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Tyler Weseling

2 December 2015

High Speed Railway Comparative Static Analysis

Introduction

As society becomes more globalized, families and friends continue to disperse over a greater

geographic landscape contributing to an increase demand for efficient and comfortable means of

travel. Due to this increasing dispersion of the population, one must rely on a form of

transportation other than biking or walking. Currently, an individual has the rational options of

utilizing an automobile, bus, conventional train, high speed railway (in Europe and Asia), and

airplane to cover the travel distances. Unfortunately, in the United States, the dependency on the

airplane industry as well as the need to possess an automobile, leaves many Americans at the

bidding of these industries.

In Japan, the Japan Rail Company has interconnected the entire country with efficient high speed

railways that are more environmentally friendly, comfortable, time-efficient, and provide the

Japanese citizens an additional option of traveling the country instead of relying upon the

collusion ability of the airline industry in order to travel longer distances domestically. In

Europe, one can travel the majority of the continent through railways, allowing for a more

personal travel experience by being able to witness the beauty of the pass-through country

landscapes. Unfortunately, this form/method of travel does not exist in the United States besides

the Amtrak Acela and the proposed maglev train route from Washington D.C. to New York City.

Thus, I am going to utilize comparative statics to analyze what variables contribute to the

demand to travel via a high speed railway travel.

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Model Specification

The mathematical model of interest to conduct this analysis would be the market model, utilizing

a demand and supply function, in regards to identifying the variables that determine the demand

and supply for high speed railway traveling as can be viewed through ticket pricing. Thus, the

two equations utilized in the model in general form are shown below:

Qd=Qs

Qd= D (P, Y0, A0, N0) ((∂D∂P ¿<0;( ∂ D∂Y 0)>0; ( ∂ D∂ A0 )>0;( ∂ D∂N 0

)>0)Qs= S (P, A0, T0) (( ∂S∂ P )>0 ;( ∂ S∂ A0 )<0 ;( ∂S∂T0

)>0¿

Within the demand equation, P is representative of the price demanded for consumers to travel

on high speed railways. Y0 is representative of a person’s discretionary income. A0 is

representative of the price for an equivalent route via an airplane. An airplane ticket is used

specifically to represent a subsitute product instead of automobiles because high speed railway in

Europe and Japan is primarily competitive with routes over 500km. Therefore, it appeared

rational to directly compare ticket prices between airlines and the high speed railway. Lastly, N0

is representative of population density/number of customers within certain city districts.

In regards to the supply equation, P is represetative of the price suppliers are willing to supply

towards in regards to high speed railway travel. A0 is representative of the price of airline tickets,

which serves as a substitute product for suppliers. In addition, T0 is representative of technology

and its impact on the supply of high speed railway travel. Lastly, in regards to both equations,

price and quantity are the endogenous variables in this model.

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Partial Derivative Signage

∂D∂P

<0

As price increases by one incremental unit or dollar, the demand for traveling on a high speed

railway is inversely affected. Individuals, if rational, will desire to utilize a different method of

transportation for travel if cheaper. This is an inverse relationship.

∂ D∂Y 0

>0

As an individual’s personal discretionary income rises, the demand to travel for high speed

railways. High speed railways offer a more comfortable/luxury method of traveling, not serving

as a necessity, and will be directly correlated with a peron’s change in discretionary income. This

is a positive relationship.

( ∂ D∂ A0 )>0

Due to airline travel serving as a close substitute, an increase in the costs associated with

traveling via an airline lead to an increase in the demand for high speed railway travel, thus, they

are directly/positively related.

∂ D∂N 0

>0

Due to high speed railways being historically more productive in countries with densely

populated cities due to minimal stopping of high speed trains, an increase in the population

density or number of customers in a certain area will lead to a direct relationship with demand.

Thus, a positive relationship exists between number of cutomers/population density and demand.

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∂S∂P

>0

The higher the price customers are willing to pay to travel, the more suppliers desire to be in the

industry or supply their products. Thus, an increase in tickets, more likely an increase in

frequency of trains, and load capacity of trains, which exhibits a positive relationship.

