The Effects of International Transportation Costs on Production Location in Developing Countries: A Theoretical Approach

Working Paper

THE EFFECT3 OF INTERNATIONAL TRANSPORTATION COSrS ON PRODUCTION LOCATION IN DEWELOPING COUNTRIES:

A THEORFXICAL APPROACH

Cynthia G r i f i

October 1984 WP-84-79

International Institute for Applied Systems Analysis

A-2361 Laxenburg, Austria

NOT FOR QUOTATION WITHOUT PERMISSION OF

THE

AUTHOR

THE

EFFECIS

OF INTERNATIONAL TRANSPORTATION COSrS ON PRODUCXON LOCATION

IN

DEVELOPING COUNTRIES:

A THiEORFllCAL APPROACH

Cynthia GrifKn

October 1984 WP-04-79

Working Papers a r e interim reports on work of t h e International Institute f o r Applied Systems Analysis and have received only limited review. Views o r opinions expressed h e r e i n do n o t necessarily r e p r e s e n t those of t h e Institute or of i t s National Member Organizations.

INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria

The objective of the Forest Sector Project a t IIASA is to study long- term development alternatives for t h e forest sector on a global basis.

The emphasis in the Project is on issues of major relevance to industrial and governmental policy makers in different regions of the world who a r e responsible for forestry policy, forest industrial strategy, and related trade policies.

The research program of the Project includes an aggregated analysis of long-term development of international trade in wood pro- ducts, and thereby analysis of the development of wood resources, forest industrial production and demand in different world regions.

This paper is an outgrowth from the author's participation in the Young Scientists Summer Program a t IIASk The work was carried out in conjunction with the UNDIO/IIASA project on transprtation costs of forest products and their influence on t h e location of various processing activi- ties. In particular, this project aims to provide t h e transportation cost data for the Global Trade Model, which is under development in the Forest Sector Project.

Markku Kallio Project Leader

Forest Sector Project

Many developing countries a r e i n t e r e s t e d in determining t h e optimal level of forest sector production and an overall production pro- gram. This paper offers a theoretical approach t o t h e aspects involved in determining such a production optimum which includes t h e develop m e n t of a global model of forest resources, wood a n d fiber processing and international t r a d e in forest products a s a partial equilibrium economic model cast in t h e mathematical programming framework with linear constraints and a nonlinear objective function. The proposed model has been under investigation by the IIASA Forest Sector Project over the last several years. Therefore, r a t h e r than focusing o n t h e continued develop m e n t of t h e core model, the emphasis is now placed on proposing basic theoretical questions surrounding t h e forest s e c t o r t o determine how transport costs, production costs, tariff and non-tariff barriers to trade, natural resource endowments and marketing costs will affect the loca- tion of various processing activities. This initial development will iden- tify t h e need for a more s t r u c t u r e d empirical analysis in resolving these questions.

CONTENTS

INTRODUCTION

GLOBAL TRADE MODEL FOR FOREST PRODUCTS THE TRANSPORTATION PROBLEM

SPATIAL DISPERSION MODAL CHOICE

NETWORK CHARACTERISTICS AND ROUTE CHOICE CONCLUSION

-

vii

-

THE WFXCTS OF INTERNATIONAL TRAETSPORTATION COSTS ON PRODUCTION IDCATION IN DEYELOPING COUNTRIES:

A THEORFllCAL APPROACH

Cynthia Griffin

INTRODUCTION

Many developing c o u n t r i e s in t h e forest s e c t o r a r e investigating t h e possibility of substituting t h e i r raw export activity with processed pro- d u c t s s u c h a s sawnwood a n d furniture with a n i n t e r e s t i n improving t h e i r overall economic competitive advantage in t h e world forest products m a r k e t . Quite simply, policy makers for t h e s e developing c o u n t r i e s a r e seeking t o develop a production plan t o motivate m i n i m u m t o t a l costs of commodity flows f r o m t h e forest t o t h e consumer.

