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Measurement of Value of Time for Freight Trips and its Benefit by Market Information
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Transportation Research Procedia 00 (2017) 000–000
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Transportation Research Procedia 25 (2017) 5144–5159 www.elsevier.com/locate/procedia
World Conference on Transport Research - WCTR 2016 Shanghai. 10-15 July 2016 Measurement of Value of Time for Freight Trips and its Benefit by Market Information
Hisa Morisugi* N h on U n iv e rsit y , T ok y o, J ap an
Abstract This paper focuses on how value of time and its accrued time saving benefits for freight trips can be expressed and measured in terms of market information such as freight fee, freight time, observable freight service consumption and observable freight service production within a framework of general equilibrium rather than individual user analysis which most previous studies adopt. In order to satisfy the weak complementarity assumption, we specifies freight time h 0 externality as product form of quality index of freight time and freight consumption s (t ) x and s (t ) X into the utility function of non-business users and production function of business users, respectively. And for freight operators the product of quality index of freight time and labour and capital inputs s (t )l , s (t )k are introduced into the production function of freight operators. Then we define VOT as marginal substitution rate between price and time and expressed it as the freight service fee per freight time multiplied by the elasticity of quality level indicator with respect to freight time for each of non-business and business users, respectively. The freight service providers’ VOT is expressed as average labour and capital cost per freight time. Thus we can measure three VOTs in terms of market information such as freight fee, freight time and average labour and capital cost and the elasticity of quality level indicator with respect to freight time although we need to estimate quality index of freight time by estimating freight demand functions of users. Next we derive an “origin” formula to measure social freight time saving benefits including its repercussion effects expressed by the change in equilibrium freight price, wage and capital rent, where we adopted the concept of equivalent variation as the social benefit definition. The proposed origin (or shortcut) formula is the weighted general equilibrium consumers’ surplus of the freight demand multiplied by VOTs with respect to freight time. The weight after the change can be approximately expressed as 1 + (income effect of marginal freight time saving benefit for non-business), so that it is also possible to calculate from the market information. Furthermore VOT to be applied to the benefit measurement formula is a sum of VOT for shippers and operators. Therefore we can say that conventional trapezoidal formula with the weight of
* Corresponding author. Tel.: 81-22-777-5620; fax: 81-22-777-5620. E-mail address: [email protected] 2214-241X © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of WORLD CONFERENCE ON TRANSPORT RESEARCH SOCIETY.
2352-1465 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of WORLD CONFERENCE ON TRANSPORT RESEARCH SOCIETY. 10.1016/j.trpro.2018.02.043
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Author name / Transportation Research Procedia 00 (2017) 000–000
Hisa Morisugi / Transportation Research Procedia 25 (2017) 5144–5159 2 name / Transportation Research 00 (2017) 000–000the weight one and VOT for only operators Author underestimates the benefit in two Procedia points. First is missing
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emB and second is
neglecting VOT for shippers. B one and VOT for only operators underestimates the benefit in two points. First is missing the weight em and second is neglecting VOT for shippers. © 2017 The Authors. Published by Elsevier B.V. © 2017 The Authors. Published by Elsevier B.V. Peer-review WORLD CONFERENCE CONFERENCE ON ON TRANSPORT TRANSPORT RESEARCH RESEARCH SOCIETY. SOCIETY. Peer-review under under responsibility responsibility of of WORLD © 2017 The Authors. Published by Elsevier B.V. Peer-review under of WORLD CONFERENCE ON TRANSPORT RESEARCH Keywords: value of responsibility time for freight service users, value of time for freight service providers,SOCIETY. freight time saving benefit measurement, marginal time saving benefit, marginal utility ratio of income, general equilibrium Keywords: value of time for freight service users, value of time for freight service providers, freight time saving benefit measurement, marginal time saving benefit, marginal utility ratio of income, general equilibrium
1. Introduction 1. Introduction The subjective value of time for freight is the marginal rate of substitution between freight time and freight cost under constant utility or profit, or cost level, which is defined for freight service consumers as well as freight service The subjective value ofreferred time fortofreight is the marginal rate of ofvalue substitution freight time cost providers. It is commonly as the monetary appraisal of timebetween or the willingness to and pay freight for savings under constant in freight time. utility or profit, or cost level, which is defined for freight service consumers as well as freight service providers. It is commonly to as appraisaltime of value time or the pay savings This paper focuses on referred how value of the timemonetary and its accrued savingofbenefits for willingness freight tripstocan befor expressed in time.in terms of market information such as freight fee, freight time, observable freight service consumption andfreight measured paper focuses how value of time and its accrued timeofsaving for freight be expressed and This observable freight on service production within a framework generalbenefits equilibrium rathertrips thancan individual user and measured in terms of market information such as freight fee, freight time, observable freight service consumption analysis which most previous studies adopt. andFor observable freight service production within a framework of society general composed equilibrium than individual user general equilibrium analysis we suppose a simple economic of rather households, two sectors of analysis which most previous studies adopt. production, firms producing composite goods, and firms producing freight transportation with a given level of freight Foreven general equilibrium analysis suppose a simple economic society composed households, two sectors of time for freight providers. Thewe reason of above simplicity is for simple and clearofderivation of VOT and time production, firms producing composite goods, and firms producing freight transportation with a given level of freight saving benefit measurement formula. It is quite possible to construct a more realistic complicated economy with many time even for time freight providers. reason of above simplicity is for government, simple and clear derivation VOTregions and time goods, labor, constraint andThe capital, heterogeneous households, investment andofmany as saving benefit measurement formula. It is quite possible to construct a more realistic complicated economy with many SCGE (spatial computable general equilibrium) modeling. Furthermore for freight transportation production sectors goods, timetoconstraint capital, networks heterogeneous households, and many regions as it is alsolabor, possible introduce and multi-mode for both person andgovernment, freight trips,investment With this complication the basic SCGE (spatial computable general equilibrium) modeling. Furthermore for freight transportation production sectors formula is exactly same as the simple case above except for appearance of many types of VOT and time saving benefit it is also possible to introduce multi-mode networks for both person and freight trips, With this complication the basic formula. formula same as the above except for appearance many types of VOT time saving benefit Nextisinexactly order to satisfy thesimple weakcase complementarity assumption, weofspecifies freight timeand externality as product formula. h 0 s (t )we x and s (t ) Xfreight into time the utility function of nonformNext of quality index of freight time and freight consumption in order to satisfy the weak complementarity assumption, specifies externality as product business users and production function of business users, respectively. And for0 freight operators the product of quality h form of quality index of freight time and freight consumption s (t ) x and s (t ) X into the utility function of nonindex of freight time and labour and capital inputs s (t )l , s (t )k are introduced into the production function of freight business users and production function of business users, respectively. And for freight operators the product of quality operators. index of freight time and labour and capital inputs s (t )l , s (t )k are introduced into the production function of freight Then we define VOT as marginal substitution rate between price and time and expressed it as the freight service operators. fee per freight time multiplied by the elasticity of quality level indicator with respect to freight time for each of nonThen users we define VOT as marginal substitutionThe ratefreight between priceproviders’ and time and expressed it as the freight service business and business users, respectively. service VOT is expressed as average labour fee per freight multiplied the elasticity ofelasticity quality level indicator respectwith to freight for each of Thus nonand capital costtime per freight timeby multiplied by the of quality levelwith indicator respecttime to freight time. business users and business users, The freightsuch service providers’ VOT time is expressed labour we can measure three VOTs in termsrespectively. of market information as freight fee, freight and sumas of average average labour and capital time multiplied the elasticity quality level withalthough respect to time. Thus and capital cost cost per andfreight the elasticity of qualitybylevel indicator of with respect to indicator freight time wefreight need to estimate we can measure three VOTs in terms of market information such as freight fee, freight time and sum of average labour quality index of freight time by estimating freight demand functions of users. The author believes it is the first time to and capital cost and VOT the elasticity of quality levelexpression indicator with respect to freight although As we for needthetoVOT estimate succeed in defining for shippers and their by observable market time information. for quality index of freight time by estimating freight demand functions of users. The author believes it is the first time to operators, our derivation looks similar to the conventional practice. succeed defining VOT for shippers andequilibrium their expression observable information. As derive for theaVOT for Next in within the framework of general ratherby than individualmarket behaviour analysis we “origin” operators, our derivation looks similar to the conventional practice. formula to measure social freight time saving benefits including its repercussion effects expressed by the change in Next within the price, framework general than individual behaviour analysisvariation we derive “origin” equilibrium freight wage of and capitalequilibrium rent, whererather we adopted the concept of equivalent as athe social formuladefinition. to measureThe social freight saving benefits including its repercussion effects expressed by the change in benefit formula is time the integration of sum of weighted marginal freight time saving benefit functions equilibrium price, wage andonly, capital rent, weisadopted the concept of equivalent social with respect freight to simply freight time where thewhere weight the marginal utility ratio of income,variation and sumasofthe marginal benefit time definition. formula is theforintegration of sum weighted marginal time saving demand benefit functions freight savingThe benefit function non-business andofbusiness users are thefreight product of freight function with respect to simply freight time only, where the weight is the marginal utility ratio of income, and sum of marginal and sum of VOTS for users and operators. Thus proposed “origin” (or “shortcut”) formula is the weighted general freight time consumers’ saving benefit function forfreight non-business businessby users arewith the product equilibrium surplus of the demandand multiplied VOTs respect of to freight freight demand time. Thefunction author and sum of VOTS for users and operators. Thus proposed “origin” (or “shortcut”) formula is the weighted general equilibrium consumers’ surplus of the freight demand multiplied by VOTs with respect to freight time. The author
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believes it is also the first time to succeed in origin formulation of only consumers’ surplus of freight demand function in spite of that there are repercussion effects originated from the freight market toward all the market. We note that all the repercussion effects expressed by the change in freight price and wage disappear due to demand-supply equilibrium. Also we note that the marginal utility ratio of income after the change can be approximately expressed as 1 + (income effect of marginal freight time saving benefit for non-business), so that it is also possible to calculate from the market information. Furthermore VOT to be applied to the benefit measurement formula (8.8) is a sum of VOT for shippers and operators. Therefore we can say that conventional trapezoidal formula with the weight of one and VOT for only operators underestimates the benefit in two points. First is missing the weight
e B m and second is neglecting VOT for shippers.
