Effects of market-based measures on a shipping company: Using an optimal control approach for long-term modeling

Research in Transportation Economics 73 (2019) 63–71

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Research in Transportation Economics journal homepage: www.elsevier.com/locate/retrec

Research paper

Effects of market-based measures on a shipping company: Using an optimal control approach for long-term modeling

T

Hidemi Tanakaa, Akira Okadab,∗ a b

Atomi University, 1-5-2 Otsuka, Bunkyo-ku, Tokyo, 112-8687, Japan Tokyo City University, 3-3-1 Ushikubo-nishi, Tsuzuki-ku, Yokohama, Kanagawa, 224-8551, Japan

ARTICLE INFO

ABSTRACT

JEL classification: R40 Q48 Q54 Q58 C61

Market-based measures for international shipping, which have been discussed by the Marine Environment Protection Committee, provide shipping companies with incentives to reduce carbon dioxide (CO2) emissions and change long-term behavior for ships, especially investment in capacity and fuel efficiency. This study presents a fundamental theoretical model to deal with the issues before applying real data. Specifically, since the literature has focused less on these issues, we develop an optimal control model to analyze the investment behavior of a shipping company, improvement of fuel efficiency, and CO2 emissions from ship operations by introducing a tax on them. The results of analysis show the classification of optimal conditions, the relationships between fuel price with emission tax and investment behavior, and the rebound effect of improving fuel efficiency of the tax. A policy mix, which consists of the tax and a direct subsidy, is deemed optimal for an international shipping company.

Keywords: Optimal control model Market-based measures Capital investment Fuel efficiency CO2 emissions

1. Introduction 1.1. Background International shipping is one of the major sectors contributing to global emissions of carbon dioxide (CO2), which is a greenhouse gas (GHG). On average, it emitted approximately 3.1 percent of the total global CO2 emissions per year during 2007–2012, according to the International Maritime Organization (IMO) (2009, 2014). Specifically, the IMO (2009, 2014) estimates the international shipping industry emitted 468 million tons (in CO2 equivalent) in 1990, 850 million tons in 2011, and 796 million tons in 2012. Based on predefined projection scenarios, the IMO predicted a continuous increase in total CO2 emissions from international shipping until 2050. The IMO (2014) projections indicate the CO2 emissions from the shipping sector will be approximately 3.5 times higher in 2050 than in 2012 as per the highest projection in the business-as-usual scenario (scenario no. 13). IMO member countries have discussed several measures to induce shipping companies to reduce their CO2 emissions. Such arguments include, for example, the Energy-Efficiency Design Index (EEDI) for regulating ship energy efficiency, the Ship Energy Efficiency Management Plan for managing shipping operations, and the Monitoring, Reporting, and Verification for correctly interpreting CO2 ∗

emissions from ships. Additionally, market-based measures (MBM) have been discussed in the Marine Environment Protection Committee (MEPC). Their discussion was initiated at MEPC 56 in 2006. However, they have not progressed since MEPC 65 in 2013. 1.2. Problem overview and literature review A shipping company confronted with MBM will have several options: decrease fuel consumption to improve ship fuel efficiency, use renewable energy, change on-board energy, increase investment to extend the load capacity for integrating more freight by ship, or alternatively shrink business. If these company behavioral options related to investment and fuel savings are not considered, it is difficult to analyze the effects of MBM on the reduction of CO2 emissions by a shipping company. Furthermore, ships are durable capital goods and the subject of long-term investment—their useful lifespans are more than twenty years. This study aims to present a fundamental theoretical model based on an optimal control approach to analyze the impact of MBM on company behaviors. Our research focuses on the development of this model and does not apply real industry data. We first review the literature on the introduction of MBM in the maritime sector. Although renewable and alternative energy are considered energy sources for ships (Brynolf, Baldi, & Johnson, 2016a, pp.

Corresponding author. E-mail addresses: [email protected] (H. Tanaka), [email protected] (A. Okada).

https://doi.org/10.1016/j.retrec.2019.01.006 Received 22 September 2017; Received in revised form 12 December 2018; Accepted 7 January 2019 Available online 27 February 2019 0739-8859/ © 2019 Elsevier Ltd. All rights reserved.

