The spatial allocation of urban public investment in Colombia, South America and exploratory linear programming model

DRAFT THE WORLD BANK DEVELOPMENT ECONOMICS DEPARTMENT URBAN AND REGTONAL ECONOMICS DIVISION URBAN AND REGIONAL REPORT No. 78-3 THE SPATIAL ALLOCATION OF bRBAN PUBLIC INVESTMENT IN COLOMBIA, SOUTH AMERICA AN EXPLORATORY LINEAR PROGRAMMING MODEL RAINER THOSS and JOHANNES F. LINN SEPTEMBER 1978 These materials are for internal use only and are circulated to stimulate discussion and critical comment. Views are those of the author and should not be interpreted as reflecting the views of the World Bank. References in publications to Reports shoild be cleared with the author to protect the tenative character of these papers. ABSTRACT This paper presents the outline 'and application of an exploratory multi-regional linear programming model for Colombia, South America. The model permits explicit consideration of multiple objectives of national, sectoral and regional policy. It focuses primarily on public investment as an instrument for spatial policy, and considers the budgetary impli- cations of public service investment. Furthermore, it is designed specifically to reflect some of the major structural conditions of the economy of a .developing country. After the description of the model frame- work, preliminary results of its application in Colombia are presented with a view to their implications for the regional distribution of public services and investment; for technological choice in service provision; for employment; and for the equity of the distribution of income. The paper concludes with a discussion of possible extensions of the model and of its limitations.. ACKNOWLEDGEMENTS The major portion of .the research for this paper was carried out in 1977 while the senior author was on leave from the University of Mnster as a consultant to the Urban and Regional Economics Division at the World Bank. The final version of the paper was prepared in 1978 while the junior author was on leave from the World Bank as a visiting researcher at the University of MUnster at the invitation of the Sonderforschungsbereich 26 Raumordnung und Raumwirtschaft, The authors are indebted to numerous staff members at the Colombian National Planning Department, especially Jose Fernando Pineda and Jairo Arias, and to Alfredo Sfeir-Younis of the.World Bank. Without their assistance the application of the model in the Colombian context would not have been possible. The authors also wish to thank Douglas H. Keare, Bertrand Renaud, and I. J. Singh for their helpful comments. All responsibility for the views expressed in this paper, however, is entirely that of the authors, and they do -not necessarily reflect the views of.the World Bank. I. INTRODUCTION Many developing countries experience high rates of growth of their largest cities, and are faced with an increased concentration of their population in a few major metropolitan areas. The governments in these countries therefore frequently raise the question whether it would be desirable to direct urban growth away from the largest cities to the inter- mediate or smaller towns, or even to the countryside. In fact, the presumption among policy makers often is that such a "decentralization" policy should indeed be pursued. One of the reasons for concern with the high degree of concentration of people in a few large urban areas is the belief that the per capita costs of providing urban services are significantly higher in large cities than in intermediate size or small towns, and that these cost differentials are not offset by higher levels of productivity. It is also commonly held that the provision of public services can be used to induce private investors and migrants to move to locations thought to be desirable in the context of a national spatial strategy. Furthermore, since public services provide direct benefits to their consumers, and generate indirect benefits through improved employment and environmental conditions, they are viewed as an instrument for alleviating income inequalities as between income groups and as between regions. It is therefore not surprising that governments are concerned with the question of how to allocate the available investment resources for urban public services across cities and regions. The Colombian government is no exception in this respect, although the concentration of urban growth is not particularly marked in Colombia.-/ In recent years, the Government 1/ See Johannes F. Linn, "Urbanization Trends, Polarization Reversal, and Spatial Policy in Colombia", Arbeitspapier Nr. 12 des Sonderforschungs- bereich 26 Raumordnung und Raumwirtschaft MUnster, 1978, pp. 26 ff. -2- has p?rsued an active policy of trying to induce private investment away from the largest four cities towards the rest of the country and has emphasized public investment ia intermediate size cities.