2. Previous studies and practice Suppose a light goods vehicle delivering cargoes ordered by some households and firms with the freight time t and freight price p. Many guidelines suggest that; (1) VOT for this transport producing sector is the imputed wage of driver if it is a non-business trip [Japan MLITT (2008), UKDOT (2011), USDOT (2011), Katoh (2013)]. VOT is the wage rate of driver plus the opportunity cost of vehicle per unit of time [Japan MLITT (2008)], or the wage rate of driver [UKDOT (2011), USDOT (2011)] if it is a business trip. But very few guidelines suggest that; (2) VOT for cargo owners, nor say that: (3) Total VOT for a light goods vehicle delivering cargoes be sum of VOT for transport producing sector plus VOT for cargo owners multiplied by the delivered cargo volume. The first statement (1) by Japan MLITT (2008) above can be shown as appropriate (see equation (6.11)), but very few previous theoretical studies say the definition and measurement of VOT for transportation producers. Rather they pay attentions to the transportation users’ VOT without explicitly introducing the transportation production [Becker(1965), De Serpa (1971), De Donnea (1972), Evans (1972), Bruzerius (1979), Gonzalez (1997), Bates and Roberts (1986), Mackie, Jara-Diaz. and Fowkes (2001), Jara-Diaz (2003), , Winston and Yang (2005) and Katoh(2013)]. Although sometimes they incorporate a simple Leontief type of household transportation production function, that is only applicable to private car transportation. Thus it needs to develop the theory of VOT for transportation producers. As for the second point of (2) there are very few studies on definition and measurement of VOT for cargo owners, which can be classified into two, opportunity cost approach and demand modeling approach [Winston(1981), Katoh(2013) ]. Japan MLITT (2008) adopts opportunity cost approach and the interest cost for the inventory of the cargo value transported for VOT for cargo owners, yet it does not justify theoretically. Feo-Valero et al.(2011) surveys 22 studies of demand modeling approach, and says that theoretically who’s VOT is not clearly distinguished yet, and practically large difference of VOTs for road, railroad, seaborne, airborne were estimated. Thus it needs to develop the theory of VOT for cargo owners, distinctly from transportation producers. Therefore we need a framework of general equilibrium including freight production sectors, but previous studies based on general equilibrium did not explicitly deal with freight time [Morisugi and Ohno(1992), Morisugi, Ohno and Miyagi(1993), Morisugi and Ohno(1995) ]. It is worthwhile to look at the concept of VOT as a commodity developed in the field of person trips as a starting reference for freight service consumers’ VOT [De Serpa (1971), De Donnea (1972), Evans (1972), Train and McFadden(1978), Bruzerius(1979), Bates and Roberts(1985), Truong and Hensher(1985), Mackie, Jara-Diaz and Fowkes (2001), Jara-Diaz (2003) and Katoh(2013)]. VOT as a commodity is derived by incorporating the total transportation time consumption to the utility function (production function) in addition to the transportation service consumption itself. Unfortunate this specification of transportation time is not fit to the application to VOT for cargo owners. Because it seems that shippers do not necessarily get a utility (productivity) from the total transportation time in addition to transportation service consumption. Rather we need the VOT as a quality instead of as a commodity. Another field we have to look at is a recreational benefit evaluation by travel cost method [Knetsch (1963), Larson
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(1992), Bockstael, Strand and Hanemann (1987), Clawson (1959), Neill, J.R. (1988), Larson and Shaikh (2004), Mäler (1974)]. Conceptually we wish to apply this method to our shippers’ VOT measurement by replacing the quality of recreation site to the quality index of freight time as shown in this paper, but it should be within a framework of general equilibrium because we need to deal with operators’ VOT as well as shippers’ VOT. 3. Economy with freight transportation production Consider an economic society composed of households, two sectors of production, firms producing composite goods with unitary price, and firms producing freight transportation with a given level of freight time. A representative household provides labor and capital services in exchange for wages and rents, purchases composite goods and freight transportation services with a given level of freight time. One representative firm produces composite goods, pays wages and rents for labor and capital input, and pays the freight service fees for freight input. Another representative firm produces freight transportation with a given level of freight time, pays wages and rents for labor and capital input, and pays for input of composite goods. It is assumed that the production functions of both firms exhibit constant returns to scale, so that zero profit condition holds at equilibrium. Society is at equilibrium in the long run with respects to composite goods, freight services, labors and capitals, so that zero profit and constant returns to scale hold. This paper assumes that freight time per trip is an externality as a kind of public goods for both of freight consumers and producers, just like case of road transportation sector with a given level of freight time, determined by the road condition constructed by the government. The reason of above simplicity is for simple and clear derivation of VOT and time saving benefit measurement formula. It is quite possible to construct a more realistic complicated economy with many goods, labor, time constraint and capital, heterogeneous households, government, investment and many regions as SCGE (spatial computable general equilibrium) modeling. Furthermore for freight transportation production sectors it is also possible to introduce multi-mode networks for both person and freight trips. With this complication the basic formula is exactly same as the simple case above except for appearance of many types of VOT and time saving benefit formula. 4. Nonbusiness freight service consumption modeling and its VOT Nonbusiness freight service consumption modeling is formulated by households’ utility maximizing behavior under the budget constraint as:
V 1, p, s h (t ), m max u z, s h (t ) x
(4.1a)
z, x
s.t. z px m
(4.1b)
, where u ( ) : direct utility function z : composite goods consumption with price 1 x : non-business freight transportation consumption with price p and time h
h
t
s (t ) : quality level indicator of freight time t . Assume s (t ) 0 and ds (t ) / dt sth 0 (4.1c) m wl rk : income level w : wage l : labour supply (fixed) r : capital rent k : capital supply (fixed)
V
: indirect utility function
h
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Note that, first, there is no time constraint because it assumes that this economy has not any person trips. h h Second, specification of utility function is as u z , s (t ) x instead of more general form as u z , x, s (t ) . The latter form is identical for measuring recreational benefit of recreational quality improvement by travel cost method under the weak complementarity assumption which says that if x 0 then marginal utility of t is zero [Mäler (1971), Mäler (1974)]. The former form this study adopts is sufficient to satisfy the weak complementarity assumption as it h is the product form of s (t ) x . This specification is often adopted with the CES utility function in the environmental economics [von Faefen and Phaneuf (2003)]. Thus this study intends to apply the recreational benefit measurement of travel cost method to the freight time saving benefit. Furthermore it will be shown in this paper that for the product form it is not necessary to apply travel cost method, since the benefit can be expressed as integral of demand function multiplied by value of time (VOT) with respect to time instead of price. As for whether or not the freight time increases the utility of freight service is an empirical matter. If as a result it shows that there is not any marginal utility of freight h h time, then set s (t ) 1 , and the theory and discussion remains as same as s (t ) 1 . With comments above to solve (4.1), let Lagrangian be as
L u z , s h (t ) x m z px
(4.2)
And its first order conditions (FOC) are
ux u sh p uz and sh x
(4.3)
, where subscriptions indicate partial differentiation with respect to the subscript while superscript h indicates household or non-business users. And its demand functions (4.4), marginal utility of income (4.5) and indirect utility function (4.6) are expressed as
z z 1, p, s h (t ), m
(4.4a)
x x 1, p, s h (t ), m
(4.4b)
1, p, s h (t ), m
(4.5)
V V 1, p, s h (t ), m
(4.6)
Envelop Theorem implies
Vm , Vp Vm x
(4.7)
h p h p st t h Vt u xs xst Vm x (4.8) h Vm x p / t ( 0) h sh x s t s ,where h (stht / s h ) : elasticity of quality level indicator with respect to freight time t . Subscriptions indicate h t
partial differentiation with respect to the subscript while superscript h indicates household or non-business users. The second equality of (4.8) is due to (4.3). Marginal freight time saving benefit of non-business freight users (=households) is defined as the value of marginal utility of time saving (Vt / Vm ) , which can be expressed as minus freight consumption multiplied by VOT, where h VOT is defined as the marginal freight time saving benefit divided by freight consumption MB / x . Therefore,
,where
MB h (Vt / Vm ) x p / t h xVOT h
(4.9)
VOT h MB h / x p / t h
(4.10)
(4.10) of VOT formula can be derived also by defining the value of time (VOT) as marginal substitution rate between price and time and apply (4.7) and (4.8)
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V x p / t V dp VOT t m p / t h dt V const V p Vm x
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h
h
(4.11)
(4.11) shows that VOT for non-business freight users can be expressed as the freight service fee per freight time multiplied by the elasticity of quality level indicator with respect to freight time t . It seems that there are three types of non-business freight service, private cars, parcel delivery service and house moving transportations. The elasticity of quality level indicator with respect to freight time t can be estimated by estimating the non-business demand functions for each type of non-business freight service. For Japanese parcel delivery service, its average delivery price is 1200 yen, and its average delivery time is 20 hours (1200 minutes). Now suppose the elasticity of the elasticity of quality level indicator with respect to freight time t is -0.2. Then VOT for shippers is 0.2 yen/minute. 、 5. Business freight service consumption modeling and its VOT Business freight service consumption modelling is formulated by cost minimization of composite goods production firms under the constant return to scale production function with inputs of labour, capital and freight transportation as:
c w, r , p, s 0 (t ) Y 0 0
min C 0 wl 0 rk 0 pX 0 0 l , k ,X
s.t. Y 0 f 0 l 0 , k 0 , s 0 (t ) X
(5.1a) (5.1b)
0
,where Y ：output level of composite goods
l 0 ：labour input w ：wage k 0 : capital input r : capital rent
X ：freight service input with its price p and time t s (t ) : quality level indicator of freight time t . Assume s 0 (t ) 0 and ds0 (t ) / dt st0 0 (5.1c) 0
f0
： composite goods production function (constant return to scale).
