Quantitative Methods and Applications in GIS - Chapter 11 (end) pps

Unlike the urban economic model built on the assumption of monocentric employ-ment see the Mills–Muth Economic Model in Appendix 6A, the Garin–Lowry model has the flexibility of simulatin

Linear Equations and Application in Simulating Urban Structure

This chapter introduces the method for solving a system of linear equations Thetechnique is used in many applications, including the popular input–output analysis(e.g., Hewings, 1985; see Appendix 11A for a brief introduction) Here, the method

is illustrated in solving the Garin–Lowry model, a model widely used by urbanplanners and geographers for analyzing urban land use structure A case study using

a hypothetical city shows how the distributions of population and employmentinteract with each other and how the patterns can be affected by the transportationnetwork The GIS usage in the case study involves the computation of a travel timematrix and other data preparation tasks

The method is fundamental in numerical analysis (NA) and is often used as abuilding block in many NA tasks, such as solving a system of nonlinear equations andthe eigenvalue problem Appendix 11B shows how the task of solving a system of linearequations is also imbedded in the method of solving a system of nonlinear equations

11.1 SOLVING A SYSTEM OF LINEAR EQUATIONS

A system of n linear equations with n unknowns x1, x2, …, x n is written as

In the matrix form, it is

220 Quantitative Methods and Applications in GIS

or simply

If matrix A has a diagonal structure, Equation 11.1 becomes

The solution is simple:

If a ii= 0 and b i= 0, x i can be any real number, and if a ii= 0 and b i≠ 0, there is no

solution for the system

There are two other simple systems with easy solutions If matrix A has a lower

triangular structure (i.e., all elements above the main diagonal are 0), Equation 11.1

becomes

Assuming a ii ≠ 0 for all i, the forward-substitution algorithm is used to solve the

system by obtaining x1 from the first equation, substituting x1 in the second equation

to obtain x2, and so on

Similarly, if matrix A has an upper triangular structure (i.e., all elements below

the main diagonal are 0), Equation 11.1 becomes

The back-substitution algorithm is used to solve the system

By converting Equation 11.1 to the simple systems as discussed above, one may

obtain the solution for a general system of linear equations Thus, if matrix A can

Solving a System of Linear Equations and Application in Urban Structure 221

be factored into the product of a lower triangular matrix L and an upper triangular

matrix U, such as A = LU, Equation 11.1 can be solved in two stages:

1 Lz = b solve for z

2 Ux = z solve for x

The first one can be solved by the forward-substitution algorithm, and the second

one by the back-substitution algorithm

Among various algorithms for deriving the LU factorization (or LU

decomposi-tion) of A, one called Gaussian elimination with scaled row pivoting is used widely

as an effective method The algorithm consists of two steps: a factorization (or

forward-elimination) phase and a solution (involving updating and back-substitution)

phase (Kincaid and Cheney, 1991, p 145) Computation routines for the algorithm

of Gaussian elimination with scaled row pivotingcan be found in various computer

languages, such as FORTRAN (Press et al., 1992a), C (Press et al., 1992b), and C++

(Press et al., 2002) In the program SimuCity.for (see Appendix 11C), the

FORTRAN subroutine LUDCOMP implements the first phase and the subroutine

LUSOLVE implements the second phase The two subroutines also call for two other

simple routines, SCAL and AXPY Free FORTRAN compilers can be downloaded

from the website http://www.thefreecountry.com/compilers/fortran.shtml and others

The author used a free FORTRAN compiler g77 (free for downloading at

http://www.gnu.org/software/fortran/fortran.html) for test running the programs

Section 11.3 discusses how the programs are utilized to solve the Garin–Lowry

model One may also use commercial software MATLAB (www.mathworks.com) or

Mathematica (www.wolfram.com) for the task of solving a system of linear equations

11.2 THE GARIN–LOWRY MODEL

11.2.1 BASIC VS NONBASIC ECONOMIC ACTIVITIES

An interesting debate on the relation between population and employment

distribu-tions in a city is whether population follows employment (i.e., workers find

resi-dences near their workplaces to save commuting time) or vice versa (i.e., businesses

locate near residents for recruiting workforce or providing services) The

Garin–Lowry model (Lowry, 1964; Garin, 1966) argues that population and

employ-ment distributions interact with each other and are interdependent However, different

types of employment play different roles The distribution of basic employment is

independent of the population distribution pattern and may be considered exogenous

