BOT Projects: Incentives and Efficiency

Larry D. Qiu and Susheng Wang1

January, 2008

Abstract: In recent years, governments are increasingly adopting Build-Operate-Transfer

(BOT) contracts for large infrastructure projects. Surprisingly, BOT contracts have received

little attention from economists. In particular, the apparent agency problem in BOT projects

has never been theoretically studied. We develop a model to examine incentives, efficiency

and regulation in BOT contracts. We show that a BOT contract with price regulation during

the concession period and with a possible extension of license is capable of achieving full ef-

ficiency. Interestingly, without price regulation, full efficiency cannot be achieved with or

without a possible extension of license.

Keywords: BOT, infrastructure, incentives, efficiency, regulation, license.

JEL Codes: H54, L14, L51

1 Qiu: School of Economics and Finance, the University of Hong Kong, larryqiu@hku.hk. Wang: Department

of Economics, HKUST, Hong Kong, s.wang@ust.hk.

Acknowledgement: We would like to thank the referee and Dilip Mookherjee (Co-Editor) for their helpful

comments and suggestions.

Page 1 of 21

1. Introduction

Large public infrastructure projects used to be funded primarily by governments. How-

ever, governments today are increasingly finding themselves either unwilling or unable to

finance a growing number of new infrastructure activities. Instead, private companies are

often empowered by governments to build and operate many large projects under so-called

BOT (build, operate and transfer) schemes. Two early examples are the Suez Canal and the

Panama Canal.2 A typical BOT arrangement or contract between a government and a private

firm specifies that the government licenses the firm to build (B) a project and then to operate

(O) the project for a certain period, normally 5 to 30 years, and finally, at the end of the con-

cession period, to transfer (T) the project at no cost to the government. The firm finances the

project and pays the costs of building and operation. During the concession period, the firm

receives revenue from its operations.3 When the concession period is to end, the government

has the option of assuming ownership or renewing the license for the firm to continue its

control of and receive revenue from the project. BOT is often adopted for public utility pro-

jects such as bridges, canals, roads, tunnels, airport terminals, and power plants. Walker and

Smith (1995, p. 27-30) listed 111 large BOT projects launched in over 31 countries and re-

gions by early 1995. The BOT practice has become popular since the 1950s and the number

of BOT projects is expected to increase dramatically over the next decade, especially in East

Asia.

Although the term BOT is relatively new, the practice has been around for several centu-

ries. The Europeans used to call such projects concessions, in which a government assumed

many responsibilities. However, a government today is little involved in planning, building,

or financing of a BOT project, and its role normally is limited to providing loans, guarantees,

tax credits, subsidies, price controls, and license renewal. The Hong Kong Cross Harbour

Tunnel opened in 1972 was the first in Hong Kong to adopt the modern BOT contract as de-

fined above. The Channel Tunnel is a BOT project granted by the governments of the United

Kingdom and France. The Dulles Toll Road Extension, costing US$250 million and begin-

ning in 1988, was reportedly the first BOT highway in the United States. The first privatized

airport terminal in Canada, a BOT project, was Terminal 3 of Pearson International Airport

in Toronto, which costs US$433 million and was completed in 1991.

2 See Walker and Smith (1995, pp. 1-2) for details.

3 In some cases, the firm is also empowered to develop the property during the concession period. When this

occurs, the contractual arrangement is called a Build-Own-Operate-Transfer (BOOT) agreement.

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BOT has become a major means of private-public cooperation in infrastructure projects.

It is different from complete privatization, complete nationalization, or joint ventures. In

particular, BOT differs from privatization in sequencing. In privatization, a government

owns the entity first and then transfers (sells) it to the private sector. In contrast, in BOT, a

private firm bears the cost of building a project and then owns it for a certain period of time

before finally transferring it to the government at no cost. “This is the fundamental attraction

of BOT. It not only takes spending off the government’s balance sheet but also brings in the

commercial skills of the private sector both in identifying viable projects and in running

them efficiently when they are built” (Walker and Smith, 1995, p. 16) and when they are ini-

tially operated.4 Moreover, BOT ensures that the government eventually retains strategic

control of large infrastructure installations.5

Despite of its attractiveness and popularity, BOT has received little attention from

economists. In particular, although the process obviously entails agency problems, there has

been basically no theoretical research on the agency problem arising from BOT projects.

Since a BOT project is to be transferred to the government unconditionally after the conces-

sion period, the firm may not make a sufficient initial investment for long-lasting quality.

The questions are: How can a BOT contract be designed to induce the firm to invest in the

best quality? If such a BOT contract exists, does it resemble a typical BOT contract found in

reality?

In this paper, we develop a BOT model based on stylized facts. A firm is licensed to build

a project that lasts for two periods. After the project has been built, the firm operates the

project in the first period, by which the firm can recoup its investment costs. In the second

period, the government may operate the project by itself or extend the license to the firm.

Here, although the quality of the project is not verifiable, we assume that it is observable ex

post. By the end of the first period, based on the observed quality of the project, the govern-

ment uses a pre-determined rule to decide whether or not to extend the license. Note that if

it is certain that the project will be transferred to the government in the second period, the

firm will not have sufficient incentives to invest for the socially optimal quality of the project.

However, although letting the firm own the project in both periods for certain can solve the

underinvestment problem (the incentive problem), private ownership generally leads to a

reduction in social welfare (the monopoly problem). Our proposed solution to this dilemma

is for the government to impose a price control and have a mixed strategy by giving the firm

4 Walker and Smith (1995, p.16) cite the Channel Tunnel as one example of a privately financed project

addressing a need that the public sector is unwilling or unable to fulfill.

5 Other benefits include relief of the government’s financial burden, relief of the administrative burden,

reduction in size of an (inefficient) bureaucracy, and better service to the public.

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the possibility to continue its operations in the second period. This possibility is dependent

on quality. We show that such a BOT contract can induce the right incentives and achieve full

efficiency. Without price regulation, however, a probabilistic license policy can never achieve

full efficiency.

In reality, two measures have been observed widely in BOT projects: price control and a

possible extension of the license. Price control is a well-known mechanism that is often used

to improve efficiency under a monopoly. It has an additional role to play in our BOT model:

It helps to mitigate the agency problem. The role of a probabilistic license extension has

never been discussed in the BOT literature. We show that it is crucial to a BOT’s success.

However, the widespread use of probabilistic license extensions does not seem to be sup-

ported by the conventional wisdom. In the view of risk aversion, an ex-post decision on the

extension of the license increases uncertainty and hence may hinder investment. On the con-

trary, our study shows that a possible extension of the license, coupled with a price control,

helps to eliminate the agency problem completely.