The development of s u c h a production plan is very complex i n n a t u r e a s i t not only is c o n c e r n e d with t h e forest s e c t o r but with several o t h e r m a r k e t s e c t o r s including t h e transportation sector. Therefore, within t h e l a s t decade t h e r e has been a notable increased i n t e r e s t i n analysis concerning t h e introduction of secondary a n d final f o r e s t pro- duction activities in developing countries. In any such analysis, one of t h e m o s t i m p o r t a n t issues t h a t is relevant t o both policy m a k e r s a n d producers alike is t h e establishment of a production program with t h e development of t h e proper relationship between t h a t program a n d t h e transportation sector. Therefore, t h e concern h a s become m o r e t h a n with just defining t h e proper mix of different levels of production activi- t i e s considering t h e development of capacities and capabilities of a par- t i c u l a r country's forest s e c t o r but also t h e r e is a concern with t h e cost of providing t h e proper transportation i n f r a s t r u c t u r e which links t h e developing c o u n t r y with t h e world forest product market.

A n u m b e r of studies have been done concerning t h e economies of t h e forest s e c t o r products on a n i n t e r n a t i o n a l level. Although s u c h stu- dies a s those done by Sedjo (1983), Wisdom (1982), a n d Dykstra a n d Kallio (1984) include innovative a n d extended discussions of t h e connection between t h e forest sector a n d t h e transportation s e c t o r , m o s t of t h e existing studies do n o t analyze in detail t h e connection a n d trade-offs between t h e development of a n optimal production m i x t o produce exports a n d t h e development of both inland a n d i n t e r n a t i o n a l transpor- tation facilities.

Yeats (1981) p r e s e n t s a m o r e detailed analysis of t h e relationship between ocean transportation costs i n p a r t i c u l a r a n d t h e level of produc- tion activity. More specifically Yeats is i n t e r e s t e d in identifying what

"factors work against domestic processing" in less developed c o u n t r i e s which includes consideration of t h e influences of ocean t r a n s p o r t a t i o n costs. In general Yeats concluded that:

-

processing lowers bulk a n d stowage factors

-

processing i n c r e a s e s product value

-

processing c a n i n c r e a s e t h e fragility of a p r o d u c t making it difficult t o handle

-

processing c a n lead t o increased freight r a t e s .

From his r e s e a r c h , Yeats found t h a t t r a n s p o r t a t i o n c o s t s actually t e n d t o increase t h r o u g h a p a r t i c u l a r processing chain. Table l a shows s u c h a tendency with wood products exported f r o m several less developed countries t o t h e United States. Similar conclusions m a y be drawn from Table l b . As t h e level of production technology i n c r e a s e s , in general. both t r a n s p o r t costs a n d t h e m o s t favored nation tariff level also increases. Therefore, t h e s t r u c t u r e of freight r a t e s facing t h e less developed countries could be s e e n a s a d e t e r r e n t t o t h e growth of pro- cessing industries.

Lipsey and Weiss (1974) look a t yet a n o t h e r a s p e c t of t h e complex relationship between transportation costs a n d t h e c h a r a c t e r i s t i c s of a p a r t i c u l a r commodity. They conclude t h a t route, vessel, a n d commodity c h a r a c t e r i s t i c s m u s t be t a k e n into a c c o u n t t o d e t e r m i n e freight r a t e s a n d could in fact lessen some of t h e "export deterrence" posed by t r a n - s p o r t costs. "The c h a r a c t e r i s t i c of t r a d e route-the balance of l i n e r a n d non-liner" t r a d e is of p a r t i c u l a r importance. Other c h a r a c t e r i s t i c s ol t h e vessel would include size a n d type s u c h a s conventional bulk c a r r i e r s a s opposed t o t h e new g e n e r a l purpose c o n t a i n e r vessels which e a s e t h e m o v e m e n t from inland t o ocean modes of transport.

I t is t h e purpose of t h i s paper t o u s e t h e existing model s t r u c t u r e proposed by Dykstra a n d Kallio (1984) a t IIASA t o develop a connection between t h e location of secondary a n d final production activities i n developing countries a n d t h e development of t h e necessary t r a n s p o r t a - tion infrastructure. The paper places emphasis on t h e t h e o r e t i c a l development of s u c h a model. However, t h e paper also m a i n t a i n s t h e overall goal ol developing a n empirically reliable planning tool for both policy m a k e r s a n d t h e forest s e c t o r producers.