Service (nonbasic) employment follows population On the other side, the population

distribution is determined by the distribution patterns of both basic and service

employment See Figure 11.1 for illustration The interactions between employment

and population decline with distances, which are defined by a transportation network

Unlike the urban economic model built on the assumption of monocentric

employ-ment (see the Mills–Muth Economic Model in Appendix 6A), the Garin–Lowry

model has the flexibility of simulating a population distribution pattern

correspond-ing to any given basic employment pattern, and thus can be used to examine the

impact of basic employment distribution on population as well as that of tation network.

transpor-The binary division of employment into basic and service employment is based

on the concept of basic and nonbasic activities A local economy (a city or a

region) can be divided into two sectors: basic sector and nonbasic sector The

basic sector refers to goods or services that are produced within the area but sold

outside of the area It is the export or surplus that is independent of the local

economy The nonbasic sector refers to goods or services that are produced within

the area and also sold within the area It is local or dependent and serves the localeconomy By extension, basic employment refers to workers in the basic sector,and service employment refers to those in the nonbasic sector The concept ofbasic and nonbasic activities is useful for several reasons (Wheeler et al., 1998,

p 140) It identifies the economic activities that are most important to a city’sviability Expansion or recession of the basic sector leads to economic repercus-sions throughout the city and affects the nonbasic sector City and regional plannersforecast the overall economic growth based on anticipated or predicted changes

in the basic activities

A common approach to determine employment in basic and nonbasic sectors is

the minimum requirements approach by Ullman and Dacey (1962) The method

examines many cities of approximately the same population size and computes thepercentage of workers in a particular industry for each of the cities If the lowestpercentage represents the minimum requirements for that industry in a city of agiven population-size range, that portion of the employment is engaged in thenonbasic or city-serving activities Any portion beyond the minimum requirements

is then classified as basic activity Classifications of basic and nonbasic sectors can

be also made by analyzing export data (Stabler and St Louis, 1990)

11.2.2 THE MODEL’S FORMULATION

In the Garin–Lowry model, an urban area is composed of n tracts The population

in any tract j is affected by employment (including both the basic and service employment) in all n tracts, and the service employment in any tract i is determined

by population in all n tracts The degree of interaction declines with distance

mea-sured by a gravity kernel Given a basic employment pattern and a distance matrix,the model computes the population and service employment at various locations

First, the service employment in any tract i, Si, is generated by the population

in all tracts k (k = 1, 2, …, n), Pk, through a gravity kernel tik, with

FIGURE 11.1 Interaction between population and employment distributions in a city.

Basic employment

Employment

where e is the service employment/population ratio (a simple scalar uniform across all tracts), d ik the distance between tracts i and k, and α the distance friction coefficient

characterizing shopping (resident-to-service) behavior The gravity kernel t ik

repre-sents the proportion of service employment in tract i owing to the influence of population in tract k, out of its impacts on all tracts In other words, the service employment at i is a result of summed influences of population at all tracts k (k = 1,

2, …, n), each of which is only a fraction of its influences on all tracts j (j = 1, 2, …, n) Second, the population in any tract j, P j, is determined by the employment in

all tracts i (i = 1, 2, …, n), E i , through a gravity kernel g ij, with

(11.3)

where h is the population/employment ratio (also a scalar uniform across all tracts)

and β the distance friction coefficient characterizing commuting

(resident-to-work-place) behavior Note that employment E i includes both service employment S i and

basic employment B i , i.e., E i = S i +B i Similarly, the gravity kernel g ij represents the

proportion of population in tract j owing to the influence of employment in tract i, out of its impacts on all tracts k (k = 1, 2, …, n).

Let P, S, and B be the vectors defined by the elements P j , S i , and B i, respectively,

and G and T the matrices defined by g ij (with the constant h) and t ik (with the

constant e), respectively Equations 11.2 and 11.3 become

Combining Equations 11.4 and 11.5 and rearranging, we have

where I is the n × n identity matrix Equation 11.6 in the matrix form is a system

of linear equations with the population vector P unknown Four parameters (the

distance friction coefficients α and β, the population/employment ratio h, and the service employment/population ratio e) are given; the distance matrix d is derived

from a road network, and the basic employment B is predefined.

Plugging the solution P back to Equation 11.4 yields the service employment vector S For more detailed discussion of the model, see Batty (1983).