This paper is organized as follows. In Section 2, we develop a BOT model based on styl-

ized facts. In Section 3, we examine two variants of BOT contracts, one with price regulation

and the other without. In Section 4, we consider two extensions. One extension has a foreign

firm, in which case only part of the profits (i.e., the profit tax) affects domestic social welfare.

The other extension has a price ceiling in the concession period, which includes the welfare-

maximizing price as a special case. We conclude the paper in Section 5. All the proofs are in

the Appendix.

2. A BOT Model

Consider a situation in which a government offers a BOT contract to a firm to build a

large project (e.g., a highway) for use by the citizens of the country. The firm must make an

investment with a fixed cost, denoted ( )k q , which is a strictly increasing and differentiable

function of the quality, ,q of the project, with (0) 0k = and ( ) 0k q′ > . 6 Once the invest-

ment has been made, the fixed cost is sunk. This describes the “B” in BOT. Assume that the

quality, after being determined in the “B” stage, does not change over time.

Once it has been built, the project lasts for two periods, which are assumed to have equal

length. The firm has the right to operate the project in the first period and earns all the prof-

6 Construction costs are crucial in the quality of large projects, such as highway projects. As Riccardo Starace,

Director of Midland Expressway (BNRR), states from experience, “Spending 2-5% more on the materials for the

subgrade and pavement can prolong the life by 50%.” (Walker and Smith, 1995, p. 62).

Page 4 of 21

its generated in that period.7 This is the concession period, i.e., the “O” in BOT. 8 By the end

of the concession period, the project is to be transferred to the government. This is the “T” in

BOT. The government may license the project back to the firm or run the project itself in the

second period. Let φ be the probability that the government licenses the project to the firm

in the second period. When this occurs, the firm runs the project and earns the second pe-

riod’s profits.

The government can observe the quality of the project after it has been in operation for

one period, but the quality cannot be verified by a third party. This implies that quality can-

not be specified in a contract. However, the government can make the probability of license

extension depend on the project quality. Let Q +⊂ R be the set of possible quality levels, q.

A license function/policy, denoted ( )qφ , is defined as a mapping from Q to [0 1], . No prior

restriction is imposed on ( )φ ⋅ .9

Let ( )x p q, be the demand function for the service provided by the project with quality

q, where p is the price charged for the service, such as a toll fee for using a road. Demand

increases with the quality but decreases with the price: ( ) 0qx p q, > and ( ) 0px p q, < . As-

sume that the demand function is the same in both periods. For each q, define the interval

[0 ( )]p q, to be the support of the demand such that ( ) 0x p q, > for [0 ( ))p p q∈ , and

[ ( ) ] 0x p q q, = . 10 Given ( )p q, , the consumer surplus is

( )

( ) ( )p q

ps p q x z q dz, = , .∫

Assume that the variable cost, ( ),c x of providing services from the project is independ-

ent of the equality level and the same in both periods. 11 Assume that ( ) 0c x′ > and

( ) 0.c x′′ ≥ Hence, a single-period operating profit is

7 We will examine the issue of profit taxes in Section 5.

8 We implicitly assume that the government can commit to not seizing the project from the firm before the

end of the concession period. It is very costly for the government to break such promises. For instance, the Suez

Canal had a 99-year concession period, and President Nasser’s acquisition of the canal 12 years before the

contract due date incurred a 28 million pound payment in compensation to the Suez Canal Company (Walker and

Smith, 1995, p. 18).

9 To implement a probability rule, imagine that the government relies on a lottery with a probability of a

certain event being ( )qφ to determine whether the firm gets a license renewal. Alternatively, think of a

bargaining process in which ( )qφ represents the bargaining power of the firm in a Nash bargaining solution.

10 If ( ) 0x p q, > for all 0p ≥ , we define ( )p q = ∞. Note that we allow ( )x p q, = ∞ when the price is

sufficiently low.

11 Although fixed investment costs are high, the cost of maintaining the project and providing the flow

services from the project will normally also be high. Even in the case of toll roads, “Direct tolling of vehicular

traffic is time consuming and expensive, adding 3 to 10% to the overall cost of the project” (Walker and Smith,

Page 5 of 21

( ) ( ) [ ( )]p q px p q c x p qπ , ≡ , − , .

In reality, there are many reasons for adopting the BOT approach in developing large

projects. In this model, we assume that the government does not have the expertise and

technology to build the project and operate it in the first period. However, once the project

has been run for a long time (the first period), the operation becomes standard and the gov-

ernment can run it as efficiently as the firm in the second period. Then, the second period

profit is always ( , )p qπ as given above, regardless of who runs the project.

The government’s BOT contract contains two basic elements. First, it empowers the firm

to build the project and guarantees that, in the first period, the firm will operate the project

and earn profits. Second, it specifies a license policy, which is the probability that the firm

continues to operate the project in the second period, otherwise the project will be trans-

ferred to the government. The contract may also contain other elements that vary in different

policy environments. For example, the government may regulate prices of the services from

the project. The main structure of a typical BOT project is illustrated in Figure 1.

The government offers a BOT contract to the firm. If the firm accepts, it then buildsthe project.

The project is transferredto the government, with probability that the project will be licensed to the firm.

φ

The firm operatesthe project in this period.

The firm or the government operatesthe project in this period.

Period 1 Period 2

Figure 1. The Structure of a BOT Project

3. Analysis

3.1. The First Best: The Benchmark Case

Consider first the benchmark case: the government is as efficient as the firm in building

and operating the project and the government does everything. In this case, the government

chooses prices and quality to maximize social welfare. Alternatively, we can think of the

1995, p. 58). For example, according to the cash flow statement for a Malaysian highway project, the construction

costs were M$383,047, highway maintenance costs were M$83,426, toll operation costs were M$96,263,

management expenses were M$26,342 and administration expenses were M$57,758.

Page 6 of 21

benchmark case as the one in which the project quality is contractible and so the government

can offer the firm a contract that specifies both price and quality.12

Since the two periods are identical in the sense that the project quality, demand and

costs remain the same, the government will choose the same price level for both periods.

Hence, the government’s optimization problem is

[ ]max 2 ( ) ( ) ( )B

p qW s p q p q k qπ

,≡ , + , − , (1)

where the superscript B signifies the first-best outcome. The corresponding first-order con-

ditions (FOCs) are

( ) ( ) 01( ) ( ) ( )2

p p

q q

s p q p q

s p q p q k q

π

π

, + , = ,

′, + , = .