Table la. Analysis of the Ad Valorem Indices of international transport costs in Indian exports to U.S. (%) (adapted from Yeats 1981).

SITC Exporters Primary Intermediate

242-251-641 Wood in t h e rough India 13.1 24.8 40.0 paper pulp, Philippines 22.1 n. a. 37.2 paper and board

242-243-631 Wood in rough, India 13.1 16.1 34.9

wood shaped, Malaysia n.a. 32.2 23.3

plywood Philippines 22.1 n. a. 37.9

Table lb. South African Exports to U.S.

Processng chain

Estimated Ad Valorem Rate (%) Transport

M.F.N.

_{tariffs Total}

Paper

Paper pulp (25 1) -14.4 0.0 14.4

Paper and board (641) 69.7 3.0 72.7

Paper articles (642) 34.8 7.2 42.0

Wood

Rough wood (242) 29.6 0.0 29.6

Shaped wood (243) 23.9 0.0 23.9

Wood m a n u f a c t u r e s (632) 48.6 5.7 54.3

GLOBAL TRADE MODEL

FOR

FDRF2T PRODUCE

During t h e last two years. t h e Global Trade Model

(GTM)

has been developed by t h e Forest Sector Project core t e a m a t t h e International Institute for Applied Systems Analysis (IIASA). The initial GTM permits the subdivision of t h e world into a s many as 18 regions and 13 forest pro- duct categories. Using data from 57 timber producing regions, t h e GTM can be u s e d t o investigate potential structural changes in t h e forest sec- tor over a 50-year time horizon. The basic GTM model s t r u c t u r e is derived from a model of pulp and paper sector of North America t o Buon- giorno (1981) and Buongiorno a n d Gilless (1983a and b).

It

also can be viewed a s an extension of t h e model developed by Adams and Haynes (1963) t o study t h e s t r u c t u r e of forest products m a r k e t in North Amer- ica. The GTM includes extensions t h a t account for the interdependences between forest products which a r e developed from a common raw material base. The model contains implicit supply functions for every product within each producing region. These supply functions are assumed t o be sensitive t o prices of all products under consideration.

The s t r u c t u r e of the

GTM

comprises five components:

-

Demand Lor e n d products

-

Supply of timber

- Supply of recycled paper

-

Production - World trade

I t is assumed t h a t t h e major forces influencing international exchange a n d production of forest products can be formulated a s a model of a com- petitive m a r k e t and proflt maximizing producers which a r e subject t o m a r k e t imperfections such a s quotas, tariffs, bilateral trade a g r e e m e n t s a n d inertia of trade flow. The impetus for such a focus is t h e transporta- tion costs. The world m a r k e t price equilibrium is then derived while tak- ing into account the c u r r e n t m a r k e t imperfections.

Therefore, t h e GTM is s e t in a partial equilibrium economy c a s t i n t h e mathematical framework of Samuelson (1952) with linear constraints a n d a nonlinear objective function. The problem of solving t h e m a r k e t equilibrium for t h e regional profit is found by maximizing t h e s u m of consumer surplus and producer surplus subject t o materials balance constraints and is formulated as follows:

i

^{cu Ytm}

¹

m a i m i z e _{c & ,Ytmleuk}

c

_{i , 0} (ce )dc*

-

_{i m}

x ^j

₀ ^~i~ (yim ) d ~ k m

- xzij*

^{' (1)}

i j k

I

subject t o

where indices i and j r e f e r t o regions, k t o products, m t o production activities, and

cik

- -

consumption of product k in region i

Pfi(cik)

=

price of product k in region i associated with a partic- ular level of consumption cil,

vim

- -

level of annual production by process m in region i associated with a particular level of production activity m

Q ( , )

=

marginal cost of production by process m in region

i

associated with a particular level of production yim

'ijk

- -

unit cost of transporting commodity k from region i t o

region j

eijk

- -

quantity of commodity k exported from region i t o region j

& =

n e t o u t p u t of product k per u n i t of production for pro- c e s s m in region i

4, =

production capacity associated with process m in region i

bjk,Uijk

=

lower a n d u p p e r bounds on t r a d e flows of product k between region i a n d j.