The following subsection offers a simple example to illustrate the model

k

n

k ik k

n

jk j

=

∑

1

11.2.3 AN ILLUSTRATIVE EXAMPLE

See Figure 11.2 for an urban area with five (n = 5) equal-area square tracts The

dashed lines are roads connecting them Only two tracts (say, tracts 1 and 2, shaded

in the figure) need to be differentiated, and carry different population and

employ-ment Assume all basic employment is concentrated in tract 1 and normalized as 1, i.e., B1 = 1, B2 = B3 = B4 = B5 = 0 This normalization implies that population and

employment are relative, since we are only interested in their variation over space

The distance between tracts 1 and 2 is a unit 1, and the distance within a tract is defined as 0.25 (i.e., d11 = d22 = d33 = … = 0.25) Note that the distance is the travel

distance through the transportation network (e.g., d23 = d21 + d13 = 1 + 1 = 2) For

illustration, define constants e = 0.3, h = 2.0, α = 1.0, and β = 1.0.

From Equation 11.2, after taking advantage of the symmetric property (i.e., tracts 2,

3, 4, and 5 are equivalent in locations relative to tract 1), we have

5 4

1 21 1 31 1 41 1 51

1 22 1 32 1 42 1 52

1 21 1 31 1 41 1 51

1 22 1 32 1 42 1 52

1 2

where tracts 3 and 5 are equivalent in locations relative to tract 2 Noting ,the above equation is simplified as

(11.8)Similarly, from Equation 11.3 we have

(11.9)

(11.10)

Solving the system of linear equations (Equations 11.7 to 11.10), we obtain

P1 = 1.7472, P2 = 0.8136; S1 = 0.4123, S2 = 0.2720 Both the population and serviceemployment are higher in the central tract than others

11.3 CASE STUDY 11: SIMULATING POPULATION

AND SERVICE EMPLOYMENT DISTRIBUTIONS IN

A HYPOTHETICAL CITY

The hypothetical city is here assumed to be partitioned by a transportation networkmade of 10 contiguous circular rings and 15 radial roads See Figure 11.3 Areasaround the city center form a unique tract CBD, and thus the city has 1 + 9*15 =

136 tracts For convenience of network distance computation, we assume that eachtract (except for the CBD, which is represented by the city center) enters or exitsthrough the node intersected by the radial and the inner ring road In other words,any non-CBD tracts are represented by these nodes on the road network Thehypothetical city does not have any geographic coordinate system or unit fordistance measurement

The following datasets are provided for the case study:

1 A polygon coverage tract contains 136 tracts of the city

2 A road network coverage road is made of the same lines from thepolygon coverage tract, but has the line and node topologies that havebeen built

3 A point coverage trtpt, representing 135 non-CBD tracts, is extractedfrom the nodes contained in the road coverage road

4 A point coverage cbd contains a single point for the location of CBD

d

d

23 1 13

1 23 1 33 1 43 1 53

1 3

242

1 24 1 34 1 44 1 54

The computation of road network distances (times) mainly utilizes the pointcoverage trtpt and the road network coverage road The arc (line) attribute tablefor the road network coverage road (road.aat) contains a standard itemlength, which will be used to define the impedance values in the network traveldistance computation In addition, road.aat contains an additional itemlength1, which is defined as 1/2.5 of length for the seventh ring road, and thesame as length for others This item will be used to define the new impedancevalues when we examine the impact of a suburban beltway on the seventh ring road.For instance, when the travel speed on the beltway is assumed to be 2.5 times thespeed on others, its travel time or impedance is 1/2.5 of others The attribute tablefor the point coverage trtpt (or cbd) contains an item trtid identifying eachtract and an item trt_perim as the perimeter of each tract Tract perimeters will

be used to calculate the average within-tract travel distances, approximated as 1/4

of the tract perimeters

11.3.1 TASK 1: COMPUTING NETWORK DISTANCES (TIMES) IN ARCGIS

In the basic case (i.e., the reference case used to compare with others), travel speed

is assumed to be uniform on all roads Travel time is equivalent to travel distance,and the length on each road segment measures the travel impedance Since the road

FIGURE 11.3 Spatial structure of a hypothetical city.