These two equations jointly determine the first-best outcome, ( )B Bp q, . It can be shown that

[ ( )]B B Bp c x p q′= , , (2)

2 ( ) ( )B B Bqs p q k q′, = . (3)

Equation (2) is the standard efficiency formula stating that the price must equal the marginal

cost in equilibrium. Equation (3) is the formula for quality: quality is set so that the marginal

social benefit of improving the quality is equal to the marginal social cost, where the mar-

ginal profit is always zero by (2).

3.2. Regulated BOT

Now, return to the BOT model in which quality is not contractible and the government

has to rely on the firm to build the project and operate it in the first period. In this section,

we assume that the government can regulate the prices charged in both periods. Then, the

BOT contract can be characterized as 1 2 , , ( )p p φ ⋅ , where ip is the price in period i that the

firm must charge.

Given a contract, the firm decides whether to reject it or accept it. To make this decision,

the firm looks at the optimal profit derived from the project, which is obtained by choosing

the quality to maximize its expected profit:

1 2 1 2( , , ) ( ) ( ) ( ) ( )p p q p q q p q k qπ φ πΠ ≡ , + , − . (4)

12 The government must provide full or partial investment funding in this latter case, so that the firm is

willing to enter into the contract. Assume that any such subsidy is in the form of a lump sum.

Page 7 of 21

If the above maximum profit is negative, the firm will reject the contract. This possibility in-

troduces the issue of individual rationality (IR) or the participation constraint. Let us ignore

the IR at the moment. That is, suppose that the above maximum profit is always positive and

so the firm always accepts the contract. Then, the government designs the contract

1 2 ( )p p φ, , ⋅ to maximize social welfare, which is the sum of the consumer surplus and prof-

its. Let 2p′ be the price set by the government in the second period if the government runs

the project by itself. Then, the government’s problem is

1 2 2

1 1 2 2( )

2 2

1 2 1 2

max ( ) ( ) ( )[ ( ) ( )]

[1 ( )][ ( ) ( )] ( )

s t IC ( , , ) ( , , ) for all

p p p qW s p q p q q s p q p q

q s p q p q k q

p p q p p q q Q

φπ φ π

φ π

∗

′, , , , ⋅= , + , + , + ,

′ ′+ − , + , −

. . : Π ≥ Π ∈ .

(5)

Let 1 2 2 ( )p p p q φ∗ ∗ ∗ ∗ ∗′Ω ≡ , , , , ⋅ be the solution to the above optimization problem, where a

star indicates the optimal solution under price regulation. The incentive-compatibility (IC)

condition ensures that the firm will be induced to choose the quality level q∗ with the con-

tract 1 2 ( ).p p φ∗ ∗ ∗, , ⋅ 13

The following lemma indicates that once the regulated price has been set at the first-best

level, Bp , the operating profit is highest when the quality is also at the first-best level, Bq .

This observation is useful for deriving the optimal contract.

Lemma 1. Concerning the one-period operating profit, ( , ),p qπ we have

( ) ( ), for all .B B Bp q p q q Qπ π, ≥ , ∈

We now return to the IR or the participation constraint. In order to focus on the effi-

ciency and incentive issue, i.e., whether or not it is feasible to design a BOT contract to in-

duce the first-best outcome, we assume that the participation constraint is always satisfied

given that 1p is set in a reasonable range. In particular, assume that the IR holds when

1Bp p= . We provide two justifications for this assumption. First, suppose the regulated first

period price is above the average cost so that the firm has positive operating profits in every

single year. Then, we consider the situation in which the first period is sufficiently long, i.e.,

covering many years (e.g., 30 years, which is normal in BOT projects), so that the total first

period operating profit is large enough to cover the investment cost. Second, if the profits are

not sufficiently large to cover the investment costs, then we allow a lump-sum transfer from

the government to the firm such that the IR holds. Since this transfer does not affect the

firm’s incentives for quality investment, we do no specify it explicitly, to simplify the model.

13 The IC condition in the optimization problem is not written in terms of FOCs. This means that we do not

rely on the troublesome first-order approach.

Page 8 of 21

Such a transfer is common in reality. For example, in a highway project, the government may

allow the firm to develop the land at the intersection locations along a highway to build

warehouses or retail space. The derived value from such land development is usually large

enough to cover part or all the investment cost.14

Define the elements in Ω :

1 2 2

( ) ( ) for 0( )

( ) ( ) for .

B

B

B B B

B B B B

p p p pq q

k q p q q qq

k q p q q qπ

φπ

∗ ∗ ∗

∗

∗

⎧ ′⎪ = = = ,⎪⎪⎪ = ,⎪⎪⎨ ⎧⎪ ⎪ / , ≤ ≤ ,⎪ ⎪⎪ = ⎨⎪ ⎪⎪ / , >⎪⎪ ⎩⎩

(6)

Because ( ) ( )B B Bp q k qπ , ≥ by assumption and ( ) ( )Bk q k q≥ for 0 Bq q≤ ≤ , we have

0 ( ) 1qφ∗≤ ≤ for all q Q∈ .

Proposition 1 below gives an optimal BOT contract under price regulation.

Proposition 1. Ω defined in (6) is an optimal solution to problem (5). Specifically,

1 2 p p φ∗ ∗ ∗, , is an optimal BOT contract which induces the first-best quality, Bq q∗ = .

The optimal BOT contract in Proposition 1 completely solves the incentive and effi-

ciency problems: it induces the right incentive and leads to full efficiency ( Bp and Bq ). Even

though the government cannot specify the desired quality level in the contract, the first-best

outcome is achieved.

The license policy plays a critical role to restore the efficiency. Since it can regulate

prices, the government wants to set the prices at the first-best level (i.e.,

1 2 2Bp p p p∗ ∗ ∗′= = = ). However, given the first-best price, ,Bp the firm may not choose the

first-best quality, .Bq The firm’s quality choice is dictated by the IC condition in (5). With an

additional policy tool, ( )φ ⋅ , the government can influence the firm’s quality choice. The idea

is to propose a probability function, ( ),qφ such that arg max ( , )B Bq p qπ= +

( ) ( , ) ( )Bq p q k qφ π − . On one hand, an increase in ( )φ ⋅ induces more quality investment

and vice versa. On the other hand, a change in ( )φ ⋅ does not alter the welfare level because

the prices in the two periods are equal. Therefore, the government can use the license policy

tool freely to affect the firm’s quality choice. The probability function, ( ),qφ∗ given in (6)

does exactly the job. The idea behind the construction of ( )qφ∗ is simple: (a) to ensure the

first-best investment in quality, we assign a sufficiently large probability, ( ),Bqφ∗ to the

14 In Hong Kong, the MTR (subway) company was given land development rights to build residential apart-

ments and commercial complexes near and above the MTR stations. In some cases where no such land was avail-

able, the government provided a financial transfer to the company.