∂S∂ A0

<0

When the price of traveling via an airliner increases as in ticket price increases, there is an

inverse relationship with the supply for high speed railway traveling by suppliers . This is due to

the inverse relationship that exists due to more suppliers will want to supply in the airliner

industry at the higher price due to being close substitutes.

∂S∂T 0

>0

As technology improves, such as more efficient time management mechanisms, such as

minimization of accidents, improvement and advancement of rolling stock technology, and more

efficient time scheduling for frequency of trains, there is an increase in the supply of traveling

via high speed railway.

Application of Implicit-Function Rule

In order to identify whether the implicit function rule is satisfied and for this analysis to continue

onwards, the function must possess continuous derivatives and must possess an endogenous non-

zero Jacobian. In order to facilitate this process, one can rearrange the demand and supply

functions as such:

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F1(P,Q;Y0,A0,N0)= D(P,Y0,A0,N0)-Q=0 F2(P,Q;A0,T0)= S (P,A0,T0)-Q=0

Thus, as established prior in the paper, the partial derivatives are assumed to be continuous and

thus, they are assummed to be continuous in F1 and F2. Thus, the first criteria is met.

The second criteria deals with a non-zero endogenous Jacobian, which is illustrated

below:

|J|≡|∂F1

∂ P∂ F1

∂Q∂F2

∂ P∂ F2

∂Q|= |∂ D∂ P −1

∂S∂ P

−1|= ∂S∂P−∂ D∂ P >0

Thus, due to the endogenous variable producing a Jacobian greater than 0, ((∂S∂P

¿ ,is positive,

minus ∂D∂P , which is negative, thus producing a positive Jacobian), meets the second rule for the

implicit function rule. Thus, the general forms and the equilibrium states are listed below:

General forms of implicit function rule:

P*=P*(Y0,A0,N0,T0) and Q*=Q*(Y0,A0,N0,T0)Thus, equilibrium identities are:

D(P*,Y0,A0,N0)-Q*≡0 [i.e., F1(P*,Q*;Y0,A0,N0)]S(P*,A0,T0)-Q*≡0 [i.e., F2(P*,Q*;A0,T0)]Comparative Statics Analysis

[ ∂F1

∂ P¿∂ F1

∂Q¿

∂F2

∂ P¿∂ F2

∂Q¿ ][ ∂ P¿

∂Y 0

∂P¿

∂ A0

∂P¿

∂ N0

∂ P¿

∂T 0

∂Q ¿

∂Y 0

∂Q¿

∂ A0

∂Q¿

∂ N0

∂Q¿

∂T 0]=[−∂F

1

∂Y 0

−∂F1

∂ A0

−∂F1

∂N0

−∂ F1

∂T 0

−∂F2

∂Y 0

−∂F2

∂ A0

−∂ F2

∂N0

−∂ F2

∂T 0]

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The above analysis is equivalent to the one below:

[ ∂D∂P¿ −1

∂S∂P¿ −1][ ∂ P

¿

∂Y 0

∂ P¿

∂ A0

∂ P¿

∂ N0

∂ P¿

∂T0

∂Q¿

∂Y 0

∂Q¿

∂ A0

∂Q¿

∂ N0

∂Q¿

∂T0]=[−∂D∂Y 0

−∂ D∂ A0

−∂D❑

∂ N00

0 −∂S∂ A0

0 −∂ S∂T0

]Now that the matrixes are set up, one can now evaluate using Cramer’s Rule how each

exogenous variable affects each endogenous variable.

1). ∂ P¿

∂Y 0=|−∂ D∂Y 0

−1

0 −1|∂S∂P

−∂ D∂ P

=∂ D∂Y 0

∂ S∂ P

−∂ D∂P

> 0

Thus, there is a positive relationship between personal discretionary income and price for travel

on high speed railway (ticket price). Thus, for each incremental increase in personal income,

price demand for traveling on high speed railway increases as well.

2). ∂Q ¿

∂Y 0=|∂ D∂ P¿

−∂ D∂Y 0

∂S∂ P¿ 0 |∂S∂P

−∂ D∂ P

=∂S∂P¿

∂ D∂Y 0

∂ S∂ P

−∂ D∂P

> 0

There is a positive relationship between personal discretionary income and quantity demanded

and quanity supplied for high speed railway travel(tickets). As personal income rises, suppliers

are willing to supply more tickets/more opportunities to travel on high speed railways.