Superscript 0:composite production sector As same as non-business case, it is again assumed for business freight demand that shorter freight time increases 0
the productivity of freight service in the form of product form s (t ) X , which is sufficient to satisfy the weak complementarity assumption. The Lagrangian (5.2), its first order conditions (5.3), conditional input demand functions (5.4) and cost function (5.5) will be
min L0 wl 0 rk 0 pX 0 [Y 0 f 0 l 0 , k 0 , s(t ) X ] 0 l , k , X , 0
0 0 p f X 0 f s00 X s 0 w 0 f l 00 , r 0 f k00 ,
(5.2) (5.3)
l 0 l 0 ( w, r , p, s 0 (t ))Y 0
(5.4a)
k 0 k 0 (w, r , p, s 0 (t ))Y 0
(5.4b)
0
X X (w, r , p, s (t ))Y 0
0
0
0
C c ( w, r , p, s (t ))Y
0
(5.4c) (5.5)
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0 0 (w, r , p, s 0 (t ))
7
(5.6)
And by envelop theorem 0 0 c 0 ( w, r , p, s 0 (t )) Cw0 l 0 , Cr0 k 0 , C p0 X , CY
(5.7)
0
and
C 0t 0 f s00 X Xst0 X ( p / s 0 ) st0 X ( p / t ) 0 ( 0)
(5.8)
(5.8) is the marginal cost reduction of freight time saving, which is its marginal time saving benefit 0
of business users (=composite goods production sector) MB :
MB0 C 0t X ( p / t ) 0 ( XVOT 0 )
(5.9)
0
And VOT is defined as the marginal freight time saving benefit divided by freight consumption MB / X . VOT 0 MB 0 / X p / t 0 (5.10) (5.10) of VOT formula can be derived also by defining the value of time (VOT) as marginal substitution rate between price and time and apply (5.7) and (5.8) :
Ct0 dp ( p / t ) 0 VOT 0 0 dt C 0 const C p
(5.11)
(5.10) and (5.11) show that VOT for business freight users can be expressed also as the freight service fee per freight time multiplied by the elasticity of quality level indicator with respect to freight time. There are many types of business freight service for each type of cargo owner of different business sectors. The elasticity of the elasticity of quality level indicator with respect to freight time t can and should be estimated by estimating the business demand function for each type of business freight service and for each type of business shippers. Suppose that Japan-west coast of USA container shipment of 1 TEU has fee of 180,000 yen/unit, and 300 hours (180000 minutes) of shipment time, and 0.2 of the elasticity of quality level indicator with respect to freight time t . Then VOT for shipper is again 0.2 yen /minute. 6. Freight service production modelling and its VOT Freight service production modelling is formulated by cost minimization of freight service production firms under the constant return to scale production function with inputs of labour, capital and composite goods as:
c w, r , p, s(t ) Y C min wl rk Z l , k ,Z
s.t. Y f s (t )l , s (t )k , Z
(6.1a) (6.1b)
, where Y ：output of freight service, l ：labour input w ：wage rate k ：capital input r ：capital rent Z ：input of composite goods t ：freight time(exogenous) s (t ) : quality level indicator of freight time t . Assume s(t ) 0 and ds(t ) / dt st 0
f
:
(6.1c)
freight service production function (constant returns of scale).
Note (6.1b) shows that freight time reduction increases the labour and capital productivity, but not for another
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input represented by composite goods. Thus (6.1b) assumes that freight production cost is a sum of time dependent cost (represented by labour and capital cost) and distance dependent cost independent of time (represented by composite goods input cost). The Lagrangian (6.2), its first order conditions (6.3), conditional input demand functions (6.4) and cost function (6.5) will be
min L
l , k ,Z ,
wl rk Z [Y f s(t )l 0 , s(t ) k 0 , Z ] w f sl s , r f sk s , 1 f Z
(6.2) (6.3)
l l (w, r ,1, s(t ))Y
(6.4a)
k k (w, r ,1, s(t ))Y
(6.4b)
Z Z (w, r ,1, s(t ))Y
(6.4c)
C c(w, r,1, s(t ))Y
(6.5)
(w, r,1, s(t ))
(6.6)
And by envelop theorem , and
Cw l , Cr k , CY c( w, r ,1, s(t ))
(6.7)
wl rk wl rk 1 st t Ct ( f sl l f sk k ) st [( w / s )l (r / s )k )]st Y Y ( 0) t Y t s
(6.8) , where the second and forth equality are due to (6.3) and (6.4), respectively. The “~” indicates unit input so that
wl rk is the unit labour and capital cost.
(6.8) is the marginal cost reduction of freight time saving, which is its marginal time saving benefit of operators MB :
wl rk Y MB Ct YVOT ) ( t
(6.9)
And VOT is defined as the marginal freight time saving benefit divided by freight production
wl rk VOT MB / Y t
MB / Y .
(6.10)
(6.10) shows that VOT for freight operators can be expressed as the unit labour and capital cost of freight service production per freight time multiplied by the elasticity of quality level indicator with respect to freight time. It seems quite reasonable to assume s (t ) is proportional to speed (delivery distance over delivery time) in case of operator. Then the elasticity is -1. Then the VOT is:
wl rk VOT t
(6.11)
Consider a light goods vehicle delivering cargoes ordered by some households and firms with the freight time t and freight price p. Y is one because it is measured by one truck. And assume one driver for that vehicle so that labour hour is t. Another cost is truck depreciation per time, Then VOT for operator is sum of the wage rate of driver and truck depreciation per time. Thus we can justify Japan MLITT (2008) in a sense that it values the opportunity cost of car while other manuals of UKDOT (2011) and USDOT (2011) underestimates [Katoh (2013)] Another way to estimate VOT for operator is to look at fact that freight price equals to unit cost of freight service
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in the long run, and it can be divided into the freight time dependent cost and transportation distance dependent cost. VOT for operator is the former cost per freight time, which is mainly composed of some part of labour cost and capital cost (drivers and cars), which is usually around 0.6. 7. Market equilibrium Market equilibrium in the economy is for composite goods, freight transportation, labor and capital shown as,
zZ Y0
x X Y
l0 l l 0 k k k
, and for zero profit condition shown as
1 c 0 ( w, r , p, s 0 (t ))
p c(w, r,1, s(t )) , where the “~” indicates unit cost. Unknown variables are
w, r , p, Y 0 , Y with numeraire of composite goods.
8. Social benefits of freight time saving Consider a change in freight time from t indirect
utility
level
of
the
A
B
to t , which impacts to price level, wage level, profits, income, and
representative
household
denoted
as
V A V 1, p A , s h (t A ), m A to
V B V 1, p B , s h (t B ), m B , where m wl rk . In order to evaluate the welfare change in monetary term, this
study adopts the equivalent valuation as the definition of benefit. The reason is as follows: It is well-known that there are three candidates of benefit definition, Equivalent Variation (EV), Compensating Valuation (CV), and DupuitMarshall definition (DM), and that only EV is a monotone transformation of utility function[Varian(1992)]. EV is the amount of compensation necessary to sustain after-change utility level and give up a good change, which is expressed as the income level necessary to sustain after-change utility level and stay at before-change situation minus beforechange income level. So social benefits defined by EV is
EV e 1, p A , s h (t A ), V B e 1, p A , s h (t A ), V A
situation 1, p
A
h
A
,where e 1, p , s (t ), V A
B
is the income level necessary to sustain utility level V
(8.1) B
and stay at before-change
, s(t A ) . e( ) is called as expenditure function.
We will transform (8.1) to the following form:
EV e 1, p A , s h (t A ), V B e 1, p A , s h (t A ), V A
A eV 1, p A , s h (t A ), V dV VB
V
e [Vp dp Vt dt Vm dm]
(8.2)
A B V
e [ xdp xVOT h dt (ldw kdr )]
A B m
,where subscripts indicate partial derivatives while the superscript h indicates household (=non-business users) and the superscript A and B indicate before-change situation and after-change situation, respectively. The last equality is
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Hisa Morisugi / Transportation Research Procedia 25 (2017) 5144–5159 Author name / Transportation Research Procedia 00 (2017) 000–000
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due to the envelope theorem of (4.7) and (4.8). note that em can be expressed as
Vm (1, p, s h (t ), m) em eV Vm eV 1, p A , s h (t A ), V Vm (1, p, s h (t ), m) Vm (1, p A , s h (t A ), m) So
(8.3)
em may be called as the marginal utility ratio of income, and em 1 at A, and increases toward to B. Note that if
em 1 , then (8.2)
is a typical incidence formula for transportation benefit evaluation except for the introduction of
household’ VOT. It is incidence form because it shows that all the repercussion effects accrued to the household are expressed by the change in freight price and time and income. This form is , however, different from a origin form of transportation benefit formula in a sense that latter can be expressed as the consumers’ surplus with respect to value of time multiplied by freight demand. In order to do so, pay attention to the non-profit condition:
0 1Y 0 C 0 0 pY C
(8.4a) (8.4b) And total differentiate them and apply the envelope theorem (5.7), (5.8), (6.7), (6.8) and (6.9),
0 1dY 0 dC 0 1dY 0 (Cw0 dw Cr0 dr C p0 dp Ct0 dt CY00 dY 0 ) (1 c 0 ( w, r , p, s 0 (t ))dY 0 l 0 dw k 0 dr Xdp XVOT 0 dt
(8.5a)
l 0 dw k 0 dr Xdp XVOT 0 dt
0 pdY Ydp dC pdY Ydp (Cwdw Cr dr Ct dt CY dY ) ( p c( w, r ,1, s(t ))dY Ydp ldw kdr YVOTdt Ydp ldw kdr YVOTdt
(8.6b)
Next insert (8.6)(which is zero) into (8.2),
EV e 1, p A , s h (t A ), V B e 1, p A , s h (t A ), V A
e [ xdp xVOT h dt (ldw kdr ) l 0 dw k 0 dr Xdp XVOT 0 dt Ydp ldw kdr YVOTdt ]
A B m
e [( x X Y )dp (l l 0 l )dw (k k 0 k )dr ) ( xVOT h XVOT 0 ( x X )VOT )dt ]
A B m
e [ x(VOT h VOT ) X (VOT 0 VOT )dt ]
A B m
(8.7) Note that (8.7) is a typical origin formula for transportation benefit evaluation except for the introduction of cargo owners’ and operators’ VOT, where all the repercussion effects expressed by the change in freight price and wage and capital rent disappear due to demand-supply equilibrium shown as the third right hand side of (8.7) and only external effects remains shown as the last right hand side [Morisugi and Ohno (1995)]. In order to apply (8.7) practically it needs to express
em by observable market information and to express (8.7) as the well-known Trapezoidal formula by
second order approximation of (8.7) as
EV
e [ x(VOT h VOT ) X (VOT 0 VOT )]dt
A B m
A hA A A 0A A B A 1 [ x (VOT VOT ) X (VOT VOT )] B (t t ) 2 e m [ x B (VOT hB VOT B ) X B (VOT 0 B VOT B )]
(8.8)
Hisaname Morisugi / Transportation Research Procedia 25 (2017) 5144–5159 Author / Transportation Research Procedia 00 (2017) 000–000
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,where for the last equality we used the fact that e B m
B
e [ x (VOT
hB
B
A
m
11
1 and that the first order approximation of integrand as
B
VOT ) X (VOT 0 B VOT B )]
emA [ x A (VOT hA VOT A ) X A (VOT 0 A VOT A )]
.