Legend

Hypothetical city with no scale or orientation

Nodes CBD Roads

network nodes are used to represent tract locations, the network travel distances arefairly easy to obtain If necessary, refer to instructions in Section 2.3 The followingprovides a brief guideline:

1 Compute the network travel distances from the nodes defined in trtpt

to the same nodes defined in trtpt through the road network road,and add the intratract travel distances at the origin and destination tracts

to obtain the total travel distances between no-CBD tracts

2 Compute the travel distances between the CBD tract and other non-CBDtracts (trtpt) as the Euclidean distances (travel distances through theradial roads are equivalent to the Euclidean distances), and add theintratract travel distances at the origin and destination tracts to obtain thetotal travel distances between them

3 Compute the intratract travel distance within the CBD tract

Output all distances to an external file odtime.txt, a space-separated textfile containing 136 × 136 = 18,496 records with three variables: origin tract ID,destination tract ID, and distance between them The file odtime.txt is sorted

by the origin tract ID and the destination tract ID and saved as a space-delimitedtext file odtime.prn For convenience, an AML program rdtime.aml forcomputing the travel distance (time) matrix and the data file odtime.prn are bothenclosed in the CD for reference

Repeat the task by using length1 as the travel impedance values, and output thetravel times to a similar text file odtime1.txt Similarly, the file odtime1.txt

is sorted by the origin tract ID and the destination tract ID, and saved as a delimited text file odtime1.prn (also enclosed in the CD) Note that in this case,travel impedance is defined as travel time instead of distance because the speed onthe seventh ring road is faster than others

space-11.3.2 TASK 2: SIMULATING DISTRIBUTIONS OF POPULATION AND

SERVICE EMPLOYMENT IN THE BASIC CASE

The basic case, as in the monocentric model, assumes that all basic employment(say, 100) is concentrated at the CBD In addition, the basic case assumes that α = 1.0

and β = 1.0 for the two distance friction coefficients in the gravity kernels The values

of h and e in the model are set equal to 2.0 and 0.3, respectively, based on data from the Statistical Abstract of the United States (Bureau of Census, 1993) If PT, BT, and

S T are the total population and total basic and service employments, respectively, we have ST = eP T and PT = hE T = h(B T + S T), and thus PT = (h/(1 – he))B T As BT is normalized to 100, it follows that PT = 500 and ST = 150 Keeping h, e, and B T constantthroughout the analysis implies that the total population and employment (basic andservice) remain constant Our focus is on the effects of exogenous variations in thespatial distribution of basic employment and in the values of the travel frictionparameters α and β, and on the impact of building a suburban beltway

The FORTRAN program simucity.for in Appendix 11C (also enclosed inthe CD) reads the travel distance (time) matrix odtime.prn, uses the

LU decomposition method to solve the Garin–Lowry model, and outputs the results

(numbers of population and service employment) to an external file basic.txt.Since the values are similar (or identical) for tracts on the same ring, 10 tracts fromdifferent rings along the same radial road (e.g., trtids = 11 to 19) are selected andshown in Table 11.1 Figure 11.4 and Figure 11.5 show the population and serviceemployment patterns, respectively

With a Suburban Beltway

Basic Case

Uniform Basic Employment α,β = 2

With a Suburban Beltway

FIGURE 11.4 Population distributions in various scenarios.

0 2 4 6 8 10 12 14 16 18

Location

Basic case Alpha, beta = 2 With a suburban beltway Uniform basic employment

11.3.3 TASK 3: EXAMINING THE IMPACT OF BASIC EMPLOYMENT PATTERN

To examine the impact of basic employment pattern, this project simulates thedistributions of population and service employment given a uniform distribution ofbasic employment In this case, all tracts have the same amount of basic employment,i.e., 100/136 = 0.8088 In the program simucity.for, one only needs to modifythe statements for coding the basic employment pattern and rerun it to obtain theresult for a uniform basic employment pattern The results are also shown in Table11.1 and in Figure 11.4 and Figure 11.5 Note that both the population and serviceemployment remain declining from the city center even when the basic employment

is uniform across space That is to say, the declining patterns with distances fromthe CBD are largely due to the location advantage (better accessibility) near theCBD shaped by the transportation network instead of job concentration in the CBD.The job concentration in CBD does enhance the effect; i.e., both the population andservice employment exhibit flatter slopes in this case (uniform basic employmentdistribution) than in the basic case (a monocentric pattern) In general, serviceemployment follows population, and their patterns are similar to each other.One may design various basic employment patterns and examine how the pop-ulation and service employment respond to the changes See Guldmann and Wang(1998) for more scenarios of different basic employment patterns

11.3.4 TASK 4: EXAMINING THE IMPACT OF TRAVEL FRICTION COEFFICIENT

Keep all parameters in the basic case unchanged except the two travel frictioncoefficients α and β Compared to the basic case where α = 1 and β = 1, this new

case uses α = 2 and β = 2 The travel friction parameters indicate how much people’s

travel behavior (including both commuting to workplace and shopping) is affected

FIGURE 11.5 Service employment distributions in various scenarios.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Location

Basic case Alpha, beta = 2 With a suburban beltway Uniform basic employment

by travel distance (time) As transportation technologies as well as road networksimprove over time, these two parameters have generally declined In other words,the new case with α = 2 and β = 2 may correspond to a city in earlier years.

