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.

Profit P ≥ 0

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

qq

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

Outcomes of alternative government regulations

DemandDemand

* * 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.