d A [em { x A (VOT hA VOT A ) X A (VOT 0 A VOT A )}](t B t A ) dt Next, in order to express em by observable market information, differentiate (8.7) with respect to m,
emB emA
m
[e 1, p A , s h (t A ), V B e 1, p A , s h (t A ), V A ] m e [ x(VOT h VOT ) X (VOT 0 VOT )]dt
A B m
e [ x(VOT h VOT ) X (VOT 0 VOT )]dt em [ xm (VOT h VOT )]dt
A B mm
0 A h A A h A 1 emm [ x(VOT VOT ) X (VOT VOT )] em [ xm (VOT VOT )] B A B (t t ) 2 emm [ x(VOT h VOT ) X (VOT 0 VOT )]B emB [ xm (VOT h VOT )]B
1 emA [ xm (VOT h VOT )]A emB [ xm (VOT h VOT )]B (t B t A ) 2
(8.8) ,where for the third equality we used that X, VOTs are not a function of m, and the last equality we used that B emm
A A emm emmm (m B m A ) 0 , because e Am 1 , e Amm 0 and e Ammm 0 .
(8.9)
The last equality of (8.8) implies that
1 1 [ xm (VOT h VOT )]A (t B t A ) 2 emB emA 1 1 [ xm (VOT h VOT )]B (t B t A ) (8.10) 2 1 1 {[ xm (VOT h VOT )]A [ xm (VOT h VOT )]B }(t B t A ) 2 B Thus we can express em as (8.10) by the market information of non-business freight demand, and VOT for non-
business freight customers and freight operators. The value of
emB is greater than 1 because second term is positive
and it is probably less than 1.3 because the second term can be approximately twice of household expenditure share of the freight consumption if we specify utility function as Cobb-Douglas type. Furthermore a quasi-linear utility function, therefore,
emB is one if we assume
xm 0 , which the conventional practice presumes.
With the derivation of (8.8) and (8.10), we can express the social freight time saving benefit as trapezoidal formula with the weight
emB . Furthermore VOT to be applied to the benefit measurement formula (8.8) is a sum of VOT for
shippers and operators. Therefore we can say that conventional trapezoidal formula with the weight of one and VOT for only operators underestimates the benefit in two points. First is the weight shippers.
e B m and second is neglecting VOT for
Author name / Transportation Research Procedia 00 (2017) 000–000
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Hisa Morisugi / Transportation Research Procedia 25 (2017) 5144–5159
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9. Concluding remarks This paper focused on how value of time and its accrued time saving benefits for freight trips can be expressed and measured in terms of market information such as freight fee, freight time, observable freight service consumption and observable freight service production within a framework of general equilibrium rather than individual user analysis which most previous studies adopt. For general equilibrium analysis we supposed a simple economic society composed of households, two sectors of production, firms producing composite goods, and firms producing freight transportation with a given level of freight time even for freight providers. The reason of above simplicity is for simple and clear derivation of VOT and time saving benefit measurement formula. It is quite possible to construct a more realistic complicated economy with many goods, labor, time constraint and capital, heterogeneous households, government, investment and many regions as SCGE (spatial computable general equilibrium) modeling. Furthermore for freight transportation production sectors it is also possible to introduce multi-mode networks for both person and freight trips, With this complication the basic formula is exactly same as the simple case above except for appearance of many types of VOT and time saving benefit formula. Next in order to satisfy the weak complementarity assumption, we specified freight time externality as product h
0
form of quality index of freight time and freight consumption s (t ) x and s (t ) X into the utility function of nonbusiness users and production function of business users, respectively. And for freight operators the product of quality index of freight time and labour and capital inputs s(t )l , s(t )l are introduced into the production function of freight operators. Then we defined VOT as marginal substitution rate between price and time and expressed it as the freight service fee per freight time multiplied by the elasticity of quality level indicator with respect to freight time for each of nonbusiness users and business users, respectively. The freight service providers’ VOT is expressed as average labour and capital cost per freight time. Thus we can measure three VOTs in terms of market information such as freight fee, freight time and average labour and capital cost and the elasticity of quality level indicator with respect to freight time although we need to estimate quality index of freight time by estimating freight demand functions of users. The author believes it is the first time to succeed in defining VOT for shippers and their expression by observable market information. As for the VOT for operators, our derivation is similar to the conventional practice. Next within the framework of general equilibrium rather than individual behaviour analysis we derived a “origin” formula to measure social freight time saving benefits including its repercussion effects expressed by the change in equilibrium freight price, wage and capital rent, where we adopted the concept of equivalent variation as the social benefit definition. The formula is the integration of sum of weighted marginal freight time saving benefit functions with respect to simply freight time only, where the weight is the marginal utility ratio of income, and sum of marginal freight time saving benefit function for non-business and business users are the product of freight demand function and sum of VOTS for users and operators. Thus proposed origin (or shortcut) formula is the weighted general equilibrium consumers’ surplus of the freight demand multiplied by VOTs with respect to freight time. The author believes it is also the first time to succeed in origin formulation of only consumers’ surplus of freight demand function in spite of that there are repercussion effects over all the market from the freight market. We noted that all the repercussion effects expressed by the change in freight price and wage disappear due to demand-supply equilibrium. Also we noted that the marginal utility ratio of income after the change can be approximately expressed as 1 + (income effect of marginal freight time saving benefit for non-business), so that it is also possible to calculate from the market information. Furthermore VOT to be applied to the benefit measurement formula (8.8) is a sum of VOT for shippers and operators. Therefore we can say that conventional trapezoidal formula with the weight of one and VOT for only operators underestimates the benefit in two points. First is missing the weight
e B m and second is neglecting VOT for shippers.
As for the generalization of product form of quality index of freight time, we can say to be able to express VOT
and marginal time saving benefit by the demand functions with the exactly same method as travel cost method. However in practice many cases specify our product form to satisfy the weak complementarity, so that it seems to be
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not very important to generalize it. Acknowledgements This work was supported by JSPS KAKENHI Grant numbers 23360218, 26289168. References Bates, J. and Roberts. M (1986) Value of Time Research: Summary of Methodology and Findings, Paper presented at the 14 th PTRC Summer Annual Meeting, University of Sussex, UK, pp14-18, July 1986 Bates, J., Polak, J., Jones, P. and Cook, A. (2001) The Valuation of Reliability for Personal Travel, Transportation Research Part E: Logistics and Transportation Review, Vol.37, Issue.2-3, pp.191-229. Becker, G.S. (1965) A Theory of the Allocation of Time, The Economic Journal, Vol.75, No.299, pp.493-517. Blayac, T. and Causse, A. (2001) Value of Travel Time: A Theoretical Legitimization of Some Nonlinear Representative Utility in Discrete Choice Models, Transportation Research Part B: Methodological, Vol.35, Issue.4, pp.391-400. Bockstael, N.E., I.E. Strand and W.M. Hanemann (1987) Time and the Recreation Demand Model, American Journal of Agricultural Economics, Vol.69, No.2, pp.293-302. Bruzelius, N.(1979) The Value of Travel Time: Theory and Practice, Groom Helm London, Surrey. .Cesario, F.J. (1967) Value of Time in Recreation Benefit Studies, Land Economics, Vol.52, No.1, pp.32-41 . Chiswick, B.R. (1967) The Economic Value of Time and the Wage Rate: Comment, Western Eonomic Journal, March 1967, pp.294-295. .Clawson, M. (1959) Methods of Measuring the Demand for and Value of Outdoor Recreation, Resources for the Future, Reprint No.10, Washington, DC. . De Donnea, F.X. (1972) Consumer Behaviour, Transport Mode Choice and Value of Time: Some Micro-economics Models, Reginal Science and Urban Economics, Vol. 1. No. 4, pp.355-382 Department for Transport UK (UKDOT) (2011) Value of Time and Operating Costs, TAG Unit 3.5.6. Transport Analysis Guidance, April 2011
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De Serpa, A.C. (1971) A Theory of the Economics of Time, The Economic Journal, Vol.81, No.324, pp.828-846. Evans, A. (1972) On the Theory of the Valuation and Allocation of Time, Scottish Journal of Political Economy, Vol.19. No.1, pp.1-17 Feo-Valero, M., Garcia-Menendez, M., and Garrido-Hidalgo, R. (2011) Valuing Freight Transport Time Using Transport Demand Modeling: a Bibliographical Review, Transport Reviews, Vol.31, No.5, pp.625-651. Gonzalez, R. M. (1997) The Value of Time, a Theoretical Review, Transport Reviews, Vol.17, No. 3, pp. 245-266 Hensher, D.A. (2001) Measurement of Valuation of Travel Time Savings, Journal of Transport Economics and Policy, Vol.35, Part.1, pp.71-98. Jara-Diaz, S.R. (2003) On the Goods-Activities Technical Relations in the Time Allocation Theory, Transportation, Vol.30, No.3, pp.245-260. Jara-Diaz, S.R. and Guevara, A. (2003) Behind the Subjective Value of Travel Time Savings: The Perception of Work, Leisure and Travel from a Joint Mode Choice-Activity Model, Journal of Transport Economics and Policy, Vol.37, No.1, pp.29-46. Jiang, M. and Morikawa, T. (2004) Theoretical Analysis on the Variation of Value of Travel Time Savings, Transportation Research Part A: Policy and Practice, Vol.38, Issue.8, pp.551-571. Knetsch, J.L. (1963) Outdoor Recreation Demands and Benefits, Land Economics, Vol.39, No.4, pp.387-396 .Katoh, H.(2013) Value of Travel Time: Theory and Practice, Gihodoh Larson, D.M. (1992). Further Results on Willingness to Pay for Nonmarket Goods, Journal of Environmental Economics and Management, Vol.23, pp. 101-122. Larson, D.M. and S. Shaikh (2001). Empirical Specification Requirements for Two-Constraint Models of Recreation Choice, American Journal of Agricultural Economics, Vol. 83, No.2, pp.428-440. Larson, D.M. and S.L Shaikh (2004). Recreation Demand Choices and Revealed Values of Leisure Time, Economic Inquiry, Vol.42, No.2, pp.264-278. Mackie, O.J., Jara-Diaz, S. and Fowkes, A.S. (2001) The Value of Travel Time Savings in Evaluation,