Page 9 of 21

quality level at ;Bq (b) to discourage overinvestment, we assign the same (or a lower) prob-

ability, ( ) ( )Bq qφ φ∗ ∗= to Bq q> (when the quality is higher than the first-best level); and

(c) to discourage underinvestment, we assign a lower probability, ( )qφ∗ to Bq q< (when the

quality is below the first-best level). To understand case (b), note from Lemma 1 that the

firm’s operating profit is lower if Bq q> than if Bq q= and so it is not necessary to assign a

strictly lower probability to discourage overinvestment.

It is worthwhile pointing out that in the normal case, i.e., when ( , ) ( )B B Bp q k qπ > , our

license probability, ( )Bqφ∗ , for the first-best quality is strictly less than 1. If, however, * ( ) 1Bqφ = , then the “T” in the BOT would not take place, which is not a commonly observed

outcome. Thus, our optimal BOT contract usually does not result in an undesirable outcome.

To see why * ( ) 1Bqφ < does the job of fully restoring the incentives, note that, given

1 2 ( )p p qφ∗ ∗ ∗, , , the firm’s expected profit is

( , , ) [ ( ) ( )] ( ) ( ), if the firm chooses ;( , , ) [ ( ) ( )] ( ) ( ), if the firm chooses .

B B B B B B B B B B

B B B B B

p p q p q k q q p q qp p q p q k q q p q q q

π φ π

π φ π

∗

∗

Π = , − + ,

Π = , − + , <

The IC condition requires ( , , )B B Bp p qΠ > ( , , )B Bp p qΠ . By comparing these two profits, we

can see that the firm’s benefit from choosing Bq q< comes from the savings in the invest-

ment cost, [ ( )k q vs. ( )Bk q ]. However, the operating profit is also lower, [ ( , )Bp qπ vs.

( , ),B Bp qπ by Lemma 1]. In addition, raising the license probability, [ ( )qφ∗ vs. ( )Bqφ∗ ], not

only increases the chance of earning the second-period profit, but it also results in having

higher second-period profits [ ( , )Bp qπ vs. ( , )B Bp qπ ]. These observations indicate that a

small increase in ( )Bqφ∗ will have a large impact on inducing the first-best quality, .Bq

It is also clear from the above discussion that the optimal contract is not unique. For ex-

ample, let ( ) ( )o q qφ φ∗= for Bq q≤ and ( )o qφ is a decreasing function for .Bq q> This

contract, 1 2 ( ),op p qφ∗ ∗, , will also lead to the first-best outcome. The most important mes-

sage from Proposition 1 is that the government can design a price-regulated BOT contract to

achieve the first-best outcome. Since the same first-best outcome is achieved, we cannot ac-

tually rank various optimal contracts. For example, the two contracts, 1 2 p p φ∗ ∗ ∗, , and * *1 2 , , op p φ , are different only in the value of the probability off the equilibrium and so they

can be viewed equivalent. If the two optimal contracts are also different in their probability

values at Bq , then the equilibria would also be different, one with a higher probability than

the other of licensing the project to the firm in the second period. However, the welfare re-

sults are the same since the government is indifferent (in welfare terms) between running

the project itself or letting the firm run the project in the second period.

Is the license policy defined in (6) time consistent? In particular, does the government

have an incentive to deviate from the license rule or renegotiate the licensing probability

with the firm ex post (i.e., after the investment has been made)? The answer is no. In the

Page 10 of 21

second period, social welfare is the same whichever party runs the project because (a) it is

assumed that the government can run the project as efficiently as the firm in the second pe-

riod, and (b) the second period price has been set at the first-best level in the ex ante contract.

Hence, the government has no incentive to deviate from its policy ex post.

3.3. Unregulated BOT

In this section, we analyze the case in which the government does not regulate prices

when the firm runs the project, but still allows for the possibility of license renewal. The ob-

jective is to highlight the role of price regulation as discussed in the preceding section.

Given a license policy, ( )φ ⋅ , the firm’s problem is to choose 1 2 p p q, , to maximize its

expected profit:15

1 2

1 2ˆ max ( ) ( ) ( ) ( ),

p p qp q q p q k qπ φ π

, ,Π = , + , − (7)

where the hat symbol refers to the case without price regulation. It is clear that the firm will

choose the same price in the two periods: 1 2p p p= ≡ . Knowing that the firm will choose a

price p and quality q according to (7), the government chooses 2 ( )p φ′, ⋅ to maximize the

expected social welfare. Thus, the government’s problem is

[ ]

22 2( )

ˆ max [1 ( )] ( ) ( ) [1 ( )] ( ) ( ) ( )

s t IC ( , , ) ( , , ) for all 0 andp p q

W q s p q p q q s p q p q k q

p p q p p q p q Qφ

φ π φ π′, , , ⋅

⎡ ⎤′ ′= + , + , + − , + , −⎣ ⎦

. . : Π ≥ Π ≥ ∈ . (8)

Let 2ˆ ˆˆ ˆ ˆ ( )p p q φ′Ω ≡ , , , ⋅ , where all the elements are defined implicitly by

2 2ˆ ˆ ˆ ˆ ˆ( ) 0 [ ( )]

ˆ ˆ ˆ( ) ( )ˆ( ) 0

p

q

p q p c x p qp q k q

ππ

φ

⎧⎪ ′ ′ ′, = , = ,⎪⎪⎪ ′, = ,⎨⎪⎪⎪ ⋅ = .⎪⎩

(9)

Proposition 2 below states the main result of the optimal BOT without price regulation.

Proposition 2. Ω as given in (9) is uniquely defined and is the optimal solution to (8). In

particular, ˆ( )φ ⋅ is the optimal contract in the absence of price regulation. The first-best out-

come cannot be obtained.

Since ˆ( ) 0φ ⋅ = , Proposition 2 says that the outcome is a pure BOT contract, in which the

firm operates the project in the first period and transfers it to the government for certain at

the end of the first period. To provide the intuition for this result, let us suppose that ( )qφ is

15 We could also consider the case in which the license option depends on prices. However, this case is

equivalent to the case with price regulation and so we will not analyze the dependence of license on prices.

