts
* *ˆ, ,T q y
* * * *
*
,
ˆ
L q t q y I yC
Tq
,T qt q y I yASC
Tq
per user per unit time or per trip
during the concession period
Assumption 4 (Power construction cost function)
, 0I y ky
1
1
decreasing returns to scale in road construction
constant returns to scale in road construction
0 1 increasing returns to scale in road construction
Decreasing, constant and increasing returns to investment
Properties of Pareto Efficient ContractsEffects of Returns to Scale in Road Construction
Properties of Pareto Efficient Contracts
Social Optimum
Monopoly optimum
q
y
y
00
Pareto-optimal solution set
1: Decreasing returns to scale in road construction
*y
q
W P
y
*
1.0
*q q Demand
Capacity
Properties of Pareto Efficient Contracts
Social optimum
Monopoly optimum
0 1: Increasing returns to scale in road construction
y
y
00
*y
q
q
W P
y Pareto optimal solution set
*
1.0
Capacity
Demand*q q
Return to Investment and Profit Properties at Social Optimum
ˆ, ,T q ySocial optimum contract:
1 ˆ1 0 1
0 1
1 ˆ1 0 0 1
Tqp
Tqp
Corresponding profit:
p B q t
Decreasing
Increasing
Constant
Return to Investment and Profit Properties at Pareto Efficient Solutions
0 1 For increasing returns to scale in road construction Profit P < 0 for certain portion of the Pareto optimal frontier
1.0 For constant or decreasing returns to scale in road construction, profit P ≥ 0 at any Pareto-efficient solution.
Government Regulation for Achieving a
Predetermined Pareto-Efficient Contract
, ,T q yBOT contract
p B q t q y
ROR , ,P T q y I y
Toll
Rate of return on investment
Return on output
Several definitions on regulatory issue
2ROO: , ,P T q y Tq p
Markup chargeThe amount of profit earned from each unit of realized demand (each trip) during the concession period
Government Regulation for Achieving a
Predetermined Pareto-Efficient Contract
Regulatory mechanisms on highway projects
Regulatory regimes implementation
Price-Cap Setting a maximum toll charge (price cap)
ROR Setting a maximum rate of return on investment (“fair” rate )
Capacity Setting a minimum level of capacity (investment level)
Demand Setting a minimum level of demand
Markup Setting a maximum markup charge
Alternative Government Regulations
* *ˆ, ,T q yConsider a target Pareto-efficient contract
* * * *ROR ROR , ,T q y * * * *-p B q t q y
Regulation regime Outcome BOT Contract
Price-Cap
ROR
Capacity
Demand
Markup
*p p
*ROR ROR
*y y *p p ˆT T*y y *p p ˆT T
*y y *y y *p p ˆT T
*q q
*2 2p p
* *ˆ, ,T q y
* *ˆ, ,T q y
* * * *2
ˆ ˆ, ,p P T q y Tq
Numerical Examples
4
0, 1.0 0.15t q y t q y
Link travel time function (BPR)
1 lnB q q Q Inverse demand function (negative exponential)
10000 veh/hQ 0.04
0 0.5 hourt
4ˆ 30 years 30 4380 1.314 10 hoursT
The operating hours per year is assumed to be hours
exp Bq Q
12 365
0.69
58.50C
Constant volume-capacity ratio:
Average social cost per user:
ˆ, , ,970,1405.8T q y T
ˆ, , ,357,517.4T q y T
Socially optimum BOT contract:
Monopoly optimum BOT contract:
0I y kt y 61.2 10 HK$/(h veh/h)k
Case 1: Constant Returns to Scale in Road Construction
6.80 HK$p
31.80HK$p
1.0
Case 1: Constant Returns to Scale in Road Construction
Social Welfare
Profit
610 HK$
610 HK$
6.80,31.80p
(HK$)
Constant Returns to Scale in Road Construction
0
Volume-Capacity Ratio
Tra
ffic