Research in Transportation Economics 73 (2019) 63–71

H. Tanaka and A. Okada

295–339; 2016b, pp. 399–418), almost all energy sources of ships consist of three types of energy—heavy fuel oil, marine gas oil, or marine gas (IMO, 2014). Hereafter, fossil fuel is assumed to be the main energy source of ships in this study. The literature on the effects of MBM on shipping companies mostly employs analysis based on short-term models. However, these models cannot analyze how much a shipping company invests in capacity and how much effort it expends to improve fuel efficiency, that is, reduce CO2 emissions. For instance, Corbett, Wang, and Winebrake (2009) and Kapetanis et al. (2014) analyze the impacts of fuel taxation on ship speed and CO2 emissions based on an annual profit model. Moreover, the impact of an Emission Trading System (ETS) and Maritime Emission Trading System on carrier outputs, choice of operational and technological measures, and fuel consumption on shipping markets are analyzed by Wang, Fu, and Luo (2015) and Huang, Shi, Wu, and zhao (2015) using short-term models. Although the short-term models in these articles incorporate the discounted costs of technologies for ships as a means to overcome the gap between the long and short terms, they do not focus on investment in capacity. As ships are durable goods requiring long-term investment, as mentioned above, an optimal control model that includes the time aspect as opposed to the static model that discards the time aspect is preferable to analyze the effects of MBM on a shipping company. The IMO (2010) model may be a long-term simulation model based on growth rate. However, the model only calculates the effects of MBM on the environment, financial burden for the shipping industry, and cost of reductions (USD per ton CO2 abated) for the evaluation years and cannot clarify the impact of MBM on the investment and purchase of ships. Additionally, the model is also vague in terms of the relationship between investment and efforts to improve fuel efficiency and the rebound effect of improving fuel efficiency induced by MBM and the EEDI policy. On the other hand, Kosmas and Acciaro (2017) first analyze the economic and environmental effects of bunker levies on the shipping market, modeled by the Cobweb theorem. Specifically, they investigate an optimal speed under bunker levies, their impact on a change in ship speed on the market, fuel consumption, and profit and levy cost allocation between ship-owners (or operators) and shipper. Their model used the Cobweb theorem to consider the dynamic aspects of the international freight market. However, they did not examine an effect of bunker levies on the capital investment of shipping companies, since their profit function did not contain any variables on capital investment. Additionally, their model on fuel consumption did not capture the improvement in fuel efficiency through an effort towards the continuous introduction of various technologies by a shipping company. As seen above, the long-term impacts of MBM on investment in capacity and effort to improve fuel efficiency have not been discussed simultaneously.

In other words, our study analyzes the impact of emission tax on capital investment, effort on fuel efficiency, and CO2 emissions of a shipping company based on an optimal control model that is an application of Hartl and Kort (1996a, b) and Kort (1995). To this end, we develop an optimal control model, which includes investment in ships, efforts for improving fuel efficiency, and voyage distance of cargo as variables to clarify the long-term impacts on these variables. This dynamic model derives the conditions for optimal solutions and improvement in fuel efficiency considering the long-term aspect. We attempt to use computer simulation for numerical analysis because the model cannot be solved analytically. The results enable us to understand the movement of the equilibrium and several variables by changing MBM intensity. The results of these analyses can also provide the policy implications of MBM for the international shipping sector in the long term. The remainder of this paper is organized as follows. Section 2 explains the fundamental structure of the optimal control model and solves it. We also identify several implications of solving the model and investigate four classified cases derived from the solution process of the model. Section 3 calibrates the model to study the dynamic relationships between capital and the efforts to improve fuel efficiency and between the level of emission tax and volume of CO2 emissions. Section 4 discusses the implications of the model solution and the results of the computer simulation. Policy implications based on the discussion are also provided here. Finally, Section 5 concludes the paper. 2. Analytical framework 2.1. Model structure Let us assume the revenue of a shipping company is described as an increasing function of capital (or cargo capacity) and voyage distance.1 The ships owned by a shipping firm are regarded as the capital owned by a regular business firm. The shipping capital, or number of ships, is identified as the capacity for cargo K. Therefore, the capital (cargo capacity) increases with productive investments and revenue increases with the capital. Furthermore, expansion of total distance for the firm also increases the revenue but requires more fuel and raises emissions. The shipping companies are forced to pay tax for a quantity of emissions.2 To avoid these taxes, the companies make effort A to improve fuel efficiency or reduce the amount of fuel consumed.3 Therefore, the 1 The productivity of a ship is presented by, for example, Stopford (2009) as ton-miles or ton-kilometers of cargo per annum, that is, tonnage multiplied by the voyage distance of transported goods. As such, the same elements influence the profit of the shipping company. We deal with tonnage and distance as independent variables and not as totalized ton-miles. The meaning of distance includes not only changes in the voyage distance of the ship by changing the destination, but also the company's total voyage distance by changing the frequency and number of ships without changing the voyage distance. Therefore, we can flexibly express the plural long-term behaviors of shipping companies. 2 We mainly consider the effects of the emission tax. Specifically, we analyze the effects of fuel price, including the tax on emissions. When considering emissions trade, the cost and method of implementation are significantly different for a policy measure, although the prices of emission permits and emission tax are theoretically equal. See Kort (1995), Hartl and Kort (1996c), and Xepapadeas (1997) for details. 3 We use the concept of effort to decrease environmental emissions of Hartl and Kort (1996a, b). They define emissions as the production function of capital multiplied by a parameter, whereas in this paper, emissions arise from fuel consumption, thus the environmental effort is not explicitly linked to production and capital. For shipping, Eide et al. (2011) consider potentially realizable efforts as follows: operational measures such as slow steaming (existing relationship between speed and fuel consumption as the cubic law (Corbett et al., 2009; Lindstad, Asbjornslett, & Stromman, 2011; Wang et al., 2015)) and cleaning and maintenance of hulls and engines; technical options such as optimizing hulls and engines, and design changes; alternative fuels; and structural