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3). ∂P ¿

∂ A0=|−∂ D∂ A0

−1

−∂S∂ A0

−1|∂S∂P

−∂ D∂ P

=∂D∂ A0

− ∂ S∂ A0

∂S∂P

− ∂ D∂ P

> 0

There is a positive relationship between airline ticket prices and the prices demanded and

supplied for high speed railway travel. Thus, as ticket prices rise for airplanes, the ticket prices in

relation to high speed railway travel also increases in accordance with this relationship being a

positive relationship.

4). ∂Q¿

∂ A0=|∂ D∂ P¿

−∂ D∂ A0

∂S∂ P¿

−∂S∂ A0

|∂S∂P

−∂ D∂ P

=[( ∂ D∂ P¿ )(−∂S∂ A0 )]−[( ∂S∂ P¿ )(−∂D∂ A0 ) ]∂S∂P

−∂ D∂ P

= 0

The relationship between airliner ticket price and that of quantity of high speed railway travel

can be inversely related,equal to zero, or be positively related based on the magnitude of the

affect of the change in price of the airline tickets. Quantity demanded may substantially increase

if airline prices raise to high but quantity supplied may also raise if the price or airlines

diminishes. Thus, the magnitude is important in this situation.

5). ∂ P¿

∂N 0=|−∂ D∂N0

−1

0 −1|∂S∂P

−∂ D∂ P

=∂ D∂N 0

∂ S∂ P

−∂ D∂P

> 0

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This is representative of a positive relationship and thus, as number of customers or the

population density of the servicing cities or geographic area, price related to traevling on high

speed railway will increase incremently.

6). ∂Q¿

∂N 0=|∂ D∂ P¿

−∂ D∂ N0

∂S∂ P¿ 0 |∂S∂P

−∂ D∂ P

=∂S∂P¿

∂ D∂N 0

∂ S∂ P

− ∂ D∂P

> 0

As evidenced in the above comparative static, there is a positive relationship between number of

customers/population density and quantity. Thus, as number of customers/population density

increases, so does the quantity for high speed railway travel.

7). ∂ P¿

∂T 0=| 0 −1

−∂S∂T 0

−1|∂S∂P

−∂ D∂ P

=−∂S∂T0

∂ S∂ P

−∂ D∂P

< 0

There is an inverse relationship between tehnology and price. As technology advancements (such

as improved time tables, railways, or frequencies of trains), the price associated with traveling on

high speed railway will react inversely.

8). ∂Q ¿

∂T 0=|∂ D∂ P¿ 0

∂S∂ P¿

−∂S∂T 0

|∂S∂P

−∂ D∂ P

= (∂ D∂ P¿ )(

−∂ S∂T0

)

∂S∂P

−∂ D∂ P

> 0

There is a positive relationship between that of technology and the quantity of travel on high

speed railways (increase in tickets available or individuals desire to travel). Thus, an increase in

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technology utilized (such as more efficient time schedules, railways, etc…), there is an increase

in the quantity of high speed railway travel.

Possible Extensions to the Model

The model can be extended to include a variety of more variables due to the complexity of high

speed railway travel. Specifically, environmental impact, such as the reduction in the amount of

carbon dioxide levels and less of a dependence on oil, especially during this heightened focus on

global warming, could be a variable taken into account. Despite that being a significant variable,

cultural tastes or values is an extremely significant factor that was left out of this comparative

static analysis. In order for myself to have been able to try to “quantify” how demand and supply

would be affected, I would have needed to focus on country specific factors. In Japan, high speed

railway travel has been the major source of transportation for over 50 years but in the United

States, automobiles and aiplanes have dominated travel for U.S. citizens. Thus, cultural tastes

would significantly dictate the quantity demanded and quantity supplied for high speed railway

travel based on regional cultural factors.

In addition, supply of high speed railway is affected by factors such as already

established infrastructure/convertability of established tracks (exclusive exploitation, mixed high

speed, mixed conventional, and fully mixed), frequency of routes, impact of freight travel, and

government policy, such as subsidies and/or taxes. Lastly, geographic dispersion of the cities and

route times (such as time per day, day per week) are critical variables to consider in doing a

proper analysis.

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