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Transportation Research Part E: Logistics and Transportation Review, Vol.37, Issue.2-3, pp.91-106. Mäler, K. (1971). A Method of Estimating Social Benefits from Pollution Control, Swedish Journal of Economics, Vol. 75, pp.121-133. Mäler, K. (1974) Environmental Economics: a Theoretical Inquiry, John Hopkins University Press for the Resource for the Future, Baltimore McConnell, K.E. (1975) Some Problems in Estimating the Demand for Outdoor Recreation, American Journal of Agricultural Economics, Vol.57, No.2, pp.330-334. McConnell, K.E. and I.E. Strand (1981) Measuring the Cost of Time in Recreation Demand Analysis: An Application to Sportfishing, American Journal of Agricultural Economics, Vol.63, No.l, pp.153-156. Ministry of Land, Infrastructure, Transportation and Tourism Japan (JAPANMLITT) (2009) , Technical Guidance for Cost Benefit Analysis of Infrastructure Projects Morey, E.R. (1984) The Choice of Ski Areas: Estimation of a Generalized CES Preference Ordering with Characteristics, Review of Economics and Statistics, Vol.66, No.4, pp.584-590. Morisugi , H and E. Ohno (1992) A Benefit Incidence Matrix for Urban Transport Improvement Papers in Regional Science: The Journal of the Regional Science Association International, Vol.71, No.1, pp.53-70 Morisugi,H., E. Ohno and T. Miyagi(1993) Benefit Incidence of Urban Ring Road －Theory and Case Study of the Gifu Ring Road Transportation 20, pp.285-303, Kluwer Academic Publishers Morisugi , H and E. Ohno (1995) Proposal of Benefit Incidence Matrix for Urban Development Projects Regional Science and Urban Economics, No.25, pp.461-481 Neill, J.R. (1988). Another Theorem on using Market Demands to Determine Willingness to Pay for Nontraded Goods, Journal of Environmental Economics and Management, Vol.15, pp.224-232. Small, K. A., C. Winston and J. Yang (2005) Uncovering the distribution of Motorists7 Preferences for Travel Time and Reliability, Econometrica, Vol.73, No.4, pp.1367-1382 Smith, V.K., W.H. Desvousges and M.P. McGivney (1983) The Opportunity Cost of Travel Time in Recreation Demand Models, Land Economics, Vol.59, No.3, pp.259-277. Economic Journal, Vol.64, No.3, pp.511-527. Train, K. and McFadden, D. (1978) The Goods/Leisure Trade off and Disaggregate Work Trip Mode Choice Models,
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Transportation Research, Vol.12, pp.349-353 . Truong, T.P. and Hensher, D.A. (1985) Travel Time Values and Opportunity Cost from a discrete Choice Model, The Economic Journal, Vol.95, No.324, pp.828-846 . U.S. Department of Transportation(USDOT)(2011) The Value of Time Savings: Departmental Guideline for Conducting Economic Evaluations Revision 2 Varian, H. R. (1992) Microeconomic Analysis, Norton Wardman, M.A. (2001) Review of British Evidence on Time and Service Quality Valuation, Transportation Research Part E: Logistics and Transportation Review, Vol.37, Issue.2-3, pp.107-128. Winston, C. (1983) The Demand for Freight Transportation: Models and Applications, Transportation Research Part A. Vol.17, No.6. pp.419-427
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World Conference on Transport Research - WCTR 2016 Shanghai. 10-15 July 2016 Measurement of Value of Time for Freight Trips and its Benefit by Market Information
Hisa Morisugi* N h on U n iv e rsit y , T ok y o, J ap an
Abstract This paper focuses on how value of time and its accrued time saving benefits for freight trips can be expressed and measured in terms of market information such as freight fee, freight time, observable freight service consumption and observable freight service production within a framework of general equilibrium rather than individual user analysis which most previous studies adopt. In order to satisfy the weak complementarity assumption, we specifies freight time h 0 externality as product form of quality index of freight time and freight consumption s (t ) x and s (t ) X into the utility function of non-business users and production function of business users, respectively. And for freight operators the product of quality index of freight time and labour and capital inputs s (t )l , s (t )k are introduced into the production function of freight operators. Then we define VOT as marginal substitution rate between price and time and expressed it as the freight service fee per freight time multiplied by the elasticity of quality level indicator with respect to freight time for each of non-business and business users, respectively. The freight service providers’ VOT is expressed as average labour and capital cost per freight time. Thus we can measure three VOTs in terms of market information such as freight fee, freight time and average labour and capital cost and the elasticity of quality level indicator with respect to freight time although we need to estimate quality index of freight time by estimating freight demand functions of users. Next we derive an “origin” formula to measure social freight time saving benefits including its repercussion effects expressed by the change in equilibrium freight price, wage and capital rent, where we adopted the concept of equivalent variation as the social benefit definition. The proposed origin (or shortcut) formula is the weighted general equilibrium consumers’ surplus of the freight demand multiplied by VOTs with respect to freight time. The weight after the change can be approximately expressed as 1 + (income effect of marginal freight time saving benefit for non-business), so that it is also possible to calculate from the market information. Furthermore VOT to be applied to the benefit measurement formula is a sum of VOT for shippers and operators. Therefore we can say that conventional trapezoidal formula with the weight of
* Corresponding author. Tel.: 81-22-777-5620; fax: 81-22-777-5620. E-mail address: [email protected] 2214-241X © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of WORLD CONFERENCE ON TRANSPORT RESEARCH SOCIETY.
2352-1465 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of WORLD CONFERENCE ON TRANSPORT RESEARCH SOCIETY. 10.1016/j.trpro.2018.02.043
2
Author name / Transportation Research Procedia 00 (2017) 000–000
Hisa Morisugi / Transportation Research Procedia 25 (2017) 5144–5159 2 name / Transportation Research 00 (2017) 000–000the weight one and VOT for only operators Author underestimates the benefit in two Procedia points. First is missing
5145
emB and second is
neglecting VOT for shippers. B one and VOT for only operators underestimates the benefit in two points. First is missing the weight em and second is neglecting VOT for shippers. © 2017 The Authors. Published by Elsevier B.V. © 2017 The Authors. Published by Elsevier B.V. Peer-review WORLD CONFERENCE CONFERENCE ON ON TRANSPORT TRANSPORT RESEARCH RESEARCH SOCIETY. SOCIETY. Peer-review under under responsibility responsibility of of WORLD © 2017 The Authors. Published by Elsevier B.V. Peer-review under of WORLD CONFERENCE ON TRANSPORT RESEARCH Keywords: value of responsibility time for freight service users, value of time for freight service providers,SOCIETY. freight time saving benefit measurement, marginal time saving benefit, marginal utility ratio of income, general equilibrium Keywords: value of time for freight service users, value of time for freight service providers, freight time saving benefit measurement, marginal time saving benefit, marginal utility ratio of income, general equilibrium
1. Introduction 1. Introduction The subjective value of time for freight is the marginal rate of substitution between freight time and freight cost under constant utility or profit, or cost level, which is defined for freight service consumers as well as freight service The subjective value ofreferred time fortofreight is the marginal rate of ofvalue substitution freight time cost providers. It is commonly as the monetary appraisal of timebetween or the willingness to and pay freight for savings under constant in freight time. utility or profit, or cost level, which is defined for freight service consumers as well as freight service providers. It is commonly to as appraisaltime of value time or the pay savings This paper focuses on referred how value of the timemonetary and its accrued savingofbenefits for willingness freight tripstocan befor expressed in time.in terms of market information such as freight fee, freight time, observable freight service consumption andfreight measured paper focuses how value of time and its accrued timeofsaving for freight be expressed and This observable freight on service production within a framework generalbenefits equilibrium rathertrips thancan individual user and measured in terms of market information such as freight fee, freight time, observable freight service consumption analysis which most previous studies adopt. andFor observable freight service production within a framework of society general composed equilibrium than individual user general equilibrium analysis we suppose a simple economic of rather households, two sectors of analysis which most previous studies adopt. production, firms producing composite goods, and firms producing freight transportation with a given level of freight Foreven general equilibrium analysis suppose a simple economic society composed households, two sectors of time for freight providers. Thewe reason of above simplicity is for simple and clearofderivation of VOT and time production, firms producing composite goods, and firms producing freight transportation with a given level of freight saving benefit measurement formula. It is quite possible to construct a more realistic complicated economy with many time even for time freight providers. reason of above simplicity is for government, simple and clear derivation VOTregions and time goods, labor, constraint andThe capital, heterogeneous households, investment andofmany as saving benefit measurement formula. It is quite possible to construct a more realistic complicated economy with many SCGE (spatial computable general equilibrium) modeling. Furthermore for freight transportation production sectors goods, timetoconstraint capital, networks heterogeneous households, and many regions as it is alsolabor, possible introduce and multi-mode for both person andgovernment, freight trips,investment With this complication the basic SCGE (spatial computable general equilibrium) modeling. Furthermore for freight transportation production sectors formula is exactly same as the simple case above except for appearance of many types of VOT and time saving benefit it is also possible to introduce multi-mode networks for both person and freight trips, With this complication the basic formula. formula same as the above except for appearance many types of VOT time saving benefit Nextisinexactly order to satisfy thesimple weakcase complementarity assumption, weofspecifies freight timeand externality as product formula. h 0 s (t )we x and s (t ) Xfreight into time the utility function of nonformNext of quality index of freight time and freight consumption in order to satisfy the weak complementarity assumption, specifies externality as product business users and production function of business users, respectively. And for0 freight operators the product of quality h form of quality index of freight time and freight consumption s (t ) x and s (t ) X into the utility function of nonindex of freight time and labour and capital inputs s (t )l , s (t )k are introduced into the production function of freight business users and production function of business users, respectively. And for freight