Assigning new values to α and β in simucity.for yields new distributionpatterns of population and service employment, also shown in Table 11.1 and inFigure 11.4 and Figure 11.5 Note the steeper slope in the case of larger α and β.That explains the flattening population density gradient over time, an importantobservation in the study of population density patterns (see Chapter 6)

11.3.5 TASK 5: EXAMINING THE IMPACT OF THE TRANSPORTATION NETWORK

Finally, we examine the impact of the transportation network, in this particular case,the building of a suburban beltway Assume that the seventh ring road is the fasterbeltway: if the average travel speed on other urban roads is 30 mph, the speed onthe beltway is 75 mph Task 1 has already generated a different travel time dataset,odtime1.prn, based on the new assumption Changing the input file fromodtime.prn to odtime1.prn in the program simucity.for yields the resultunder the new assumption, also shown in Table 11.1 and in Figure 11.4 and Figure11.5 The distribution patterns of population and service employment are similar tothose in the basic case, but both are slightly flatter than those in the basic case (notethe lower values near the CBD and higher values near the 10th ring) In other words,building the suburban beltway narrows the gap in location advantage between theCBD and suburbia, and thus leads to flatter population and service employmentpatterns See Wang (1998) for the impacts of different travel speeds and differentlocations of the suburban beltway

11.4 DISCUSSION AND SUMMARY

The concept of basic and nonbasic activities emphasizes different roles in theeconomy played by basic and nonbasic sectors The Garin–Lowry model uses theconcept to characterize the interactions between employment and population distri-butions within a city In the model, basic (export-oriented) employment serves asthe exogenous factor, whereas service (locally oriented) employment depends onthe population distribution pattern; on the other side, the distribution pattern ofpopulation is also determined by that of employment (including both basic andnonbasic employment) The interactions decline with travel distances or times asmeasured by gravity kernels Based on this, the model is constructed as a system oflinear equations Given a basic employment pattern and a road network (the latterdefines the matrix of travel distances or times), the model solves for the distributions

of population and service employment

Applying the Garin–Lowry model in analyzing real-world cities requires thedivision of employment into basic and nonbasic sectors Various methods (e.g., theminimum requirements method) have been proposed to separate basic employmentfrom the total employment However, the division is unclear in many cases, as mosteconomic activities in a city serve both the city itself (nonbasic sector) and beyond(basic sector) The case study uses a hypothetical city to illustrate the impacts of

basic employment patterns, travel friction coefficients, and transportation networks.The case study helps us understand the change of urban structure under variousscenarios and explain many empirical observations in urban density studies.For example, suburbanization of basic employment in an urban area leads todispersion of population as well as service employment, but the dispersion does notchange the general pattern of higher concentrations of both population and employ-ment toward the CBD As the improvements in transportation technologies and roadnetworks enable people to travel farther in less time, the traditional accessibility gapbetween the central city and suburbs is reduced and leads to a more gradual decline

in population toward the edge of an urban area This explains the flattening densitygradient over time reported in many empirical studies on urban density functions.Suburban beltways “were originally intended primarily as a means of facilitatingintercity travel by providing metropolitan bypass, it quickly became apparent thatthere were major unintended consequences for intracity traffic” (Taaffe et al., 1996,

p 178) The simulation in the case with a suburban beltway suggests a flatterpopulation distribution pattern In a study reported by Wang (1998, p 274; withmore ring roads and even faster speed on the beltway), the model even generates asuburban density peak near the beltway when all basic employment is assumed to

be located in the CBD

The model may be used to examine more issues in the study of urban structure.For example, comparing population patterns across cities with different numbers ofring roads sheds light on the issue of whether large cities exhibit flatter densitygradients than smaller cities (McDonald, 1989, p 380) Solving the model for a citywith more radial or circular ring roads helps us to understand the impact of roaddensity Simulating cities with different road networks (e.g., a grid system, a semi-circular city) illustrates the impact of road network structure