The objective function is t h e maximization of c o n s u m e r a n d producer s u r p l u s solving for t h e equilibrium consumption, production a n d level of export flow. Equation (2) r e p r e s e n t s a type of m a t e r i a l s balance con- s t r a i n t where consumption is s e t equal t o production less n e t exports.

Equation (3) a n d (4) r e p r e s e n t r e s o u r c e and t r a d e capacity c o n s t r a i n t s respectively.

Formulated in this m a n n e r t h e objective function m e a s u r e s t h e ' n e t social welfare' of a region. More precisely, t h e solution c a n indicate t h e q u a n t i t y of timber t o be h a r v e s t e d in e a c h region, t h e q u a n t i t y of har- v e s t e d raw materials t o be t r a d e d between regions or converted i n t o final products within each region a s well a s t h e quantity of t h e final products t o be traded among t h e regions considered. The dual solution t o this problem gives a n indication of m a r g i n a l prices of both raw m a t e r i a l s a n d final products.

However, for t h e purpose of t h i s p a p e r t h e focus is now drawn towards t h e last t e r m of t h e objective function which r e p r e s e n t s t h e transportation costs. The GTM a s s u m e s t h a t t h e r e is a competitive world m a r k e t where t h e price differential between regions r e s u l t s in t r a d e flows. Under this assumption, a n equilibrium solution c a n be derived assuming t h e r e is one world m a r k e t price a n d t h e price differentiation between nations is based solely on transportation costs which c a n include tariffs, subsidies, a n d inland, ocean a s well a s port t r a n s p o r t a t i o n costs. To introduce t h i s c u r r e n t t r e a t m e n t of transportation costs in t h e GTM i t is useful t o consider t h e following optimality conditions t h a t explain t h e trading s e c t o r in t e r m s of transportation costs z , , ~ p e r unit of commodity k , price i n region i , n*, a n d t h e shadow price, dijk, of t r a d e :

(-zQk

- xik +

nlk

-

bijk)(eijk

- 4 .

l k )

=

⁰ for all i , j . k 6 ² 0 for a l l i , j , k

~ l +

t h e s e conditions m a y be i n t e r p r e t e d a s follows:

(a) if ei;,,

> bjk.

^{t h e n dijk}

⁼

^njkriik

-

zilk

(b) if Uijk

>

e;k

> kjk.

^{t h e n}

^nik -

n*

=

^zijk

(c) if e&

= 4j*,

t h e n dijk

= o

^and njk

-

^ne

^c

z i j k .

Further economic implications can be s t a t e d from t h e above condi- tions when we i n t e r p r e t t h e Lagrangian multiplier, bijk. associated with t h e condition (a) i t is apparent t h a t if a commodity flow exceeds its lower bound then t h e marginal profit of trade will be t h e import price a t region j less t h e export price a t region i and transport costs. If t r a d e is t o occur, as condition (b) states, t h e price difference between region i and j m u s t be a t least equal to t h e transportation cost. And, if flow is on its lower bound, t h e r e is no incentive to trade since t h e price difference is less than transportation cost as stated in condition (c).

This presentation of transportation cost is very similar t o t h e tran- sportation problem of linear programming a s will be seen in t h e following sections. However, i t is difficult for a planner or production analyst to ascertain m u c h of t h e interaction between t h e transportation sector and t h e forest product s e c t o r in this simple term. Especially in developing countries, one of t h e most important questions would be t h e proper allo- cation of capital and scarce resources between production development and transportation infrastructure improvements. A change in produc- tion technology may bring about new needs in transportation modes, time and capacity on all 'links' of the transportation network used by t h e producer a n d i n some cases this would include port facilities as well. It is the purpose of t h e remainder of this paper t o focus on developing a m o r e complete description of the interaction between production levels a n d t h e efficient use of transportation infrastructure. The following sec- tions develop a theoretical approach to integrating freight network con- cepts with t h e forest product trade analysis in t h e GTM t o conceptualize an initial formulation of an integrated forest commodity flow-transportation network model.