Page 11 of 21

differentiable, although this supposition is not needed in the proof of Proposition 2. Then,

the FOCs for (7) are

( ) 0p p qπ , = , (10)

[ ]1 ( ) ( ) ( ) ( ) ( )qq p q q p q k qφ π φ π′ ′+ , + , = . (11)

These two equations determine the firm’s optimal choice, ˆ ˆ( )p q, . Condition (10) implies that

ˆ ˆ ˆ[ ( )] 0;p c x p q′− , > that is, the price is set higher than the marginal cost of production. This

monopoly pricing is in sharp contrast to the marginal-cost pricing in the regulated BOT in

(2). As shown by (11), the firm chooses the quality such that the expected marginal profit of

improving the quality is equal to the marginal cost of investment. This quality setting is dif-

ferent from that in the first-best condition in (3).

The two well-known distortions associated with a monopoly (Spence, 1995) are present

in the case without price regulation: (a) the price is higher than the marginal cost, and (b)

quality is not at the socially optimal level. When the government has two independent con-

trol instruments, price regulation and a license policy, as described in Section 3.2, the two

distortions can be eliminated completely. In the absence of price regulation, only one control

instrument (i.e., the license policy) is left and, hence, it is not surprising that the first-best

outcome cannot be restored. However, what is surprising from Proposition 2 is that the re-

maining instrument has no role to play, i.e., ( ) 0φ ⋅ = . The reason is as follows. Recall that

with price regulation, the government can easily eliminate the price distortion. But that will

also reduce the firm’s incentive to invest in quality. To restore the incentive, the firm is given

a probability to run the project in the second period. Without price regulation, price distor-

tion cannot be eliminated directly. If the quality chosen by the firm is lower than Bq , then

the license policy may help to reduce or eliminate this quality distortion. However, this will

induce the firm to charge a higher price, worsening the price distortion. If the quality chosen

by the firm is higher than Bq , then the license policy will worsen both the price distortion

and quality distortion. Thus, the government chooses ( ) 0φ ⋅ = so that the price distortion

will not be made worse.

4. Extensions

The main result established in the preceding section is that the first-best outcome can be

achieved by a BOT contract with price regulation. In this section, we reexamine this result

with some modifications to the earlier model. The analysis shows that the result remains un-

changed.

Page 12 of 21

4.1. A Foreign Firm

Suppose that the firm in the BOT is a foreign firm. This is a realistic extension because

BOT projects usually welcome bids from all over the world. The winners may be foreign

firms, especially on BOT projects in developing countries. This change in the model has

strong implications for the welfare function and therefore the first-best outcome. The firm

faces an economy-wide profit tax, which is exogenously given as (0, 1)t ∈ . Suppose that the

government can also levy a lump-sum tax, ,τ specifically on the project. Since only part of

the firm’s profit, via taxes, goes to domestic welfare, in the absence of the moral hazard prob-

lem in quality, the government will surely run the project by itself in the second period. Thus,

the first-best problem, in which quality q is contractible, is

[ ]1 2,

1 1 2 2,= max ( , ) ( , ) ( ) ( , ) ( , ) ,F

p p qW s p q t p q k q s p q p qπ π τ+ − + + +

where the superscript F denotes the first-best outcome when a foreign firm is in the BOT.

The FOCs define the first-best outcome, 1 2( , , ),F F Fp p q which is jointly determined by

1

1

11 1

1

2 2

11 2 1

1

( , )( , ) (1 ) ,( , )

( , ) ,

(1 ) ( , )( , ) ( , ) ( ) ( , ).( , )

F FF F F

F Fp

F F F

F FF F F F F F F

q q qF Fp

x p qp c x p q tx p q

p c x p q

t t x p qs p q s p q tk q x p qx p q

⎡ ⎤′= + −⎣ ⎦

⎡ ⎤′= ⎣ ⎦−′+ = −

Now suppose that quality, ,q is not contractible but there is price regulation. Similar to

the analysis in Subsection 3.2, the government’s problem becomes

1 2 2

1 1 2 2, , , , ( )

2 2

1 2 1 2

max ( , ) [ ( , ) ( )] ( )[ ( , ) ( , )]

[1 ( )][ ( , ) ( , )]

s.t. IC: ( , , , ) ( , , , ) for all ,

p p p qW s p q t p q k q q s p q t p q

q s p q p q

p p q p p q q Q

φπ φ π

φ π τ

τ τ

∗

′ ⋅= + − + +

′ ′+ − + +

Π ≥ Π ∈

(12)

where the firm’s expected profit is given as

1 2 1 2( , , , ) (1 )[ ( , ) ( ) ( , ) ( )] .p p q t p q q p q k qτ π φ π τΠ = − + − −

Define 1 2= ( , , , , ( )) :F p p q τ φ∗ ∗ ∗ ∗ ∗Ω ⋅

1 1 2 2 2

2

2

2

,

( ) ( ) for 0( )

( ) ( ) for ,

= (1 ) ( ) ( , ).

F F

F

F F F

F F F F

F F F

p p p p pq q

k q p q q qq

k q p q q q

t q p q

πφ

π

τ φ π

∗ ∗ ∗

∗

∗

∗ ∗

⎧ ′⎪ = = = ,⎪⎪⎪ = ,⎪⎪⎪⎪ ⎧⎪ / , ≤ ≤ ,⎨ ⎪⎪ = ⎨⎪ ⎪⎪ / , >⎪⎪ ⎩⎪⎪⎪ −⎪⎩

(13)

Page 13 of 21

The functional form of the above license function is the same as that in (6). Because

2( ) ( )F F Fp q k qπ , ≥ by assumption and ( ) ( )Fk q k q> for 0 Fq q≤ ≤ , we have

0 ( ) 1qφ∗≤ ≤ for all q Q∈ . We find that this license policy under price regulation and a

lump-sum tax, * ,τ can achieve the first-best outcome.

Proposition 3. FΩ given in (13) is uniquely defined and is an optimal solution to problem

(12). Furthermore, 1 2 , p p τ φ∗ ∗ ∗ ∗, , is an optimal BOT contract that induces the first-best

quality, Fq q∗ = .

The use of the lump-sum tax is to maintain the first-best welfare level. Unlike the case

when a domestic firm is in the BOT, the welfare when the project is run by a foreign firm is

different from that when it is run by the government. To induce incentives, the foreign firm

is given a chance to run the project in the second period, but that will reduce the welfare. A

way out is to let the lump-sum tax restore the welfare level without distorting the incentives.