APPENDIX 11A: THE INPUT–OUTPUT MODEL

The input–output model is widely used in economic planning at various levels of

governments In the model, the output from any sector is also the input for allsectors (including the sector itself), and the inputs to one sector are provided bythe outputs of all sectors (including itself) The key assumption in the model is that

the input–output coefficients connecting all sectors characterize the technologies for

a time period and remain unchanged over the period The model is often used toexamine how a change in production in one sector of the economy affects all othersectors, or how the productions of all sectors need to be adjusted in order to meetany changes in demands in the market

We begin with a simple economy of two industrial sectors to illustrate the model

Consider an economy with two sectors and their production levels: X1 for auto and

X2 for iron and steel For each unit of output X1, a11 is used as input (and thus a total

amount of a11X1) in the auto industry itself; for each unit of output X1, a 12 is used

as input (and thus a total amount of a12X2) in the iron and steel industry In addition

to inputs that are consumed within industries, d1 serves the final demand to

consum-ers Similarly, X2 has three components: a21X1 as the total input for the auto industry,

a22X2 as the total input for the iron and steel industry, and d2 for the final demand.

It is summarized as

where the aij are the input–output coefficients.

In matrix,

IX = AX + D where I is an identity matrix Rearranging the equation yields

(I – A)X = D which is a system of linear equations Given any final demands in the future D and the input–output coefficients A, we can solve for the productions of all industrial sectors X.

APPENDIX 11B: SOLVING A SYSTEM OF NONLINEAR EQUATIONS

We begin with the solution of a single nonlinear equation by Newton’s method Say

f is a nonlinear function whose zeros are to be determined numerically Let r be a

real solution and let x be an approximation to r Keeping only the linear term in the

Taylor expansion, we have

(B11.1)

where h is a small increment, such as h = r – x Therefore,

If x is an approximation to r, should be a better approximation

to r Newton’s method begins with an initial value x0 and uses iterations to graduallyimprove the estimate until the function reaches an error criterion The iteration isdefined as

The initial value assigned (x0) is critical for the success of using Newton’s

method It must be “sufficiently close” to the real solution (r) (Kincaid and Cheney,

1991, p 65) Also, it only applies to a function whose first-order derivative has adefinite form The method can be extended to solve a system of nonlinear equations.Consider a system of two equations with two variables:

(B11.2)

Similar to Equation B11.1, using the Taylor expansion, we have

(B11.3)

This system of linear equations provides the basis for determining h1 and h2

The coefficient matrix is the Jacobian matrix of f1 and f2:

Therefore, Newton’s method for Equation B11.2 is

where the increments h 1,n and h 2,n are solutions to the rearranged system of linearequations (Equation B11.3):

( , )( , )

∂ ∂∂

f x( h x, h) f x x( , ) h x f h xx f

f x

2 2 1

f x f x

f

n f

1

1 1

1 2 2

1 1 2

2 1 2 ,

,

, , , ,

Solving this system of linear equations uses the method discussed in Section 11.1.Solution of a larger system of nonlinear equations follows the same strategy, andonly the Jacobian matrix is expanded For instance, for a system of three equations,the Jacobian matrix is

APPENDIX 11C: FORTRAN PROGRAM FOR SOLVING THE

GARIN–LOWRY MODEL

*******************************************************************

*******************************************************************

* ALPHA, BETA: distance friction coefficients

* H: population / employment ratio

* E: service employment / population ratio

* N: total number of tracts the city is divided into

* D(i,j): distance between tracts i and j

* A(i,j), B(i,j): matrices G, T in the Garin-Lowry model

* BEMP(i): basic employment vectors [known]

* POP(i), SEMP(i): population & service employment vectors

* [variables to be solved]

* other variables: intermediates for computational purposes,

* defined where they appear first

x f x f x

f x

1 2 2 2 3 3

1 3 2 3 3

* Step 1 Define parameters & Input data

* Input the values of ALPHA, BETA, H, E

DATA ALPHA /1.0/

DATA BETA /1.0/

DATA H /2.0/

DATA E /0.3/

* Input the distribution of Basic Employment

* In the basic case, all employment (100) is assumed to be at CBD

* Step 2 Build matrices A, B, I-A*B, based on D(i,j)

* Derive matrix Aij, Bij first