- However, the question of what is the optimal allocation of public investment in urban services across cities in Colombia remains to be addressed int a systematic and comprehensive analytical framework. The central purpose of this paper is to ddvelop an analytical framework with which to determine the optimal allocation of investment in urban public services across cities and between various types of services in a developing country (LDC). This allocation must of course be made in accordance with the government's primary policy objectives of efficiency in resource allocation, of economic growth, and of equity in the distribution of income. Furthermore, it has to allow for potential constraints on the availability of foreign exchange and on the country's savings capacity, and for possible underutilization of existing resources, especially labor. Moreover, the framework has to allow for the particular structural conditions of LDC economies and for the limitations on data availability frequently encountered there. Subsidiary goals of the model proposed here are the testing of the budgetary implications of alternative public investment strategies; the determination of the opportunity cost (in terms of welfare foregone) of a number of policy constraints which may be imposed by the government, e.g. minimum service standards, environmental standards, minimum targets of improved inter-regional equity; and the determination of the optimal mix of labor intensive and cApital intensive technologies for public service investments. 1/ Liin, op. cit., pp. 50 ff. -3- Any decision regArding optimal spatial allocation of public investment involves numerous normative judgments, as well as assumptions regarding behavioral and technical conditions. The model developed in this paper can be used to test the sensitivity of the optimal spatial strategy to variations in these normative judgments and factual parameters. The model should, however, be seen only as a first step in a program of developing more complex quantitative tools of regional policy analysis. In any case, policy conclusions derived from models of this kind should be viewed only as providing additional guidance and information, rather than hard and fast prescriptions. The model was developed for, and tentatively applied to the case of Colombia. However, the analytical framework is applicable to other countries. The method of analysis consists in the formulation, solution and sensitivity testing of a multi-regional linear programming model of the Colombian economy, giving special attention to public investment.and the provision of public services. The model disaggregates the Colombian economy into eight metropolitan regions and the rest of the country, and distinguishes nine types of publicly provided infrastructure (water supply, sewerage, electricity, telephones, solid waste disposal, public roads, education, health, and administration).- It determines the optimal distribution of production and consumption activites across cities in line with the primary policy objectives (efficiency, growth, and equity), constraints (foreign exchange, savings), resources endowments, and existing technological choices. By introducing explicitly the major urban services which have to be provided by the public sector in accordance 1/ In, the preliminary application of the model, only four regions and two services were explicitly incorporated. 11 . . .. . . .. . -4- with this optimal spatial distribution of economic activity, the need for public investment in these services in-the various cities can be determined, while allowing for differentials in the relative costs and benefits of service provision in alternative locations. None of the individual components of the model presented below are by themselves new to the research community. On the contrary, much of the refinements of other highly sophisticated studies with partial! focus cannot be incorporated in the present study because the necessary detail of analysis would make the model unmanageable. On the other hand, it appears that the combination of (a) a regionally disaggregated LP model; (b) consideration of multiple objectives of national, sectoral, and regional policy; (c) explicit concern with public investment instruments; (d) consideration of budgetary implications of public service investments; and (e) explicit design for the structural conditions of an LDC economy; represents a new departure aimed at making existing theoretical tools relevant for policy formulation. The remainder of this paper is divided into three sections. The first presents the structure of the model; the second section reports on the results of an initial attempt to apply the model in Colombia; the last section discusses some possible extension of the model and its major limitations. II. THE STRUCTURE OF THE MODEL A model designed for the purpose of regional public investment policy has to allow for choice between locations and between labor intensive and capital intensive production processes. It has to contain the targets -5- to be attained and the resources available, and it must take into account the interralationships between the different activities in order to describe not only the direct but also the indirect effects which the measures to be taken may have on the well-being of different segments in the population. This section explains the essential design characteristics of this model. The model at present distinguishes in each region between two groups of people (P ) according to their share in total income: One group (g 1) g consists of the lower 