THE

TRANSPOHTATION PROBLEM

To begin t h e transportation sector development i t is beneficial t o theoretically displace t h e transportation t e r m from t h e

GTM

objective function a n d initially observe transportation costs exogenous t o t h e model. For t h e purposes of this paper, t h e exogenously determined tran- sportation costs will t h e n be reintroduced into t h e original GTM objective function.

Extracting t h e transportation t e r m of t h e objective function, we c a n readily see t h a t we have t h e simplest case of a commodity flow problem:

t h e transportation problem of linear programming. In such a problem, we seek t o minimize transportation costs subject t o b o w n requirements for total d e m a n d This implies t h a t the producer of a good seeks those commodity flows t h a t will minimize total transportation cost. Thus t h e formulation is consistent with t h e assumption t h a t t h e producer is an overall profit maximizer. Therefore in its simplest form, t h e transporta- tion problem may be written in standard form as follows:

minimize

f f!

^{zijk e,}

i = l j =1

subject to

-xeijk 2

-X;,

for all i , k

j

e . - 1 0 for a l l i , j , k

I l k

where

X;, =

t h e total output of commodity k from region i

- t h e total import d e m a n d for commodity k from region i Yik

-

zijk

=

t h e transportation cost for one unit of commodity k from region i t o region j

eijk

=

t h e flow of commodity k from region i t o region j a n d t h e Lagrangian becomes:

t h e first derivative with r e s p e c t t o eijk is found

therefore

Here Q and

pjk

a r e t h e dual variables of t h e t r a n s p o r t a t i o n prob- lem. They a r e i n t e r p r e t e d h e r e as shadow prices o r location rents. From t h e above we t h e n have t h e following Kuhn-Tucker condition:

(a) Ifyijkei17,

=

0: t h e n yijk

=

⁰ and/or eijk

=

0 Therefore

(b) ~f eijk

>

0: t h e n yijk

= o

^{a n d}

pir: -

^a*

=

ziIk

(c) If eijk

=

^0; t h e n yijk 2 O andBjk

-

^Q

+

yijk

=

zijk o r

Bjk -

aik

s

zijk.

Condition (b) implies t h a t for flows eijk t o be positive, t h e differential i n shadow prices m u s t be equal t o t h e transportation costs. Therefore, t h e r e n t s are determined by the costs. Condition (c) s t a t e s t h a t if flows do n o t occur, t h e r e n t or 'price' difference is less t h a n t h e transportation cost and trade could n o t occur. These a r e similar t o t h e conditions derived with t h e

GTM.

Briefly, from looking a t t h e dual of t h e above formulation and apply- ing t h e fundamental theorem of linear programming we know the objec- tive function of the primal and dual a r e equal so that:

for

Therefore, the r e n t s a r e set so t h a t t h e profit made by traders buy- ing a t %, paying transportation cost zijk and selling a t

pjk

are exactly zero. Thus the transportation problem can be used a s a model of a corn- petitive spatial system a p a r t from t h e original GTM objective function.

Since we also require

then t h e number of independent constraints is (2R - 1) where R is the n u m b e r of regions involved. Therefore, i t is impossible to solve for all t h e Q and

pjk

even if we know which eijk

>

0. So we m u s t s e t one Q

equal t o zero and then find

Pjk

^for eijk

>

^0, and continue t o solve for all optimal values.

Although for initial estimations t h i s simple formulation i s useful, i t is widely accepted t h a t t h e cost-minimizing solution given by t h e tran- sportation problem does n o t really reflect t h e r e a l world commodity flow pattern. This is due mainly t o t h e fact t h a t such a cost minimizing solu- tion does not allow for shipments of a single product between regions in both directions since one of these two shipments might not be necessary a n d would be considered economically inefficient. Therefore, many more t h a n (2R

-

1) flows c a n occur. This 'crosshauling' of commodities occurs because of two main reasons:

-

Aggregation of commodities have often classified distinctly different groups of goods in t h e s a m e commodity class which could appear a s crosshauling

-

Actual crosshauling of a good due to imperfect information, established trade p a t t e r n s and product differentiation due to marketing and advertising.