1.3. Research purpose Although MBM, which has been discussed in MEPC, would impact the optimal speed, fuel consumption, profit and financial burden of shipping companies (IMO, 2010; Kosmas & Acciaro, 2017), the literature has focused less on the behaviors of shipping companies, especially investment in the capacity and fuel efficiency of ships from a long-term viewpoint. Therefore, we analyze these issues using a general and environment-economic optimal control model, which is different from the above maritime-specific models. Concretely, we use the models of Hartl and Kort (1996a, b) and Kort (1995), which explain the effects of the introduction of emission tax, emission standards, and emissions trading from a general dynamic perspective. Since they focus on the efforts towards pollution abatement by an individual company, their models are appropriate for analyzing the impact of MBM on company behavior, that is, a company's investment and efforts to improve fuel efficiency and the influence of CO2 emissions. 64

Research in Transportation Economics 73 (2019) 63–71

H. Tanaka and A. Okada

instantaneous cash flow for a shipping firm is defined as4

R (x , K )

rZ(A, x )

C(I )

C(A)

T(Z(A, x ) )

value of its cash flow over an infinite planning horizon is given as (1)

Max

where R is the revenue of the firm, which increases with distance x and capital K, and r the fuel price per unit. It is assumed the amount of fuel consumption Z decreases due to effort A to improve fuel efficiency or reduce the amount of fuel used, but increases with voyage distance x. It is also assumed that cost functions of productive investment I and the efforts of improving A are defined as different forms C (I) and C (A), respectively, and tax of emission T can increase with fuel amount Z. The investment cost C (I) can be considered as increasing.5 The investment cost comprises acquirement and internal adjustment costs. The latter include installation costs, worker training, and stresses imposed on the managerial and administrative capabilities of existing staff. On the other hand, the cost of effort A to improve fuel efficiency is defined as C (A) = A , where is the marginal cost parameter of the improved effort. Fuel efficiency f and emission e from consuming fuel are assumed as follows:

x /Z

f(A) = (f1

e (Z ) = Z =

f0 )(1 x

f(A)

=

(f1

f0 )(1

. e gA ) + f0

aK

bx

s. t .

K=I

2R

x2

(x , K )

rZ(A, x )

C(I )

aK

bx ,

0,

I

C(A) A

0,

T(Z(A,x ) ) ] dt , and x

0.

(4)

x+

2R p f R K+ + i + a = (i + a ) x K f f x

b

R K

(5)

This equation shows that the change in revenue is decomposed into change in voyage distance and capital, change rate of fuel efficiency, p pZ and interest rate i. In this equation, rewriting f = x , can be considered the average fuel cost per unit distance. Further, using f = (f1 f0 ) ge gAA , we obtain

(2)

Fuel efficiency f is defined as voyage distance x divided by the amount of fuel consumption Z as usual. Additionally, it is assumed that f is an increasing function of efforts A to improve fuel efficiency, and these improvement efforts increase fuel efficiency from f0 at the beginning to f1 at the end, where g and are coefficients. In other words, f0 is the fuel efficiency without effort and f1 is possibly the desired target fuel efficiency achieved through infinite efforts. Of course, in real business situations, since improvement methods depend on the wisdom and technologies of firms, this change may be more complex. The capital of shipping firm K increases with productive investment I and decreases due not only to spending time as usual, but also to voyage distance, whose parameters are a, b, respectively, so that the state equation of the capital can be presented as follows6:

K =I

it [R

0

Overviewing the structure of the model, the firm determines Z (amount of fuel consumption), x (voyage distance), and I (capital investment). Whereas Z is the function of A (effort to improve fuel efficiency) and x, the firm has A, x, and I as independent variables of each function. Solving the optimal problem, we seek optimal conditions that hold among the three independent variables. Finally, we consider the T effect of p = r + Z (fuel price including the charge of emission, that is, the political variable) on these variables. It is assumed that these functions and variables are continuous, differentiable, and regular analytic in the domain. Solving the problem yields the following equation (we omit the calculations for brevity):

Exp [ gA]) + f0 , x

e

x, I ,A

f = f (f1

(f1

f0 ) ge

f0 )(1

where f2 =

f0 f1

f0

gA

e

gA )

=

( )

+ f0

f1

A=

f0

f0

1

gA (1 + f2 ) e gA

A 1A

(6)

satisfies f1 > f0 > 0 and f2 > 0, which

is the reciprocal of the potential improvement rate of fuel efficiency, and A is the independent variable. When it is close to zero, fuel efficiency has a large possibility to improve, but when it is close to infinity, there is no possibility, despite infinite efforts. 2.2. Solutions for the case I > 0 and A > 0

(3)

The model can be analyzed for the following four cases, i) I > 0, A > 0; ii) I > 0, A = 0; iii) I = 0, A > 0; and iv) I = 0, A = 0. The differences between cases depend on the magnitude relation of each parameter. Therefore, they are different in each case. Considering case i), the optimal conditions are written as follows (we omit the calculations for sake of brevity, and the same applies hereafter):