4.2. Price Ceiling

Recall that in Subsection 3.2 we consider the case in which the firm must charge the

price set by the government. Let us call this case strict price regulation. In this section, we

relax this type of price regulation by considering a price ceiling. That is, the price ceiling re-

places the strict price regulation in the concession period. Moreover, the price ceiling is as-

sumed to be given exogenously, i.e., it is not a policy variable. In particular, we allow the

price ceiling to be any level higher than the welfare-maximizing price, Bp . However, the sec-

ond-period price is still determined by the government no matter who runs the project. This

is a realistic extension.

When quality is contractible, we have the first-best problem. The first-period price is

constrained by the price ceiling. This notion of the first-best outcome is weaker than that in

Section 3.1 in the sense that the firm’s monopoly power in the first period may not be fully

checked. Let 1p be the price ceiling. Since, in the first-best case, the government runs the

project in the second period, the firm’s problem is

1 1

1max ( , ) ( ).p p

p q k qπ≤

Π = −

Without loss of generality, assume that the price ceiling is less than the firm’s profit-

maximizing price level. Then, the first-period price hits the ceiling. Hence, the government’s

optimization problem becomes (the superscript C denotes the first-best outcome in the case

of price ceiling)

2

1 1 2 2max ( ) ( ) ( ) ( ) ( ).C

p qW s p q p q s p q p q k qπ π

,≡ , + , + , + , − (14)

The FOCs imply that

Page 14 of 21

2 2[ ( )],C C Cp c x p q′= , (15)

1 1 2 2( ) ( ) ( ) ( ) ( ),C C C C C C Cq q q qs p q p q s p q p q k qπ π ′, + , + , + , = (16)

which determine the first-best price, 2 ,Cp and quality, .Cq

We now consider the second-best problem in which quality is not contractible. Given a

contract, 1 2( , , ( )),p p φ ⋅ the firm’s profit is

1 2 1 2( , , ) ( , ) ( ) ( , ) ( ).p p q p q q p q k qπ φ πΠ ≡ + − (17)

The government’s optimization problem is16

1 21 1 2 2,

1 2 1 2 1 1

max ( ) ( ) ( ) ( ) ( )

s.t. ( , ) ( , ), , .p p q

W s p q p q s p q p q k q

p p q p p q p p q Q

π π∗

,≡ , + , + , + , −

′ ′ ′ ′Π , ≥ Π , ≤ ∈ (18)

Define the elements in ∗Ω :

1 1 2,

( ) ( ) for [0, ]( )

( ) ( ) for .

C

C

C C C

C C C C

p p p pq q

k q p q q qq

k q p q q qπ

φπ

∗ ∗

∗

∗

⎧⎪ = = ,⎪⎪⎪ = ,⎪⎪⎨ ⎧⎪ ⎪ / , ∈ ,⎪ ⎪⎪ = ⎨⎪ ⎪⎪ / , >⎪⎪ ⎩⎩

(19)

The above license function is the same as that in (6). Because ( ) ( )C C Cp q k qπ , ≥ by assump-

tion and ( ) ( )Ck q k q> for 0 Cq q≤ ≤ , we have 0 ( ) 1qφ∗≤ ≤ for all q Q∈ . We find that the

first-best outcome can also be achieved using a license policy with a price ceiling.

Proposition 4. Suppose that the demand function, ( , ),x p q is concave in ,q the cost func-

tion, ( ),c x has a constant marginal cost, (e.g., ( )c x cx= ),17 and the price ceiling is not

higher than the firm’s profit-maximizing price. Then, ∗Ω defined in (19) is an optimal solu-

tion to problem (18). Furthermore, 1 2 p p φ∗ ∗ ∗, , defined in (20) is an optimal BOT contract

that induces the first-best quality, Cq .

Proposition 4 shows that, even without strict price regulation in the concession period,

the incentive problem can be resolved completely by a similar license policy as in Proposition

1. However, there is a crucial difference between Proposition 4 and Proposition 1. In

Proposition 1, with a strict price control in both the concession period and the extended pe-

riod (namely, the second period when the license is extended), the government can com-

pletely eliminate monopoly power. In Proposition 4, with a limited price control in the form

16 In the second period, no matter who manages the project, since the objective is the same, the government

will set the same price.

17 A constant marginal cost is unnecessary. This assumption substantially reduces the complexity of the

proof.

Page 15 of 21

of an exogenously given price ceiling, the firm is allowed to exercise a certain degree of mo-

nopoly power, which is captured by the difference between the price ceiling and the welfare-

maximizing price. As a result, full efficiency is generally not achieved in Proposition 4, where

full efficiency in the context of our model means a complete resolution of both monopoly

power and the agency problem.

There is also a crucial difference between Proposition 4 and Proposition 2. In

Proposition 2, the government cannot exercise the price control in the extension period, but

in Proposition 4, the government can exercise full price control in the extension period.

Without the price control, a license extension cannot improve social welfare, but with the

price control, a proper license extension can improve social welfare.

5. Concluding Remarks

Incentives and monopoly power are two key issues inherent in BOT projects. This study

analyzes the effectiveness of the two policy instruments, i.e., price regulation and licensing,

in dealing with those two issues. Through price regulation, the government can impose a

competitive price during the concession period. Using a licensing policy, the government can

induce the firm to invest at the socially optimal level. Hence, the first-best outcome can be

reached by a properly designed BOT contract with price regulation and licensing. However, if,

for whatever reason, the price is not regulated during the concession period, there is no role

for a license policy to play. In other words, a license policy can work effectively only in a

regulated environment.

The above results are obtained using a simplified model that aims at capturing only the

key features of BOT contracts. There are many potentially interesting extensions of this

model. In Section 4, we study two extensions. In one case, the firm in the BOT belongs to a

foreign country and so the after-tax profits for the foreign firm are not part of the domestic

welfare. We show that the first-best outcome is still achievable under a contract with price

regulation and licensing, supplemented by a lump-sum tax. In the other extension, a price

ceiling replaces strict price regulation in the concession period. Again, the first-best outcome

is also achievable.

Another possible extension is to consider the length of the concession period as a con-

tracted variable. When a BOT contract specifies a longer duration for the concession period,

the firm’s marginal benefit from its quality investment will increase, and so it will improve

the quality of the project. Because of this, with price regulation, an optimal BOT contract

should extend the concession period; but, without price regulation, an optimal BOT contract

should shorten the concession period.

Page 16 of 21

We could also specify a minimum level of quality at the end of the concession period,

provided that such a minimum level is verifiable. One feature of BOT contracts is that the

quality of a project may deteriorate over time. If a proper penalty is introduced and some

risk in achieving a targeted level of quality is involved, a minimum quality requirement may

force the firm to invest sufficiently in quality.