Therefore, it is necessary t o find a 'suboptimal' or differently con- s t r a i n e d solution t o this problem which recognizes these factors to develop a more realistic description of commodity flows. One approach introduced by Erlander (1977) is t o observe the present level of crosshauling and constr,uct our model t o have a similar level by adding a constraint t h a t is a measure of t h e level of dispersion or crosshauling expression in an entropy function:

where

=

a scalar measure of t h e level of dispersion in an observed commodity flow matrix.

If unconstrained by o t h e r factors the range of

&

is

o ^{& <} -

In (R) ( 2 0 )

where R represents t h e number of alternatives choices. Then, defining ( 0 ) In ( 0 ) t o be zero,

&

is equal to zero if all shipments a r e made on one alternative. Sk reaches its maximum value if the proportion of shippers choosing each alternative is equal. Therefore t h e magnitude of Sk is determined partly by t h e number of alternatives.

Adding equation ( 1 9 ) to t h e original transportation problem t h e new Lagrangian is

L

=

z z z z i j k e i j k +

c

⁺ z C e i j k In ( e i j k ) )

i j k k Fk i j

We now combine a modified equation (2) with t h e transportation model to describe t h e relationship t h a t can be developed between t h e transportation sector and the forest product sector. To do so i t is neces- sary to make these preliminary assumptions:

-

Total exports of commodity k ,

E;I,

and total imports,

&,

a r e known for e a c h region

-

As with t h e original

GTM,

transportation costs alone determine interregional commodity flows

-

_Final demand in a given region i s known.

The objective function of such a combined model m u s t reflect t h e total interregional transportation costs associated with export/import activity. Therefore t h e transportation problem now takes on t h e follow- ing form:

minimize

~zzzijkeijk ⁺ z

&(&

+ I , )

i j k i E$

subject t o

E ^&

^{2 Ei for all k}

a €9

z ^&

^{4 Ik for all k}

i E*

-zzeijk

^In ( e i j k ) ²

&

^{for all k}

i j

where

@

and a r e t h e estimated amounts of exports a n d imports respectively of commodity k through ports in region i

$ denotes t h e subset of regions t h a t have adequate port facilities

&

is t h e cost of ex/importing one unit of k through a port in region i, with shipping costs a n d plus any other costs associated with port activity.

Equation ( 2 3 ) and ( 2 4 ) a r e introduced as target levels for exports and imports. The remaider of t h e constraints a r e already familiar. The new Lagrangian is formed as follows:

L = zzzzweijk ⁺ z

&(Eirc

+

I f i )

+

z p i r c ( - J k +

z

i j k i €$ k i E$

Differentiating with respect t o eiik,

&

^and

1'

t h e following optimality conditions a r e obtained:

And

Plus t h e original constraints. All these conditions a r e interpreted in t h e following manner:

-

if ei,k

>

0, t h e n

kjk =

0 and in eijk

=

pk(yir;

-

z i j k ; assuming zijk 1s finite, t h e n esjk

> --

-

if

Eire >

0, t h e n ^{& I}

=

^{0 and} yik

=

uk

-

dik

-

if

4 =

0, t h e n

&

² 0 and yik ² ok

- &

solving for zijL we now have: ^'

MODAL CHOICE

Next we consider a possible choice between modes and routes.

I t

is i m p o r t a n t t o introduce t h e concept of t h e mode a n d route choice t o more realistically develop a model of interregional transportation. For simplicity we now a s s u m e t h e r e is one specific commodity k such t h a t we drop t h e commodity notation temporarily a n d introduce t h e following notation:

e i j , is t h e cost from region i t o region j by mode m eijm is t h e flow from region i t o region j by mode m

Pijm is t h e proportion of flow from region i t o region j on mode m and is equal t o eij,/

E., E. =

C C e i j ,

i j

Qijm is a n a priori probability of a shipment from region i t o region j on mode m

Then we c a n redefine

Sk

a s

a n d t h e original problem becomes

minimize

C

C C z i j m p i j m i j r n

subject t o

Ei

C C P i j r n _{j m}

= ,

^{for all i}

where

z i j m

=

min ( z i j m ) m

From this s t r u c t u r e of the model, t h e following properties a r e observed from the original optimality conditions and by solving for eijm we find:

and t h e following conditions are determined:

-

Shipments are proportional t o

4. I j

and the exponential of the transportation costs

-

Destination and mode chosen a r e determined by an exponential function

-

The denominator of equation ( 4 2 ) summarizes the accessibility from region i t o all regions j via all modes m .