Further, considering the non-negativity condition I ≥ 0, A ≥ 0, and x ≥ 0, the problem faced by the shipping firm to maximize the present (footnote continued) changes such as improving contract and logistic planning. We define all activities to comprehensively improve fuel efficiency as "efforts" so as not to discriminate between these. By discriminating between factors A1, A2, … in the model, we will be able to analyze the properties of the efforts in more detail. 4 There are liners and trampers in shipping transport. Liner transport markets can be considered imperfect competitive markets due to the existence of the Shipping Conference and mega container-shipping mergers and acquisitions and global alliances, although there are also independent actions and service contracts in the U.S. shipping act that encourage competition. On the other hand, the tramper transport markets can be said to be a free competitive market between consignors and tramper companies (Girgin, Karlis, & Nguyen, 2018; Stopford, 2009). Although the structure is different for both markets, we analyze the behavior of shipping companies in partial equilibrium under given freight rates, but do not focus on problems due to market structure difference related to such freight rate negotiation between shippers and shipping companies. 5 This assumption arises from the internal cost of adjustment. Hartl and Kort (1996a) state that these examples are preventing the production line from installing new machines or the stresses imposed upon the managerial and administrative capabilities of existing staff. Additionally, this assumption is well known in studies such as that of Xepapadeas (1997, p.89). This concept also seems to apply to a ship's investment case. 6 Despite K, A, x, and I being functions of time and subscript "t" ideally being indicated, it has been omitted to improve legibility. Moreover, it is assumed that the capital of a shipping company diminishes by time spent and voyage distance, while a variety of other factors actually influence capital.

bCI = b =

C , and I

= CA = p = As tance;

p , f (A )

R (x , K ) x

where CI

is a co-state variable,

f g x f e gA = px 2 f2 A f0 [(1 + f2 ) e gA

1]2

(7-1)

,

where CA

C , and A and x are independent variables A p stated above, f f pZ x and p 2 A = x f

=

pZ x f A

(7-2)

means the average fuel cost per unit dis-

(Z ) means the average fuel cost multiplied by

the change in improved fuel efficiency and equals the monetary amount of fuel savings. Therefore, from the above equations, the optimal solutions prove that the marginal cost of investment equals the change in revenue per unit distance minus average fuel cost, and the marginal cost of efforts to improve the fuel efficiency equals the monetary amount of fuel savings. Their equations can thus be rewritten as

R (x , K ) C 1 f =b + x I x f

A

65

C 1 (1 + f2 ) e gA = bCI + A x g

1

CA

(8)

Research in Transportation Economics 73 (2019) 63–71

H. Tanaka and A. Okada

Therefore, for the optimal solution, the marginal revenue of distance equals the marginal cost of investment plus the marginal cost of efforts to improve fuel efficiency. Equations (7-2) transforms into

x (A) =

[(f1 CA f 2 = p f p

f0 )(1 (f1

A

=

f0

(1 + f2

e

gA)

f0 ) ge

1)

f p x K + + i+a+ x K f f

x =i+a R

x K

b

(13)

Substituting (3), (6), and (10) into (13) yields

+ f0 ]2

A = A

gA

gA )2e gA

e

(

(

e gA

f

p (a + i ) f2

a +a+i

0 (1 + f2 ) e

(1 + f2 ) e gA + 1

1)

(1 + f2 ) e gA

1

gA

x (A)

I(A, K )

1 R(x (A) , K ) f

gA + p f2

K

e gA

0 (1 + f2 ) e

gA

+

bx (A) K

(1

1

)

gA 1

(1 + f2 ) e gA

1

(9)

(14)

which is the relational equation of x and A, and it should be essentially expressed as A=A(x). However, it is impossible for the equation to be represented as an explicit function. It is therefore expressed in the inverse function form, as above. From (9), the relationship between A and x can be clearly characterized in the optimal solution. Considering f0 = f1 f0 , optimal voyage distance7 increases with the possibility of f2 improving the fuel efficiency (or decreasing f2). Further, the derivative with respect to the efforts of improving A yields

where I(A, K) is the function presented in (12) and x(A) in (9), and A and K are the independent variables of the equation. Further, the relationship between the variables in (3) can be described as follows:

f2 pg

x 2 f = A f A

1 f A

2f 2ge x= A2 1 + f2

gA

e

gA

K = I(A, K )

=

1

2f

f A

A2

A = 2

g (1 + f2 ) e gA

e (Z ) = e A

+g x>0

1

2bcI(A, K ) =

p

f0 p = q (1 + f2 f f2 pg f2 f0 (1 + f2

e

gA )

=

pg

[(1 + f2 ) e gA

1] ,

= p [(1 + f2 ) e gA] g > 0. (A is the independent variable)

(10)

bCI = b =

CA = p

(11)

x f f02 A

R (x , K ) x

+

p f0 A

A =0

>

(17-1)

p xg . (x is the independent variable) f0 f2

gA) 2e gA

(17-2)

Equation (17-2) shows that, when the marginal cost of the efforts to improve fuel efficiency is large, no improvement effort is optimal. In case iii), using λI as the Lagrange multipliers for I to solve this optimal problem, we obtain,

bCI = b + b

I

>b =

R (x , K ) x

p f (A )

1

e

(16)