Appendix

Proof of Lemma 1

By Taylor’s expansion,

2

[ ( )] [ ( )]1[ ( )][ ( ) ( )] ( )[ ( ) ( )]2

[ ( )][ ( ) ( )] (since ( ) 0)[ ( ) ( )] (by (2))

B B B

B B B B B B B B

B B B B B

B B B B

c x p q c x p q

c x p q x p q x p q c x p q x p q

c x p q x p q x p q cp x p q x p q

ζ

ζ

, − ,

′ ′′= , , − , + , − ,

′ ′′≥ , , − , ≥

= , − , ,

where ζ is a value between q and .Bq Then,

( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 0.B B B B B B B B B B Bp q p q p x p q x p q p x p q x p qπ π ⎡ ⎤ ⎡ ⎤− ≥ − + − =⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Proof of Proposition 1

Our strategy is to find a solution that achieves the first-best outcome. Such a solution

can be found through three steps. First, let 2 2p p′ = . Then, the government’s problem be-

comes

1 21 1 2 2( )

1 2 1 2

max ( ) ( ) ( ) ( ) ( )

s t IC ( , , ) ( , , ) for allp p q

W s p q p q s p q p q k q

p p q p p q q Qφ

π π∗

, , , ⋅= , + , + , + , −

. . : Π ≥ Π ∈ . (20)

Second, since ( )φ ⋅ does not appear in the above objective function, the IC condition in

(20) can be ignored for the moment when we focus on choosing the quality and prices. By

the structure of the objective function, we must have 1 2p p= for the optimal prices. Let

1 2p p p≡ = . Then, the problem is reduced to finding the optimal ( )p q∗ ∗, that satisfy

[ ]max 2 ( ) ( ) ( )p q

W s p q p q k qπ∗

,= , + , − .

This optimization problem is identical to the first-best problem given in (1). Hence, the first-

order conditions (FOCs) (2)–(3) hold, and therefore Bp p∗ = and Bq q∗ = .

Page 17 of 21

Finally, we can show that, given ( )φ∗ ⋅ as defined in (6), the firm’s choice of quality based

on the IC condition in (20) is Bq . With the contract ( )Bp φ∗, ⋅ , the firm’s profit becomes

( , , ) ( ) ( ) ( ) ( )B B B Bp p q p q q p q k qπ φ π∗Π = , + , − .

Then, for Bq q≤ , invoking Lemma 1 and (6) yields

( , , ) ( , , )B B B B Bp p q p p qΠ − Π

[1 ( )] ( ) [1 ( )] ( ) ( ) ( )B B B B Bq p q q p q k q k qφ π φ π∗ ∗= + , − + , − +

[1 ( )] ( ) [1 ( )] ( ) ( ) ( )B B B B B Bq p q q p q k q k qφ π φ π∗ ∗≥ + , − + , − +

( )[ ( ) ( )] [ ( ) ( )]B B B Bp q q q k q k qπ φ φ∗ ∗= , − − − (21)

( )[ ( ) ( )] ( ) [ ( ) ( )]B B B B B Bp q k q k q p q k q k qπ π= , − / , − − (22)

0,=

where we have applied the definition of )qφ∗( in (6) to go from (21) to (22). For Bq q> ,

since ( ) ( )B B Bp q p qπ π, ≤ , by Lemma 1, the above derivation still holds up to (21). That is,

( , , ) ( , , ) ( )[ ( ) ( )] [ ( ) ( )]B B B B B B B B Bp p q p p q p q q q k q k qπ φ φ∗ ∗Π − Π ≥ , − − − .

Again, using the definition of )qφ∗( in (6) for Bq q> , ( ) ( )Bq qφ φ∗ ∗= , implying that

( , , ) ( , , ) ( ) ( ) 0B B B B B Bp p q p p q k q k qΠ − Π ≥ − > .

That is, given ( )φ∗ ⋅ in (6) and price Bp , the firm will voluntarily choose Bq as its optimal

quality. Hence, Ω defined in (6) constitutes a solution to (20) and W ∗ reaches the first-best

welfare level, .BW

Proof of Proposition 2

A simple version of the problem is18

[ ] [ ]

[ ]

22 2( )

ˆ max [1 ( )] ( ) ( ) [1 ( )] ( ) ( ) ( )

s t ( ) 0

1 ( ) ( ) ( ) ( ) ( )

p q p

p

q

W q s p q p q q s p q p q k q

p q

q p q q p q k q

φφ π φ π

π

φ π φ π

, , , ⋅= + , + , + − , + , −

. . , = ,

′ ′+ , + , = .

(23)

This problem can be solved in two steps. First, consider the following problem:

18 A more sophisticated version involves second-order conditions for the IC conditions and the proof will be

much more complicated. This proof here has illustrated the key element of the more completed proof. Actually, in

the more completed proof, we need to control under-investment by [ ]1 ( ) ( ) ( ) ( ) ( )qq p q q p q k qφ π φ π−′ ′+ , + , ≥

at q only; it is easy to control over-investment by a convex ( ).k ⋅

Page 18 of 21

[ ] [ ]2

2 2,ˆ max (1 ) ( ) ( ) (1 ) ( ) ( ) ( )

s t ( ) 0

(1 ) ( ) ( ) ( ).

p q p r

p

q

W r s p q p q r s p q p q k q

p q

r p q p q k q

δπ π

π

π δπ

, , ,= + , + , + − , + , −

. . , = ,

′+ , + , =

Second, given a solution 2ˆˆ ˆ ˆ ˆ( , , , , )p p q r d from the above problem, find a piecewise differenti-

able function : [0, 1]φ + →R such that ˆ ˆ( )q rφ = and ˆˆ( ) .q dφ′ =

We must have ˆ 0.r = If ˆ 0,r > we can take an arbitrary 0 ˆ(0, )r r∈ and easily design a

piecewise differentiable function 0 : [0, 1]φ + →R such that 0 0ˆ( )q rφ = and

00

ˆ ˆ ˆ( ) (1 ) ( )ˆ( ) .

ˆ ˆ( )qk q r p q

qp q

πφ

π

′ − + ,′ =

, Since social welfare 2 2ˆ ˆ ˆ ˆ( ) ( )s p q p qπ, + , under the govern-

ment’s control is higher than that ˆ ˆ ˆ ˆ( ) ( )s p q p qπ, + , under the firm’s control, the total social

welfare is increased when r is reduced from r to 0r while all other variables have been re-

tained. This is a contradiction. Hence, we must have ˆ 0r = or ˆ ˆ( ) 0.qφ =

Hence, the government’s optimization problem (23) becomes

2

2 2ˆ max ( ) ( ) ( ) ( ) ( )

s t ( ) 0

( ) ( )

p p q

p

q

W s p q p q s p q p q k q

p q

p q k q

π π

π

π

′, ,′ ′= , + , + , + , −

. . , = ,

′, = .