Further analysis develops a general form for the z i j ' s as:

w 1

z i j

=

--ln CqVm exp ( - p - . )

P m ^%I m

where

Zij

is seen as a weighted function of t h e modal costs.

NETWORK - S C AND ROUTE CHOICE

So far we have considered the entropy constraint in t e r m s of mode choice where t h e resulting model is a combined destination and route choice model with z i j m assumed known. Now we move to a more realistic level of modeling where zijm is not known but instead depends upon the transportation 'link' costs which in t u r n depend on the amount flow transversing a particular link. Reducing the number of modes to one for simplicity in presentation, we can assume t h a t transport cost can be defined directly in terms of operating costs and a r e reflected in rates charged for a particular service. Several approaches have been

developed for estimating t h e level of freight rates. For now we simply assume t h a t e a c h link of t h e transportation network has a unit user cost v,, such t h a t i t is an increasing function of annual flow, f ,:

where t h e flow associated with a particular link a i s defined as follows:

and

t?ijkR

=

t h e a m o u n t of commodity k shipped from region i t o region j on route

R

"" ⁼ i

¹ if link a is included in r o u t e R from region i t o region j for commodity k

0 otherwise

Since we have assumed t h a t t h e shipper is a cost minimizer in gen- eral, if link costs a r e fixed t h e shipper will choose t h e route or combina- tion of Links t h a t traverse t h e transportation network with minimal accu- mulated link cost. This behavior could congestion on t h e lowest cost link and therefore t h e basis for t h e assumption of costs being an increasing function of t h e flow on a link. One link can serve many different routes so i t s is easily seen t h a t flows between each origin-destination pair are very interdependent between links. If, even under congestion we con- tinue t o assume t h a t t h e wareowner seeks t h e minimum cost path, then a network equilibrium problem results. When t h e travel costs over all route pairs a r e equal and when no unused route h a s a lower transport cost we have Wardrop's (1952) user equilibrium. The solution to this problem results in e a c h shipper minimizing his total transport costs.

The problem of network equilibrium for a single mode was intro- duced by Beckman e t al. (1956) and made computationaly efficient by LeBlanc e t al. (1975). The mathematical formulation of t h e model minimizes t h e total a r e a under t h e link cost function t o obtain an equili- brium solution. Therefore, our transportation problem objective func- tion now becomes:

minimize C J v a ( t ) d t

+

x x & ( &

+ ^I*)

a 0 i j

subject t o

Equations (23)-(29), and (34).

The solution t o this problem does not minimize total transport cost unless link costs a r e assumed constant. However, if link costs a r e not constant, a solution c a n be found by replacing equation (44) with

where

d(va ( f a

))f

a

m a w ^{= =}

marginal cost of link a

af

a

and a t t h e optimal solution for all used routes, R, connecting region i to region j :

Ultimately, equation (47) would replace the transportation t e r m in equation (1). Several programming techniques such as bi-level mathematical programming reviewed by Kolstad and Lasdon (1983) and the nonlinear complementarity approach used by Ksk and Boyce (1983) have been applied t o the integration of commodity flows a n d the tran- sportation network activity. However, t h e consideration of these developments is beyond the scope of this paper. For now we seek to maintain the development of t h e transportation costs in a separate pro- gram and reintroduce these costs into the original objective function thereby maintaining the original formulation and optimality conditions of the original GTM while leaving the possibility of integrating these two problems as a future research opportunity.

However, t h e problem remains to identify cost functions for t h e link which in actuality is the cost function for a particular modal choice. The original user cost estimations were from Hassan and Wilson (1982) and Sedjo (1983). In general most research done in this area suggests that transportation costs a r e determined by t h e following network mode characteristics:

-

Distance of voyage

-

Size of shipment

-

Loading and unloading terms

-

Registry of ships (in t h e case of ocean transport)

-

Volume of trade on a particular route

-

Value of commodity

-

Totalroutecharacteristics

Current research in this area provides many viable alternatives for determining a cost function for each link.