The rest of the cases are ii) I > 0, A = 0; iii) I = 0, A > 0; and iv) I = 0, A = 0. The optimal solutions include the corner solution in all three cases. In case ii), using λA as the Lagrange multipliers for A to solve this optimal problem, we obtain

where q, , , c, are constant. Each of q, , is a parameter, presented as the revenue.8 Additionally, c and represent the marginal cost parameters of investment and of improving effort, respectively. Then, using (9) and (11), transforming (7-1) provides the following equation:

R x

x f

2.3. Solution of the extra cases

Obviously, the rate of change in the optimal distance is proportional to that of the efforts to improve fuel efficiency. Henceforth, the revenue of firm R(x, K) and cost functions C(I) and C(A) are specified as follows:

R (x , K ) = qx K , C (I ) = cI 2, C (A) = A

(15)

Here, e is inversely proportional to fuel price p, including the emission tax, and is directly proportional to the marginal cost of improving fuel efficiency C (A) = v . The first order derivative of (16) with respect to A is positive, implying that emissions increase with efforts to improve fuel efficiency, as this situation could be responsible for increasing the optimal distance in (9).

+g A

(1 + f2 ) e gA + 1 A gA A (1 + f2 ) e gA 1

bx (A) . (A and K are independent variables)

The structure of the model is described as a system comprising (14) and (15). Further, the quantity of emissions from consuming fuel is

where A is the independent variable. Therefore, the optimal distance increases with the efforts of improving A, but decreases with fuel price p, including the tax on emissions. Further, the marginal cost of improving fuel efficiency C (A) = v raises optimal distance, and can be interpreted as the firm desiring to increase distance to recover spending cost and raise revenue. Further, differentiating (9) with respect to time yields

f x (A ) =2 x (A ) f

aK

K

CA = p

f g x f = px 2 f2 A f0 [(1 + f2 ) e gA e gA

1]2

.

(A and x are independent variables)

(12)

(18-1)

(18-2)

Equation (18-1) shows that, when the marginal investment cost is large, no investment is optimal. In case iv), we similarly obtain (18-1) and (23-2). The following equation also satisfies the above mentioned three cases:

As shown in (12), the relationship between optimal investment I (A, K) and improvement efforts A can be described as an explicit expression. Equation (5) is thus rewritten as follows: 7

The optimal distance in this model means minimizing the reduction in capital and optimizing fuel consumption. 8 Parameter q is constant and includes all factors influencing revenue, except for capacity and distance. Considering that revenue is ton-miles times the freight rate, parameter q could be regarded as the freight rate itself. Naturally, the revenue of firms is in reality determined by the complex contract between shipping firms and owners of goods and is not easily described by a simple function. However, the function of this model is used with some adaptability and versatility. This formulation is thus presented as a comprehending function with linear and power components.

R C 1 f
A

C A

In other words, marginal revenue is less than all marginal costs, comprising capital investment and efforts to improve fuel efficiency. In each case above, by examining whether equations (9), (10), (12) and (13) are satisfied, we evaluate the structure of the model. In case ii), (9) and (10) do not hold, and A = 0 in (2) and (6). Therefore, fuel efficiency f will not change over time, so there is no relationship 66

Research in Transportation Economics 73 (2019) 63–71

H. Tanaka and A. Okada

between x and A. From (17-1), the optimal investment is presented as follows, which could be considered a replacement for (12):

2bcI(x. K ) = qx

p f0

1K

of A = 0 in (29) and (30). Evidently, the value of the parameters is different in each case, meaning the obtained emission values are eventually different, but the equations have a similar expression.

(19)

Consequently, (13) transforms into

3. Dynamic behavior in the system

p x K x 1) + + (i + a) =i+a x K f0 R

(

b

x K

+

bx K

The previous section indicates there are four cases in the optimal solution, namely i) I > 0, A > 0; ii) I > 0, A = 0; iii) I = 0, A > 0; and iv) I = 0, A = 0. Here, we analyze only the first case of the interior solution I > 0, A > 0 as an interesting case from the economic perspective.9 The structure of the model is written as

(20)

Substituting (3) into it, we obtain

x = x

p (i f0

a +a+i

I(x , K )

x R(x , K )

+ a)

K

(1

1

)

(21)

1

where I(x, K) is the function presented in (19). x and K are the independent variables in the equation. Therefore, the relationship between the variables in (3) can be described as

K = I(x, K )

aK

A = A

x . (x is the independent variable) f0

(

1)

e gA

f

p (a + i) f2

a +a+i

0 (1 + f2 ) e

K = I(A, K )

(1 + f2 ) e gA + 1 (1 + f2 ) e gA

1

e gA

f2

gA + p f

0 (1 + f2 ) e

+

bx (A) K

(1

1

gA

gA

1

(1 + f2 ) e gA

1

) (24)

where x(A) is presented in (9), and A and K are independent variables. Therefore, the relationship between the variables in (3) can be described as

K =

aK

(25)

bx (A)

The structure of the model is described by a system comprising (24) and (25). Further, the emissions quantity from consuming fuel is

e (Z ) =

1 + f2 x = f pg e

e

gA

gA

(A is the independent variable)

(26)

Finally, in case iv), (12) does not hold, and I = 0 in (3). Further, (9) and (10) do not hold and A = 0 in (2) and (6). Therefore, fuel efficiency f will not change over time, and there is no relationship between x and A. As a result, (13) transforms into