operators the product of quality operators. index of freight time and labour and capital inputs s (t )l , s (t )k are introduced into the production function of freight Then we define VOT as marginal substitution rate between price and time and expressed it as the freight service operators. fee per freight time multiplied by the elasticity of quality level indicator with respect to freight time for each of nonThen users we define VOT as marginal substitutionThe ratefreight between priceproviders’ and time and expressed it as the freight service business and business users, respectively. service VOT is expressed as average labour fee per freight multiplied the elasticity ofelasticity quality level indicator respectwith to freight for each of Thus nonand capital costtime per freight timeby multiplied by the of quality levelwith indicator respecttime to freight time. business users and business users, The freightsuch service providers’ VOT time is expressed labour we can measure three VOTs in termsrespectively. of market information as freight fee, freight and sumas of average average labour and capital time multiplied the elasticity quality level withalthough respect to time. Thus and capital cost cost per andfreight the elasticity of qualitybylevel indicator of with respect to indicator freight time wefreight need to estimate we can measure three VOTs in terms of market information such as freight fee, freight time and sum of average labour quality index of freight time by estimating freight demand functions of users. The author believes it is the first time to and capital cost and VOT the elasticity of quality levelexpression indicator with respect to freight although As we for needthetoVOT estimate succeed in defining for shippers and their by observable market time information. for quality index of freight time by estimating freight demand functions of users. The author believes it is the first time to operators, our derivation looks similar to the conventional practice. succeed defining VOT for shippers andequilibrium their expression observable information. As derive for theaVOT for Next in within the framework of general ratherby than individualmarket behaviour analysis we “origin” operators, our derivation looks similar to the conventional practice. formula to measure social freight time saving benefits including its repercussion effects expressed by the change in Next within the price, framework general than individual behaviour analysisvariation we derive “origin” equilibrium freight wage of and capitalequilibrium rent, whererather we adopted the concept of equivalent as athe social formuladefinition. to measureThe social freight saving benefits including its repercussion effects expressed by the change in benefit formula is time the integration of sum of weighted marginal freight time saving benefit functions equilibrium price, wage andonly, capital rent, weisadopted the concept of equivalent social with respect freight to simply freight time where thewhere weight the marginal utility ratio of income,variation and sumasofthe marginal benefit time definition. formula is theforintegration of sum weighted marginal time saving demand benefit functions freight savingThe benefit function non-business andofbusiness users are thefreight product of freight function with respect to simply freight time only, where the weight is the marginal utility ratio of income, and sum of marginal and sum of VOTS for users and operators. Thus proposed “origin” (or “shortcut”) formula is the weighted general freight time consumers’ saving benefit function forfreight non-business businessby users arewith the product equilibrium surplus of the demandand multiplied VOTs respect of to freight freight demand time. Thefunction author and sum of VOTS for users and operators. Thus proposed “origin” (or “shortcut”) formula is the weighted general equilibrium consumers’ surplus of the freight demand multiplied by VOTs with respect to freight time. The author
5146
Hisa Morisugi / Transportation Research Procedia 25 (2017) 5144–5159 Author name / Transportation Research Procedia 00 (2017) 000–000
3
believes it is also the first time to succeed in origin formulation of only consumers’ surplus of freight demand function in spite of that there are repercussion effects originated from the freight market toward all the market. We note that all the repercussion effects expressed by the change in freight price and wage disappear due to demand-supply equilibrium. Also we note that the marginal utility ratio of income after the change can be approximately expressed as 1 + (income effect of marginal freight time saving benefit for non-business), so that it is also possible to calculate from the market information. Furthermore VOT to be applied to the benefit measurement formula (8.8) is a sum of VOT for shippers and operators. Therefore we can say that conventional trapezoidal formula with the weight of one and VOT for only operators underestimates the benefit in two points. First is missing the weight
e B m and second is neglecting VOT for shippers.
2. Previous studies and practice Suppose a light goods vehicle delivering cargoes ordered by some households and firms with the freight time t and freight price p. Many guidelines suggest that; (1) VOT for this transport producing sector is the imputed wage of driver if it is a non-business trip [Japan MLITT (2008), UKDOT (2011), USDOT (2011), Katoh (2013)]. VOT is the wage rate of driver plus the opportunity cost of vehicle per unit of time [Japan MLITT (2008)], or the wage rate of driver [UKDOT (2011), USDOT (2011)] if it is a business trip. But very few guidelines suggest that; (2) VOT for cargo owners, nor say that: (3) Total VOT for a light goods vehicle delivering cargoes be sum of VOT for transport producing sector plus VOT for cargo owners multiplied by the delivered cargo volume. The first statement (1) by Japan MLITT (2008) above can be shown as appropriate (see equation (6.11)), but very few previous theoretical studies say the definition and measurement of VOT for transportation producers. Rather they pay attentions to the transportation users’ VOT without explicitly introducing the transportation production [Becker(1965), De Serpa (1971), De Donnea (1972), Evans (1972), Bruzerius (1979), Gonzalez (1997), Bates and Roberts (1986), Mackie, Jara-Diaz. and Fowkes (2001), Jara-Diaz (2003), , Winston and Yang (2005) and Katoh(2013)]. Although sometimes they incorporate a simple Leontief type of household transportation production function, that is only applicable to private car transportation. Thus it needs to develop the theory of VOT for transportation producers. As for the second point of (2) there are very few studies on definition and measurement of VOT for cargo owners, which can be classified into two, opportunity cost approach and demand modeling approach [Winston(1981), Katoh(2013) ]. Japan MLITT (2008) adopts opportunity cost approach and the interest cost for the inventory of the cargo value transported for VOT for cargo owners, yet it does not justify theoretically. Feo-Valero et al.(2011) surveys 22 studies of demand modeling approach, and says that theoretically who’s VOT is not clearly distinguished yet, and practically large difference of VOTs for road, railroad, seaborne, airborne were estimated. Thus it needs to develop the theory of VOT for cargo owners, distinctly from transportation producers. Therefore we need a framework of general equilibrium including freight production sectors, but previous studies based on general equilibrium did not explicitly deal with freight time [Morisugi and Ohno(1992), Morisugi, Ohno and Miyagi(1993), Morisugi and Ohno(1995) ]. It is worthwhile to look at the concept of VOT as a commodity developed in the field of person trips as a starting reference for freight service consumers’ VOT [De Serpa (1971), De Donnea (1972), Evans (1972), Train and McFadden(1978), Bruzerius(1979), Bates and Roberts(1985), Truong and Hensher(1985), Mackie, Jara-Diaz and Fowkes (2001), Jara-Diaz (2003) and Katoh(2013)]. VOT as a commodity is derived by incorporating the total transportation time consumption to the utility function (production function) in addition to the transportation service consumption itself. Unfortunate this specification of transportation time is not fit to the application to VOT for cargo owners. Because it seems that shippers do not necessarily get a utility (productivity) from the total transportation time in addition to transportation service consumption. Rather we need the VOT as a quality instead of as a commodity. Another field we have to look at is a recreational benefit evaluation by travel cost method [Knetsch (1963), Larson
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(1992), Bockstael, Strand and Hanemann (1987), Clawson (1959), Neill, J.R. (1988), Larson and Shaikh (2004), Mäler (1974)]. Conceptually we wish to apply this method to our shippers’ VOT measurement by replacing the quality of recreation site to the quality index of freight time as shown in this paper, but it should be within a framework of general equilibrium because we need to deal with operators’ VOT as well as shippers’ VOT. 3. Economy with freight transportation production Consider an economic society composed of households, two sectors of production, firms producing composite goods with unitary price, and firms producing freight transportation with a given level of freight time. A representative household provides labor and capital services in exchange for wages and rents, purchases composite goods and freight transportation services with a given level of freight time. One representative firm produces composite goods, pays wages and rents for labor and capital input, and pays the freight service fees for freight input. Another representative firm produces freight transportation with a given level of freight time, pays wages and rents for labor and capital input, and pays for input of composite goods. It is assumed that the production functions of both firms exhibit constant returns to scale, so that zero profit condition holds at equilibrium. Society is at equilibrium in the long run with respects to composite goods, freight services, labors and capitals, so that zero profit and constant returns to scale hold. This paper assumes that freight time per trip is an externality as a kind of public goods for both of freight consumers and producers, just like case of road transportation sector with a given level of freight time, determined by the road condition constructed by the government. The reason of above simplicity is for simple and clear derivation of VOT and time saving benefit measurement formula. It is quite possible to construct a more realistic complicated economy with many goods, labor, time constraint and capital, heterogeneous households, government, investment and many regions as SCGE (spatial computable general equilibrium) modeling. Furthermore for freight transportation production sectors it is also possible to introduce multi-mode networks for both person and freight trips. With this complication the basic formula is exactly same as the simple case above except for appearance of many types of VOT and time saving benefit formula. 4. Nonbusiness freight service consumption modeling and its VOT Nonbusiness freight service consumption modeling is formulated by households’ utility maximizing behavior under the budget constraint as:
V 1, p, s h (t ), m max u z, s h (t ) x
(4.1a)
z, x
s.t. z px m
(4.1b)
, where u ( ) : direct utility function z : composite goods consumption with price 1 x : non-business freight transportation consumption with price p and time h
h
t
s (t ) : quality level indicator of freight time t . Assume s (t ) 0 and ds (t ) / dt sth 0 (4.1c) m wl rk : income level w : wage l : labour supply (fixed) r : capital rent k : capital supply (fixed)
V
: indirect utility function
h
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Note that, first, there is no time constraint because it assumes that this economy has not any person trips. h h Second, specification of utility function is as u z , s (t ) x instead of more general form as u z , x, s (t ) . The latter form is identical for measuring recreational benefit of recreational quality improvement by travel cost method under the weak complementarity assumption which says that if x 0 then marginal utility of t is zero [Mäler (1971), Mäler (1974)]. The former form this study adopts is sufficient to satisfy the weak complementarity assumption as it h is the product form of s (t ) x . This specification is often adopted with the CES utility function in the environmental economics [von Faefen and Phaneuf (2003)]. Thus this study intends to apply the recreational benefit measurement of travel cost method to the freight time saving benefit. Furthermore it will be shown in this paper that for the product form it is not necessary to apply travel cost method, since the benefit can be expressed as integral of demand function multiplied by value of time (VOT) with respect to time instead of price. As for whether or not the freight time increases the utility of freight service is an empirical matter. If as a result it shows that there is not any marginal utility of freight h h time, then set s (t ) 1 , and the theory and discussion remains as same as s (t ) 1 . With comments above to solve (4.1), let Lagrangian be as
L u z , s h (t ) x m z px
(4.2)
And its first order conditions (FOC) are
ux u sh p uz and sh x
(4.3)
, where subscriptions indicate partial differentiation with respect to the subscript while superscript h indicates household or non-business users. And its demand functions (4.4), marginal utility of income (4.5) and indirect utility function (4.6) are expressed as
z z 1, p, s h (t ), m
(4.4a)
x x 1, p, s h (t ), m
(4.4b)
1, p, s h (t ), m
(4.5)
V V 1, p, s h (t ), m
(4.6)
Envelop Theorem implies
Vm , Vp Vm x
(4.7)
h p h p st t h Vt u xs xst Vm x (4.8) h Vm x p / t ( 0) h sh x s t s ,where h (stht / s h ) : elasticity of quality level indicator with respect to freight time t . Subscriptions indicate h t
partial differentiation with respect to the subscript while superscript h indicates household or non-business users. The second equality of (4.8) is due to (4.3). Marginal freight time saving benefit of non-business freight users (=households) is defined as the value of marginal utility of time saving (Vt / Vm ) , which can be expressed as minus freight consumption multiplied by VOT, where h VOT is defined as the marginal freight time saving benefit divided by freight consumption MB / x . Therefore,
,where
MB h (Vt / Vm ) x p / t h xVOT h
(4.9)
VOT h MB h / x p / t h
(4.10)
(4.10) of VOT formula can be derived also by defining the value of time (VOT) as marginal substitution rate between price and time and apply (4.7) and (4.8)
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h
h
(4.11)
(4.11) shows that VOT for non-business freight users can be expressed as the freight service fee per freight time multiplied by the elasticity of quality level indicator with respect to freight time t . It seems that there are three types of non-business freight service, private cars, parcel delivery service and house moving transportations. The elasticity of quality level indicator with respect to freight time t can be estimated by estimating the non-business demand functions for each type of non-business freight service. For Japanese parcel delivery service, its average delivery price is 1200 yen, and its average delivery time is 20 hours (1200 minutes). Now suppose the elasticity of the elasticity of quality level indicator with respect to freight time t is -0.2. Then VOT for shippers is 0.2 yen/minute. 、 5. Business freight service consumption modeling and its VOT Business freight service consumption modelling is formulated by cost minimization of composite goods production firms under the constant return to scale production function with inputs of labour, capital and freight transportation as:
c w, r , p, s 0 (t ) Y 0 0
min C 0 wl 0 rk 0 pX 0 0 l , k ,X
s.t. Y 0 f 0 l 0 , k 0 , s 0 (t ) X
(5.1a) (5.1b)
0
,where Y ：output level of composite goods
l 0 ：labour input w ：wage k 0 : capital input r : capital rent
X ：freight service input with its price p and time t s (t ) : quality level indicator of freight time t . Assume s 0 (t ) 0 and ds0 (t ) / dt st0 0 (5.1c) 0
f0
： composite goods production function (constant return to scale).
Superscript 0:composite production sector As same as non-business case, it is again assumed for business freight demand that shorter freight time increases 0
the productivity of freight service in the form of product form s (t ) X , which is sufficient to satisfy the weak complementarity assumption. The Lagrangian (5.2), its first order conditions (5.3), conditional input demand functions (5.4) and cost function (5.5) will be
min L0 wl 0 rk 0 pX 0 [Y 0 f 0 l 0 , k 0 , s(t ) X ] 0 l , k , X , 0
0 0 p f X 0 f s00 X s 0 w 0 f l 00 , r 0 f k00 ,
(5.2) (5.3)
l 0 l 0 ( w, r , p, s 0 (t ))Y 0
(5.4a)
k 0 k 0 (w, r , p, s 0 (t ))Y 0
(5.4b)
0
X X (w, r , p, s (t ))Y 0
0
0
0
C c ( w, r , p, s (t ))Y
0
(5.4c) (5.5)
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0 0 (w, r , p, s 0 (t ))
7
(5.6)
And by envelop theorem 0 0 c 0 ( w, r , p, s 0 (t )) Cw0 l 0 , Cr0 k 0 , C p0 X , CY
(5.7)
0
and
C 0t 0 f s00 X Xst0 X ( p / s 0 ) st0 X ( p / t ) 0 ( 0)
(5.8)
(5.8) is the marginal cost reduction of freight time saving, which is its marginal time saving benefit 0
of business users (=composite goods production sector) MB :
MB0 C 0t X ( p / t ) 0 ( XVOT 0 )
(5.9)
0
And VOT is defined as the marginal freight time saving benefit divided by freight consumption MB / X . VOT 0 MB 0 / X p / t 0 (5.10) (5.10) of VOT formula can be derived also by defining the value of time (VOT) as marginal substitution rate between price and time and apply (5.7) and (5.8) :
Ct0 dp ( p / t ) 0 VOT 0 0 dt C 0 const C p
(5.11)
(5.10) and (5.11) show that VOT for business freight users can be expressed also as the freight service fee per freight time multiplied by the elasticity of quality level indicator with respect to freight time. There are many types of business freight service for each type of cargo owner of different business sectors. The elasticity of the elasticity of quality level indicator with respect to freight time t can and should be estimated by estimating the business demand function for each type of business freight service and for each type of business shippers. Suppose that Japan-west coast of USA container shipment of 1 TEU has fee of 180,000 yen/unit, and 300 hours (180000 minutes) of shipment time, and 0.2 of the elasticity of quality level indicator with respect to freight time t . Then VOT for shipper is again 0.2 yen /minute. 6. Freight service production modelling and its VOT Freight service production modelling is formulated by cost minimization of freight service production firms under the constant return to scale production function with inputs of labour, capital and composite goods as:
c w, r , p, s(t ) Y C min wl rk Z l , k ,Z
s.t. Y f s (t )l , s (t )k , Z
(6.1a) (6.1b)
, where Y ：output of freight service, l ：labour input w ：wage rate k ：capital input r ：capital rent Z ：input of composite goods t ：freight time(exogenous) s (t ) : quality level indicator of freight time t . Assume s(t ) 0 and ds(t ) / dt st 0
f
:
(6.1c)
freight service production function (constant returns of scale).
Note (6.1b) shows that freight time reduction increases the labour and capital productivity, but not for another
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input represented by composite goods. Thus (6.1b) assumes that freight production cost is a sum of time dependent cost (represented by labour and capital cost) and distance dependent cost independent of time (represented by composite goods input cost). The Lagrangian (6.2), its first order conditions (6.3), conditional input demand functions (6.4) and cost function (6.5) will be
min L
l , k ,Z ,
wl rk Z [Y f s(t )l 0 , s(t ) k 0 , Z ] w f sl s , r f sk s , 1 f Z
(6.2) (6.3)
l l (w, r ,1, s(t ))Y
(6.4a)
k k (w, r ,1, s(t ))Y
(6.4b)
Z Z (w, r ,1, s(t ))Y
(6.4c)
C c(w, r,1, s(t ))Y
(6.5)
(w, r,1, s(t ))
(6.6)
And by envelop theorem , and
Cw l , Cr k , CY c( w, r ,1, s(t ))
(6.7)
wl rk wl rk 1 st t Ct ( f sl l f sk k ) st [( w / s )l (r / s )k )]st Y Y ( 0) t Y t s
(6.8) , where the second and forth equality are due to (6.3) and (6.4), respectively. The “~” indicates unit input so that
wl rk is the unit labour and capital cost.
(6.8) is the marginal cost reduction of freight time saving, which is its marginal time saving benefit of operators MB :
wl rk Y MB Ct YVOT ) ( t
(6.9)
And VOT is defined as the marginal freight time saving benefit divided by freight production
wl rk VOT MB / Y t
MB / Y .
(6.10)
(6.10) shows that VOT for freight operators can be expressed as the unit labour and capital cost of freight service production per freight time multiplied by the elasticity of quality level indicator with respect to freight time. It seems quite reasonable to assume s (t ) is proportional to speed (delivery distance over delivery time) in case of operator. Then the elasticity is -1. Then the VOT is:
wl rk VOT t
(6.11)
Consider a light goods vehicle delivering cargoes ordered by some households and firms with the freight time t and freight price p. Y is one because it is measured by one truck. And assume one driver for that vehicle so that labour hour is t. Another cost is truck depreciation per time, Then VOT for operator is sum of the wage rate of driver and truck depreciation per time. Thus we can justify Japan MLITT (2008) in a sense that it values the opportunity cost of car while other manuals of UKDOT (2011) and USDOT (2011) underestimates [Katoh (2013)] Another way to estimate VOT for operator is to look at fact that freight price equals to unit cost of freight service
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in the long run, and it can be divided into the freight time dependent cost and transportation distance dependent cost. VOT for operator is the former cost per freight time, which is mainly composed of some part of labour cost and capital cost (drivers and cars), which is usually around 0.6. 7. Market equilibrium Market equilibrium in the economy is for composite goods, freight transportation, labor and capital shown as,
zZ Y0
x X Y
l0 l l 0 k k k
, and for zero profit condition shown as
1 c 0 ( w, r , p, s 0 (t ))
p c(w, r,1, s(t )) , where the “~” indicates unit cost. Unknown variables are
w, r , p, Y 0 , Y with numeraire of composite goods.