For t h e purpose of t h i s study, we a r e concerned with determining and strengthening t h e explanation of t h e relationship between commo- dity flows and transportation costs. To establish the approach presented here satisfies t h a t research we turn to a recent study done by Kim, Boyce a n d Hewings (1983) done with Korean data. In this study, 18 com- modities grouped into four categories, of which lumber can be found in the fourth category, were studied. To develop a relationship between the transportation and commodity flows, the dispersion of flow constraint, equation (19) is normalized and an expression for flow is determined:

Then, t h e t r a n s p o r t a t i o n d e t e r r e n c e factor,

pk,

a n d t h e observed level of dispersion is d e t e r m i n e d . The results can be s e e n in Table 2. From t h i s table one c a n s e e t h a t

if

Pk =

0 t h e n eiJ2 is proportional to production a n d u n r e l a t e d t o t r a n s p o r t a t i o n costs

-, = t h e n eijk approaches t h e solution of t h e simple t r a n - pk

sportation problem of linear programming.

As shown by Table 1, i t is obvious t h a t shippers of commodity g r o u p four, including paper products, a r e t h e m o s t sensitive t o transportation cost

-

which is in keeping with o u r original assumptions and o u r overall objec- tive of depicting t h e r e l a t i o n s h i p between t h e transportation a n d forest product sector.

Table 2. Results of Korean prototype commodity flow model (adapted from Kim.

Boyce and Hewings, 1984).

Commodity g r o u p

Group 1: coal, limestone, petroleum, metallic minerals Group 2: cement, non metallic materials

Group 3: agricultural products, livestock and silkworm, forest products fish Group 4: food and tabacco, textile, lumber-wood-furniture, pulp and paper,

chemicals, machinery, miscellaneous manufactured products.

CONCWSION

Therefore, t o summarize, by analyzing t h e transportation sector separately we have s t a r t e d with the simple transportation problem of linear programming and expanded it mathematically to include the con- cept of spatial interaction through the dispersion of flow constraint which allows u s to reach a 'suboptimal' solution while considering the problems of aggregation and crosshauling. Reformulation of this con- straint was shown to be useful in reflecting modal choice and established trade agreements. Then returning to the assumption of one mode we introduced network characteristics and route choice i n t o t h e expanded transportation problem objective function.

Briefly, i t was shown t h a t the materials balance constraint of equa- tion (2) in the GTM constraints could be introduced a s a constraint on the transportation problem to clarify t h e connection between the tran- sportation sector a n d t h e forest product sector. However, since i t is the overall purpose to reintroduce the new transportation costs into the GTM, this constraint could be dropped in the subprogram.

The advantages of this approach a r e mostly found in t h e increased flexibility in defining which link. port, mode or combination of the latter is the most cost efficient to the producer/shipper of a product. In con- sidering the overall objective, t h e forest sector producer in a developing country m u s t find t h e proper balance between investment in new pro- duction activities a n d the possibility of investing in t h e transportation infrastructure. To such decision makers, it is extremely important to seek out the most efficient means of transport among all model and routes.

Although in this initial analysis, the GTM objective function still uses the fixed transportation cost term, we recognize t h a t t h e assumption of constant transportation costs between regions is of little use to those who m u s t decide what priority investments in both production technol- ogy and transportation improvements should be made. Therefore. the proposed 'submodel' provides some extended description of the issues involved in t h e transportation sector t h a t could effect commodity flows.

The development and functional form of t h e model facilitates future model developments which would integrate t h e forest sector and the transportation sector more completely. Theoretically. s u c h advances a r e already being developed (Griflin, Boyce and Kim 1983) for t h e integra- tion of world trade flows and t h e transportation network i n which the hierarchical interactions between the producer-trader-shipper and in some cases a group of governmental or institutional regulations are con- sidered. Therefore, i t appears t h a t the framework of t h e model proposed here and its f u r t h e r development will provide a useful planning tool for developments, planners, producers, governments, and industries interested in t h e Forest sector either direqtly or indirectly.

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