(

1)

p x K x + + (i + a) =i+a x K f0 R

b

x K

(27)

Substituting (3) into it, we obtain

x = x

a +a+i

p (i f0

+ a)

x R(x , K )

+

bx K

(1

1

1

)

(28)

where x and K are the independent variables. The relationship between the variables in (3) can be described as

K =

aK

bx

(29)

The structure of the model is described by a system comprising (28) and (29). Further, the emissions quantity from consuming fuel is

e (Z ) =

x . (x is the independent variable) f0

0 (1 + f2 ) e

1)

(1 + f2 ) e gA + 1 (1 + f2 ) e gA

1

gA

x (A)

I(A, K )

1 R(x (A) , K ) f2

gA + p f

K

e gA

0 (1 + f2 ) e

gA

+

bx (A) K

(1

1

)

gA 1

(1 + f2 ) e gA

1

aK

bx (A)

(15-2)

where I(A, K) is the function presented in (12), and x(A) in (9), and A and K are independent variables. Although these differential equations could be solved with analytical methods that give an exact expression of the solution, the equations now under consideration are too difficult for this approach. Therefore, we need to rely on numerical analysis, even for the usual phase diagram analysis. Henceforth, we assign the following values to parameters: f1 = 50, q = 0.9, g = 0.0003, a = 0.6, b = 0.5, c = 0.1, v = 0.1, = 1.1, = 0.7, and i = 0.8.10 Hereafter, we draw the phase diagram while seeking the combination of (A, K) with A = 0 and K = 0. We check how the equilibrium solution is affected by the charge and initial efficiency of the firm, and we simulate the influence on efficiency improvement and environmental emission. In this calculation, we mainly use a computer algebra system. First, under a certain initial fuel efficiency (f0 = 8), Fig. 1 shows the effect on the equilibrium locus of changing price p = 100,120,140. The values of the axes are normalized as intersection point to A = K = 1 at p = 100. As per (2), f0 is the initial fuel efficiency without effort. In Fig. 1, the red line is the A = 0 isocline and the blue line the K = 0 isocline. Considering the change in the signs of A and K , the trajectory to the equilibrium is shown in the figure, and the equilibrium is a saddle point. In the figure, the equilibrium point shifts up and to the right and both capital K and efforts to improve A increase as fuel price p, including the tax on emissions, increases. It can be interpreted that, to avoid the increasing tax, the firm increases its effort to improve fuel efficiency. Therefore, as the increase in A expands optimal distance x, as shown in (9), and the revenue of the firm increases, capital will increase as well. Fig. 2 shows the effects of different fuel economies at the outset, with f0 = 12, 8, 6 (or 1.5, 1, 0.75 normalized in Fig. 1 case. This decrease means fuel efficiency worsens) under p = 140, a constant. The values of the axes are normalized as intersection point to A = K = 1 at f0 = 1.5 (normalized in Fig. 1). When fuel efficiency is initially worse—or, in other words, the value is smaller—the efforts necessary to improve it are larger, while, on the contrary, a better initial fuel efficiency requires less effort. Consequently, as the fuel price (including the tax on emission) increases, the firm attempts to improve fuel efficiency to avoid the

(23)

x 1 R (x (A) , K )

gA

e gA

f

p (a + i) f2

(14-2)

Next, in case iii), (12) does not hold and I = 0 in (3). However, (9) and (10) hold, thus (13) still remains. Substituting (3) into (13) results in

A = A

(

(22)

bx

The structure of the model is described by a system comprising (21) and (22). Further, the emissions quantity from consuming fuel is

e (Z ) =

a +a+i

(30)

9

The remainder of the corner solution cases ii), iii), and iv) should be investigated in other ways, considering the incentive for investment and efforts. 10 Although these parameters represent a special case for trying to substitute other values, they would represent the features of the model developed in this analysis.

Comparing these expressions for the emission in each case, it is possible to classify them into two different cases: i) and iii) in the case of A > 0 in (16) and (26), and ii) and iv) in the case 67

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Fig. 1. Effects of changing price under a specific initial fuel efficiency.

Fig. 4. Effects of changing price on fuel efficiency.

f/f0 = 1.36 at p = 160, and f/f0 = 1.47 at p = 180. Finally, to analyze the effects on emissions, the emission expression is rewritten as follows:

e (Z ) = e A

=

x f

=

1 x f A

e gA e gA

1 + f2 pg

x f f2 A

,

= p [(1 + f2 ) e gA] g > 0.

(16-2)

As stated above, emissions increase with effort A to improve fuel efficiency. Further differentiating the above equation of emission with respect to fuel price p (including tax of emission) yields

e =e p =e

Fig. 2. Effects of changing initial fuel efficiency (f0).