In the above problem, the two IC conditions determine ( )p q, . Also, since 2p′ does not affect

the IC conditions, we immediately find the FOC for 2p′ : 2 2[ ( )]p c x p q′ ′ ′= , . This completes

the proof.

Proof of Proposition 3

Since the lump-sum tax does not affect the moral hazard problem, we ignore it first and

then take it into account at the end. Our strategy is to construct a mechanism such that the

first-best outcome is obtained.

We first impose 1 1= Fp p and 2 2 2= = Fp p p′ . Then, the second-best problem becomes

1 1 2 2, ( )

2 2

1 2 1 2

max ( , ) [ ( , ) ( )] ( )[ ( , ) ( , )]

(1 ( ))[ ( , ) ( , )]

s.t. IC: ( , , ) ( , , ) for all ,

F F F F

q

F F

F F F F

s p q t p q k q q s p q t p q

q s p q p q

p p q p p q q Q

φπ φ π

φ π

⋅

+ +

+ − + +

+ − +

Π ≥ Π ∈

where the firm’s expected profit is given as

1 2 1 2( , , ) (1 )[ ( , ) ( ) ( , ) ( )].F F F Fp p q t p q q p q k qπ φ π+Π ≡ − + −

Following the same proof of Lemma 1, we can easily find:

Page 19 of 21

2 2( , ) ( , ) for all .F F Fp q p q q Qπ π≥ ∈

With a slight change in the proof, we also get the following strict inequality in the first period:

1 1( , ) > ( , ) for all .F F Fp q p q q Qπ π ∈

With the above preparation, then, following the proof of Proposition 1, we can show that the

IC constraint holds under 1 2= ( , , , ( )).F F F Fp p q φΩ ⋅

We now turn to the lump sum tax. For the τ∗ defined in (13), we have

1 1 2 2

2 2 2

1 1 2 2

= ( , ) ( ( , ) ( )) ( )[ ( , ) ( , )]

(1 ( ))[ ( , ) ( , )] (1 ) ( ) ( , )

= ( , ) ( ( , ) ( )) ( , ) ( , )

= .

F F F F F F F F F F

F F F F F F F F F F

F F F F F F F F

F

W s p q t p q k q q s p q t p qq s p q p q t q p q

s p q t p q k q s p q p qW

π φ π

φ π φ π

π π

∗ + − + +

+ − + + −

+ − + +

That is, the second-best welfare is also equal to the first-best welfare. Finally, we ensure the

participation constraint:

1 2 1 2

1 2 2

1

( , , , ) ( , , )

= (1 )[ ( , ) ( ) ( , ) ( )] (1 ) ( ) ( , )

= (1 )[ ( , ) ( )] > 0.

F F F F F F

F F F F F F F F F F F

F F F

p p q p p qt p q q p q k q t q p qt p q k q

τ τ

π φ π φ π

π

∗ + ∗Π = Π −

− + − − −

− −

Proof of Proposition 4

Our strategy is to find a solution that achieves the first-best outcome. Since the first-

price must hit the ceiling, the government’s problem is

21 1 2 2

1 2 1 2

max ( ) ( ) ( ) ( ) ( )

s.t. ( , ) ( , ), .p q

W s p q p q s p q p q k q

p p q p p q q Q

π π∗

,≡ , + , + , + , −

′ ′Π , ≥ Π , ∈ (24)

This problem can be separated into two problems. The first problem is the same as (14); the

second problem is to show that the license function in (6) ensures

1 1( , ) ( , ), .B B Bp p q p p q q QΠ , ≥ Π , ∈ (25)

With the contract 1 , ( )Bp p φ∗, ⋅ , the firm’s profit is

1 1( , , ) ( ) ( ) ( ) ( )B Bp p q p q q p q k qπ φ π∗Π = , + , − .

Then, for Bq q≤ , using (19) and the fact that ( , ) 0q p qπ ≥ yields

Page 20 of 21

( ) ( )( )

( )

1 1

1 1 1

( , , ) ( , , )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( )[ ( ) ( )] [ ( ) ( )]

( )[ ( ) ( )] ( ) [ ( ) ( )]0,

B B B

B B B B B

B B B

B B B B B

B B B B B B

p p q p p q

p c x p q x p q q p c x p q

q p c x p q k q k q

p c x p q q q k q k q

p q k q k q p q k q k q

φ

φ

φ φ

π π

∗

∗

∗ ∗

Π − Π

⎡ ⎤= − , − , + − ,⎣ ⎦− − , − +

≥ − , − − −

= , − / , − −

=

which implies (25) for .Bq q≤

For ,Bq q> given the license function in (19), ( )qφ is constant: 0( )qφ φ= for

,Bq q≥ where 0 (0, 1).φ ∈ If q maximizes the firm’s profit and ˆ ,Bq q≥ then it is deter-

mined by

1 0ˆ ˆ ˆ( ) ( ) ( ).Bq qp q p q k qπ φ π ′, + , =

For any 0,q ≥ denote the social benefit and the company’s revenue, respectively, as

( ) ( )

1 2 1 1 2 2

1 2 1 0 2 1 1 0 2 2

( , , ) ( ) ( ) ( ) ( ),( , , ) ( , ) ( , ) ( , ) ( , ).

B p p q s p q p q s p q p qR p p q p q p q p c x p q p c x p q

π π

π φ π φ

≡ , + , + , + ,

≡ + = − + −

Then, q and Bq are defined, respectively, by

1 1ˆ ˆ( , , ) ( ), ( , , ) ( ).B B B Bq qR p p q k q B p p q k q′ ′= =

Since ( , ) 0qs p q ≥ and ( , ) 0,q p qπ ≥ we have

1 2 1 2 1 2( , , ) ( , , ), for all ( , , ) 0.q qR p p q B p p q p p q≤ ≥

Since ( , )x p q is concave in ,q 1( , , )BR p p q is concave in .q As shown in the following figure,

since the curve of qB is always above the curve of qR and the latter curve is decreasing and

the curve of ( )k q′ is increasing, we must have ˆ.Bq q≥ This means that the profit function

defined in (17) cannot possibly achieve its maximum in [ , ).Bq + ∞

1( , , )BqB p p q

1( , , )BqR p p q

q Bq q

( )k q′

Hence, condition (25) is satisfied, given the contract in (19).

Page 21 of 21

References

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