8. Social benefits of freight time saving Consider a change in freight time from t indirect
utility
level
of
the
A
B
to t , which impacts to price level, wage level, profits, income, and
representative
household
denoted
as
V A V 1, p A , s h (t A ), m A to
V B V 1, p B , s h (t B ), m B , where m wl rk . In order to evaluate the welfare change in monetary term, this
study adopts the equivalent valuation as the definition of benefit. The reason is as follows: It is well-known that there are three candidates of benefit definition, Equivalent Variation (EV), Compensating Valuation (CV), and DupuitMarshall definition (DM), and that only EV is a monotone transformation of utility function[Varian(1992)]. EV is the amount of compensation necessary to sustain after-change utility level and give up a good change, which is expressed as the income level necessary to sustain after-change utility level and stay at before-change situation minus beforechange income level. So social benefits defined by EV is
EV e 1, p A , s h (t A ), V B e 1, p A , s h (t A ), V A
situation 1, p
A
h
A
,where e 1, p , s (t ), V A
B
is the income level necessary to sustain utility level V
(8.1) B
and stay at before-change
, s(t A ) . e( ) is called as expenditure function.
We will transform (8.1) to the following form:
EV e 1, p A , s h (t A ), V B e 1, p A , s h (t A ), V A
A eV 1, p A , s h (t A ), V dV VB
V
e [Vp dp Vt dt Vm dm]
(8.2)
A B V
e [ xdp xVOT h dt (ldw kdr )]
A B m
,where subscripts indicate partial derivatives while the superscript h indicates household (=non-business users) and the superscript A and B indicate before-change situation and after-change situation, respectively. The last equality is
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due to the envelope theorem of (4.7) and (4.8). note that em can be expressed as
Vm (1, p, s h (t ), m) em eV Vm eV 1, p A , s h (t A ), V Vm (1, p, s h (t ), m) Vm (1, p A , s h (t A ), m) So
(8.3)
em may be called as the marginal utility ratio of income, and em 1 at A, and increases toward to B. Note that if
em 1 , then (8.2)
is a typical incidence formula for transportation benefit evaluation except for the introduction of
household’ VOT. It is incidence form because it shows that all the repercussion effects accrued to the household are expressed by the change in freight price and time and income. This form is , however, different from a origin form of transportation benefit formula in a sense that latter can be expressed as the consumers’ surplus with respect to value of time multiplied by freight demand. In order to do so, pay attention to the non-profit condition:
0 1Y 0 C 0 0 pY C
(8.4a) (8.4b) And total differentiate them and apply the envelope theorem (5.7), (5.8), (6.7), (6.8) and (6.9),
0 1dY 0 dC 0 1dY 0 (Cw0 dw Cr0 dr C p0 dp Ct0 dt CY00 dY 0 ) (1 c 0 ( w, r , p, s 0 (t ))dY 0 l 0 dw k 0 dr Xdp XVOT 0 dt
(8.5a)
l 0 dw k 0 dr Xdp XVOT 0 dt
0 pdY Ydp dC pdY Ydp (Cwdw Cr dr Ct dt CY dY ) ( p c( w, r ,1, s(t ))dY Ydp ldw kdr YVOTdt Ydp ldw kdr YVOTdt
(8.6b)
Next insert (8.6)(which is zero) into (8.2),
EV e 1, p A , s h (t A ), V B e 1, p A , s h (t A ), V A
e [ xdp xVOT h dt (ldw kdr ) l 0 dw k 0 dr Xdp XVOT 0 dt Ydp ldw kdr YVOTdt ]
A B m
e [( x X Y )dp (l l 0 l )dw (k k 0 k )dr ) ( xVOT h XVOT 0 ( x X )VOT )dt ]
A B m
e [ x(VOT h VOT ) X (VOT 0 VOT )dt ]
A B m
(8.7) Note that (8.7) is a typical origin formula for transportation benefit evaluation except for the introduction of cargo owners’ and operators’ VOT, where all the repercussion effects expressed by the change in freight price and wage and capital rent disappear due to demand-supply equilibrium shown as the third right hand side of (8.7) and only external effects remains shown as the last right hand side [Morisugi and Ohno (1995)]. In order to apply (8.7) practically it needs to express
em by observable market information and to express (8.7) as the well-known Trapezoidal formula by
second order approximation of (8.7) as
EV
e [ x(VOT h VOT ) X (VOT 0 VOT )]dt
A B m
A hA A A 0A A B A 1 [ x (VOT VOT ) X (VOT VOT )] B (t t ) 2 e m [ x B (VOT hB VOT B ) X B (VOT 0 B VOT B )]
(8.8)
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,where for the last equality we used the fact that e B m
B
e [ x (VOT
hB
B
A
m
11
1 and that the first order approximation of integrand as
B
VOT ) X (VOT 0 B VOT B )]
emA [ x A (VOT hA VOT A ) X A (VOT 0 A VOT A )]
.
d A [em { x A (VOT hA VOT A ) X A (VOT 0 A VOT A )}](t B t A ) dt Next, in order to express em by observable market information, differentiate (8.7) with respect to m,
emB emA
m
[e 1, p A , s h (t A ), V B e 1, p A , s h (t A ), V A ] m e [ x(VOT h VOT ) X (VOT 0 VOT )]dt
A B m
e [ x(VOT h VOT ) X (VOT 0 VOT )]dt em [ xm (VOT h VOT )]dt
A B mm
0 A h A A h A 1 emm [ x(VOT VOT ) X (VOT VOT )] em [ xm (VOT VOT )] B A B (t t ) 2 emm [ x(VOT h VOT ) X (VOT 0 VOT )]B emB [ xm (VOT h VOT )]B
1 emA [ xm (VOT h VOT )]A emB [ xm (VOT h VOT )]B (t B t A ) 2
(8.8) ,where for the third equality we used that X, VOTs are not a function of m, and the last equality we used that B emm
A A emm emmm (m B m A ) 0 , because e Am 1 , e Amm 0 and e Ammm 0 .
(8.9)
The last equality of (8.8) implies that
1 1 [ xm (VOT h VOT )]A (t B t A ) 2 emB emA 1 1 [ xm (VOT h VOT )]B (t B t A ) (8.10) 2 1 1 {[ xm (VOT h VOT )]A [ xm (VOT h VOT )]B }(t B t A ) 2 B Thus we can express em as (8.10) by the market information of non-business freight demand, and VOT for non-
business freight customers and freight operators. The value of
emB is greater than 1 because second term is positive
and it is probably less than 1.3 because the second term can be approximately twice of household expenditure share of the freight consumption if we specify utility function as Cobb-Douglas type. Furthermore a quasi-linear utility function, therefore,
emB is one if we assume
xm 0 , which the conventional practice presumes.
With the derivation of (8.8) and (8.10), we can express the social freight time saving benefit as trapezoidal formula with the weight
emB . Furthermore VOT to be applied to the benefit measurement formula (8.8) is a sum of VOT for
shippers and operators. Therefore we can say that conventional trapezoidal formula with the weight of one and VOT for only operators underestimates the benefit in two points. First is the weight shippers.
e B m and second is neglecting VOT for
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9. Concluding remarks This paper focused on how value of time and its accrued time saving benefits for freight trips can be expressed and measured in terms of market information such as freight fee, freight time, observable freight service consumption and observable freight service production within a framework of general equilibrium rather than individual user analysis which most previous studies adopt. For general equilibrium analysis we supposed a simple economic society composed of households, two sectors of production, firms producing composite goods, and firms producing freight transportation with a given level of freight time even for freight providers. The reason of above simplicity is for simple and clear derivation of VOT and time saving benefit measurement formula. It is quite possible to construct a more realistic complicated economy with many goods, labor, time constraint and capital, heterogeneous households, government, investment and many regions as SCGE (spatial computable general equilibrium) modeling. Furthermore for freight transportation production sectors it is also possible to introduce multi-mode networks for both person and freight trips, With this complication the basic formula is exactly same as the simple case above except for appearance of many types of VOT and time saving benefit formula. Next in order to satisfy the weak complementarity assumption, we specified freight time externality as product h
0
form of quality index of freight time and freight consumption s (t ) x and s (t ) X into the utility function of nonbusiness users and production function of business users, respectively. And for freight operators the product of quality index of freight time and labour and capital inputs s(t )l , s(t )l are introduced into the production function of freight operators. Then we defined VOT as marginal substitution rate between price and time and expressed it as the freight service fee per freight time multiplied by the elasticity of quality level indicator with respect to freight time for each of nonbusiness users and business users, respectively. The freight service providers’ VOT is expressed as average labour and capital cost per freight time. Thus we can measure three VOTs in terms of market information such as freight fee, freight time and average labour and capital cost and the elasticity of quality level indicator with respect to freight time although we need to estimate quality index of freight time by estimating freight demand functions of users. The author believes it is the first time to succeed in defining VOT for shippers and their expression by observable market information. As for the VOT for operators, our derivation is similar to the conventional practice. Next within the framework of general equilibrium rather than individual behaviour analysis we derived a “origin” formula to measure social freight time saving benefits including its repercussion effects expressed by the change in equilibrium freight price, wage and capital rent, where we adopted the concept of equivalent variation as the social benefit definition. The formula is the integration of sum of weighted marginal freight time saving benefit functions with respect to simply freight time only, where the weight is the marginal utility ratio of income, and sum of marginal freight time saving benefit function for non-business and business users are the product of freight demand function and sum of VOTS for users and operators. Thus proposed origin (or shortcut) formula is the weighted general equilibrium consumers’ surplus of the freight demand multiplied by VOTs with respect to freight time. The author believes it is also the first time to succeed in origin formulation of only consumers’ surplus of freight demand function in spite of that there are repercussion effects over all the market from the freight market. We noted that all the repercussion effects expressed by the change in freight price and wage disappear due to demand-supply equilibrium. Also we noted that the marginal utility ratio of income after the change can be approximately expressed as 1 + (income effect of marginal freight time saving benefit for non-business), so that it is also possible to calculate from the market information. Furthermore VOT to be applied to the benefit measurement formula (8.8) is a sum of VOT for shippers and operators. Therefore we can say that conventional trapezoidal formula with the weight of one and VOT for only operators underestimates the benefit in two points. First is missing the weight
e B m and second is neglecting VOT for shippers.
As for the generalization of product form of quality index of freight time, we can say to be able to express VOT
and marginal time saving benefit by the demand functions with the exactly same method as travel cost method. However in practice many cases specify our product form to satisfy the weak complementarity, so that it seems to be
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