( e gA)( g ) A 1 A + +g p 1 + f2 e gA p p 1 + f2 1 + p 1 + f2 e

gA

g

A p

The sign of this expression depends on the magnitude of the relation between the first and second parts on the right-hand side of the equation: the former represents the direct influence of p (negative), and the latter the indirect influence of A (positive). Considering the above p

figures, an increasing p leads to an increasing A. As such, emissions e rise. However, emissions e are found to be inversely proportional to p. Therefore, the change in emissions finally depends on the magnitude of the relation between the direct influence of p and indirect influence that bypasses the effect of the changing effort A.11 Fig. 5 shows the total effect of the change p, similar to the previous two figures. The emissions shown in the figure are normalized as e = 1 at p = 100. The increasing p leads to a decreasing emission as e = 0.983 of the norm at p = 120, e = 0.973 at p = 140, and e = 0.967 at p = 180. Emission e decreases as the fuel price, including tax on emissions, p increases from p = 100 to p = 180. 4. Discussion and policy implications

Fig. 3. Effects of changing price on the efforts to improve fuel efficiency.

The analysis in the second section has some implications under optimal conditions: the marginal cost of investment equals the change in the revenue per unit of distance transported minus average fuel cost and the marginal cost of the efforts to improve fuel efficiency equals the monetary value of the fuel savings. When the marginal cost of efforts to improve fuel efficiency is large, it is optimal to expend no effort to improve fuel efficiency. Similarly, when the marginal investment cost is large, the choice not to invest is also optimal. It would be possible to

burden. Fig. 3 shows the effect of changing price p on the efforts to improve fuel efficiency A. Fig. 3 is drawn for case f0 in Fig. 1, with a discretely changing price from p = 100 to p = 180. The effort shown in the figure is normalized as A = 1 at p = 100. The increasing price p raises the necessary efforts almost proportionally. Therefore, these efforts improve fuel efficiency. Fig. 4 shows the effect of the changing price on fuel efficiency, similar to Fig. 3. The improving fuel efficiency f/f0 shown in Fig. 4 is normalized as f/ f0 = 1 at p = 100. The increasing p leads to increasing efforts and improved fuel efficiency as f/f0 = 1.13 at p = 120, f/f0 = 1.25 at p = 140,

11 According to the second expression, when the improving fuel efficiency parameter g is small and the potential impossibility of improving f2 is small as well (i.e., possibility is high), the overall sign is negative.

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the effort to improve fuel efficiency. When the initial fuel efficiency is worse, or the value is smaller, the necessary improvement effort is larger. Conversely, when the initial fuel efficiency is better, or the value is larger, the necessary improvement effort is smaller. This indicates that better initial fuel efficiency requires less effort. As the IMO has already introduced the EEDI regulation for fuel efficiency, it is possible for shipping firms to achieve higher fuel efficiency in the future. If the IMO introduces the additional regulation of MBM to further reduce CO2 emissions, the low MBM price could not induce an effort towards fuel efficiency. As a result, companies that have originally achieved this improvement make fewer efforts and those that have not improved need to make greater improvement efforts. The analysis describes the effect of MBM on CO2 emissions. In the expression of emissions (16), emission e is inversely proportional to price p. That is, the rising price p leads to a decrease in emissions, which is the direct effect. Conversely, it is positive to the first order differentiation on the expression of emission e with respect to effort A. This means that fuel consumption and emissions increase despite improvement efforts. The increasing p raises the necessary efforts almost proportionally. It is possible that the improving fuel efficiency due to effort A will lead to expanding the cargo and distance and, thus, emission e increases, which is the indirect effect or so-called rebound effect of the improvement in fuel efficiency. Therefore, improving fuel efficiency expands voyage distance and impacts the reduction of CO2 emissions. Consequently, the change in emissions ultimately depends on the magnitude of the relation between the direct influence of p and the indirect influence that bypasses the effort to change A, that is, the rebound effect. Overall, emission e finally decreases as the fuel price, including tax of emission, p increases. However, comparing Figs. 4 and 5, the level of the emission reduction is gradually smaller, although the price increases the improvement in fuel efficiency. It seems that the rebound effect is not small. Setting the tax above a certain level could have a less significant effect despite the policy for emission reduction. The reason could be the increase in the marginal costs of emission reduction, such as effort cost. Finally, a policy mix for inducing the improvement of ship fuel efficiency and reducing CO2 emissions is provided based on the above analysis. The previous discussions imply that if a shipping company fulfils the cost conditions to be classified into optimal cases i) or iii), it implements the effort of improving fuel efficiency under the emission

Fig. 5. Effects of changing price on emission.

classify these cases as two classes, that is, i) and iii) as A > 0, and ii) and iv) as A = 0. Cases ii) and iv) do not include an effort to improve due to the significantly higher marginal cost of effort. Although setting a higher tax than the marginal cost leads every firm to exert itself, a subsidy for improving fuel efficiency as an alternative policy also leads to changing the negligent behavior of firms due to a decrease in the marginal cost of effort. The source of the subsidy should typically represent revenue from the emission tax. Table 1 summarizes the results. The third section shows the relationship between emission tax, capital, effort to improve, and CO2 emissions using a numerical analysis. Consequently, as fuel price (including tax on emissions) increases, both capital K and improvement effort A increase. This can be interpreted as the firm making an effort to improve fuel efficiency to avoid the increasing tax. As the increasing effort expands optimal distance, the revenue of the firm increases, leading to an increase in capital. Cargo demand and transport distance need to be sufficient for the shipping firm, otherwise this relationship would not be satisfied. The IMO (2014, Figs. 6–8, pp. 11–12) shows a similar situation in the effect of an increase in crude oil price from 2007 to 2012. Demand ton-mile, volume of shipping capacity (fleet total dwt capacity in the figures), and fuel efficiency increase simultaneously. Furthermore, we analyzed the effects of initial fuel efficiency f0 on Table 1 Analysis results.