Load L
evel
0.5 0.6 0.7 0.8 0.9 1.0
200
400
600
800
1000
Regulation strategy
Pareto optimal solution set
SO
MO
0.67 0.6943
59.07 60.46C
Volume-capacity ratio:
Average social cost per user:
ˆ, , ,955,1375.4T q y T
ˆ, , ,368,549.24T q y T
Social optimum BOT contract:
Monopoly optimum BOT contract:
0I y kt y 60.25 10 HK$/(h veh/h)k
6.97 HK$p
31.05HK$p
1.2
Case 2: Decreasing Returns to Scale in Road Construction
Social Welfare
Profit
610 HK$
610 HK$
6.97,31.05p
(HK$)
0 0
151151
243.4 243.4392
392
608.5608.5
867867
1070.21070.2
1200.5
1200.5
1200.5
1243.3
1243.3
1255.2
Volume-Capacity Ratio
Tra
ffic
Loa
d Le
vel
2467.6 2467.6
2642.3 2642.3
2800 2800
3000
3000
3147
3147
3235
3269.5
3278.4
0.6 0.7 0.8
400
600
800
900
Pareto optimal solution set
Regulation strategy
SO
MO
10
Descreasing Returns to Scale in Road Construction
0I y kt y 61.5 10 HK$/(h veh/h)k 0.8
0.523 0.5024
53.37 52.87C
Volume-capacity ratio:
Average social cost per user:
ˆ, , ,1230,2448.2T q y T
ˆ, , ,445,850.9T q y T
Social optimum BOT contract:
Monopoly optimum BOT contract:
1.9 HK$p
27.25 HK$p
Case 2: Increasing Returns to Scale in Road Construction
Social Welfare
Profit 610 HK$
610 HK$
1.9,27.25p(HK$)
30L
Zero-profit Pareto optimal contract:
2.394p Years
HK$
* 2399.2y Veh/h* 1206q Veh/h* 0.5027
00
325325
685685
985 985
12351235
1395
13951395
1425.6
Volume-Capacity Ratio
Tra
ffic
Loa
d Le
vel
2962.5
3177.23177.2
3512.43512.4
3729.8
3860.9
3958.5
0.4 0.5 0.6 0.7
400
600
800
1000
1200
Pareto optimal solution set
Regulation strategy
SO
MO
Increasing Returns to Scale in Road Construction
* * ˆ, , ,849,1230.4T q y T
* 9.957 HK$p
Regulations and Outcomes
Target Pareto Efficient BOT contract
*ROR 12.16%
Regulation regime Outcome BOT Contract
Price-Cap
ROR
Capacity
Demand
Markup
*p p
*ROR ROR
*y y
*q q 4ˆA ,849,1230.4T
1ˆA ,723,769.2T
2ˆA ,502,1725T
3ˆA ,487,1230.4T
*2 3.34 HK$p
*2 2p p 5
ˆA ,849,1230.4T
Conclusions
Introduced the definition of the Pareto efficient cont
ract to the BOT problem
Investigated the properties of the set of Pareto effici
ent contracts
Examined the effectiveness of alternative governme
nt regulations
Relevant Research Tan, Z.J., Yang, H. (2012) Flexible build-operate-transfer contracts for road franchisin
g under demand uncertainty. Transportation Research 46B, No.10, 1419–1439.
Tan, Z.J. and Yang, H. (2012) The Impact of user heterogeneity on road franchising. Transportation Research 48E, No.5, 958–975.
Wu, D., Yin, Y. and Yang, H. (2011) The independence of volume-capacity ratio of private toll roads in general networks. Transportation Research 45B, No.1, 96–101.
Tan, Z.J., Yang, H. and Guo, X.L. (2010) Properties of Pareto efficient contracts and regulations for road franchising. Transportation Research 44B, No.4, 415-433.
Tan, Z.J., Yang, H. and Guo, X.L. (2009) Build-Operate-Transfer Schemes for Road Franchising with Road Deterioration and Maintenance Effects. Proceeding of the 18th International Symposium on Transportation and Traffic Theory (ISTTT18) (edited by Lam W.H.K., Wong, S.C. and Lo. H.K.), Springer, pp.241-261, Hong Kong, 16-18 July 2009.
Guo, X.L. and Yang, H. (2009) Analysis of a build-operate-transfer scheme for road franchising. International Journal of Sustainable Transportation, Vol.3, No.5-6, 312-338.