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tax and reduces CO2 emissions. Hence, policymakers should first focus on shipping companies in optimal cases ii) or iv) and try to turn them into cases i) or iii) by enforcing the policy mix. Such a policy mix is the emission tax and direct subsidy for the effort of improving the fuel efficiency of ships, funded by the emission tax. This policy mix can induce shipping companies in cases i) or iii) to reduce CO2 emissions (third row of Table 1) and shipping companies that have the significantly higher marginal costs of effort are subsidized to transfer from cases ii) or iv) to cases i) or iii) (second row of Table 1). The difference between this policy mix and the Efficiency Incentive Scheme and GHG fund charges proposed by IMO member states is the use of the revenue. Uses of the revenue from the proposed charges are described as purchase of approved emission reduction credits, finance of mitigation or adaptation activities in developing countries, and support of research into fuel efficiency improvement. The policy mix also has the advantages of insector MBM such as “does not cap shipping sector activity in the future” or “any revenues generated by MBM is used mainly within the shipping sector” as per the table of Annex 4 of MEPC (2011). To support this policy implication, IMO should research several topics including the marginal cost curve of efforts to improve fuel efficiency, analysis of the revenue-recycling effect of the policy mix and effect on economic welfare especially, and the rebound effect of tax introduction. The first research direction is crucial for the policy mix in terms of selecting candidate shipping companies. Whereas previous research (CE Delft, 2010; DNV, 2010; Eide, Longva, Hoffmann, Endresen, & Dalsøren, 2011; IMO, 2009; Institute of Marine Engineering, Science and Technology, 2010) depicted the marginal abatement curve in the entire shipping sector, curves of marginal cost of effort should be estimated separately by ship types.

CO2 reduction due to an increase in voyage distance to earn additional revenue. We imply, through numerical analysis, that the rebound effect may not be small. A policy mix for inducing the improvement of ship fuel efficiency and reducing CO2 emissions from the long-term perspective is provided based on the previous analysis. This policy mix includes the emission tax and a direct subsidy for effort to improve the fuel efficiency of ships funded by the revenue from the emission tax. The policy mix induces fuel efficiency improvement for a shipping company that has a higher marginal cost for effort and has advantages for in-sector MBM as well. This study has several limitations. First, our optimal control model did not calculate mitigation paths over time and did not include the revenue-recycle effect. Second, we did not analyze the effects of certain important policies such as the EEDI regulation policy on a shipping company by using the optimal control model. Third, production or cost functions with estimated parameters based on the econometric analysis of actual ship types or shipping companies are needed to elaborate on the analysis. As these topics could provide useful information to policy makers, it is necessary to explore them in further studies. Finally, we have analyzed the model in partial equilibrium, which excluded analysis on the consignor's and consignee's side. By constructing a general equilibrium model including both sides, we can also consider the influence of interaction between them and the market structure. Acknowledgments The authors are grateful to the anonymous reviewers and the editors for valuable comments on improving the paper. We also thank Dr. Takuma Matsuda of Japan Maritime Center for advising our research. All remaining errors are ours. This work was supported by Grant-in-Aid for Scientific Research (C) 17K00701 from the Japan Society for the Promotion of Science.

5. Conclusion MBM, which has been discussed at the MEPC, may provide shipping companies with incentives to reduce CO2 emissions from ships and change the long-term behavior of company decision-making (e.g., investment for ships). Since the models in the literature are not developed based on an optimal control approach, they have not included analysis of the impact of MBM on capital investment and reduction of CO2 emission in the long-term. Therefore, this study developed an optimal control theoretical model to analyze the investment behavior of a shipping company and its CO2 emissions under the introduction of MBM, and especially of a tax on CO2 emissions in the long term. The optimal control model includes capital investment in ships, effort to improve fuel efficiency, and voyage distance variables to clarify the long-term impact. We identify optimal conditions and classify them into four cases based on capital investment and the effort to improve fuel efficiency during the process of solving the model. As a result, while fuel price (including the tax on emissions) increases, both capital and improvement effort increase as well. On the other hand, the classification suggests that when the marginal cost with respect to the improvement of fuel efficiency is large, a zero effort for improving fuel efficiency is optimal. We therefore believe that a subsidy for improving fuel efficiency is an alternative policy to change the “no effort” behavior of a company. Numerical analysis based on the model is conducted to clarify the relationships between fuel price including MBM tax, capital investment, and efforts towards fuel efficiency. As the fuel price including the tax on emissions increases, the firm makes the effort to improve fuel efficiency and expand voyage distance, provided cargo demand from the market is available to the company as needed. Then, capital will be gained by way of an increase in the firm's revenue. The results also provide information on the relationship between price under MBM and CO2 emissions through the process of the rebound effect. As the fuel price (including tax on emissions) p increases, CO2 emissions decrease. The rebound effect due to the effort to improve fuel efficiency cancels the

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