40 percent of the population which in Colombia at present receive only about 10 percent of all value added. The other group (g = 2) comprises the upper 60 percent, who receive the rest of the income. Population grows in each region with the natural rate (u) and by net in- migration (M). The income of each group (Y ) determines its participation g in private consumption (C ). The rest of the output is used for public consumption, investment (DK, DQ, BQ),and exports (EX). Each group offers labor of four different qualifications, which are denoted by the index q. If this labor is not being used in any period, it goes to waste in the form of unemployment (U) and the unemployed receive no share in the primary income distribution. The persons who cannot find employment which meets their qualification (q) but accept employment in category (q + 1) will be called subemployed (Sq). Goods and private services are produced in seven sectors of production, three of which employ, in part, "modern" capital intensive and in part, "traditional" labor intensive techniques. Output of sector (i) is denoted Xi if produced in a "modern" process, and Xi, if produced labor 1/ See Montek S. Ahluwalia, "Inequality, Poverty and Development".. Journal of Development Economics, Vol. 3 (1976), p. 340. The partitioning of the population into these two broad income groups reflects the explicit policy concern of the Colombian government in the period of 1974 to 1978. -6- intensively. Similarly, DQ denotes public investment constructed with capital intensive methods, and DQ signifies investment constructed in the form of labbr intensive public works. Nine public services and their respective investments are distinguished by their index s. Their quantities are symbolized by X , DQs, and DQ s; their user charges by CHsg, where the index g ggain indicates the respective income group of inhabitants. The variable R stands for public revenue from taxation and domestic borrowing. DTG indicates the increment of land needed for waste dispesal per period. Table 1 provides a list of all variables used. Table 2 shows the meaning of the indices. As far as the definition of regions is concerned, it was concluded that, from each of the eight proposed future planning regions in Colombia, the largest city (metropolitan area) should be selected. The rest of the country would then be included as a ninth region.- Parameters are listed in Table 3. The model postulates the following relationships among regions and activities:2/ National Constraints: For each of sectors 1-6, there is a balance condition at the national level (1). These conditions serve two purposes. First, they guarantee that each private sector produces at least as much as is needed to meet the demand in the form of intermediate products, public and private investment, private consumption, and exports. Secnnd, they describe the direct and indirect impacts (multiplier effects) of an 1/ This choice of the delineation of regions is a compromise between the desire to focus the study, on the one hand, on the problem of urban poverty and urban public services, and the necessity to consider, on the other hand, the overall national resource and demand constraints and their implications for the growth process. The aggregate region "Rest of the Country" is rather heterogeneous and should in due time be subdivided into another set of eight regions, each of which consists of the hinter- land of the original eight cities. 2/ Numbers in parentheses refer to the equations and inequalities in Table 4. -7- Table 1: VARIABLES Endogenous Variables: X Production (capital intensive) X Production (labor intensive) DQ Gross Public Investment (capital intensive) DQ Gross Public Investment (labor intensive) DK Gross Private Investment C Cons-Vmption EX Total Exports Y Income U Unemployed Persons P Population M Net In-Miaration DT G Additional Territory Required for Garbage Disposal S Subemployed Persons R Tax Revenues and Loans GH Revenue from User Charges Y1 National Per Capita Incomel Exogenous Variables: QM Public Capital Imports M ~ Private Capital Imports DTB Additional Land for Building and Garbage Disposal Q(t-1) Public capital stock (at end of previous period) K(t-1) Private capital stock (at end of previous period) P(t-1) Population at end of previous period BOD Water Quality Standard B Internal Borrowing EM Emigration -8- Table 2: INUTCES Public Services: Other Sectors: s = 1 Administration i,j 1 Agriculture (export) 2 :Education 2 Agriculture (domestic) 3 Health Services 3 Industry 4 Transportation 4 Transportation 5 Electricity 5 Mining 6 Telephone 6 Services 7 Water 7 Construction 8 Solid Waste Regions: 9 Sewerage r = 1 Barranquilla Income Groups: 2 Cartagena g = 1 Lower Forty Percent 3 Medellin 2 Upper Sixty Percent 4 Bucaramanga 5 Pereira Labor Qualifications: 6 Cali q = 1 University Education 7 Bogota 2 Highschool Education 8 Ibague 3 Primary Education 9 Rest of Colombia 4 No Formal Education 9- Table 3: PARAMETERS h. ,h. input from sector i per unit of output in public service s. gi,g. input from sector i per unit of investment in public service s. isis a ,a input from sector i per unit of output in private sector i. b. demand from sector i per unit of private investment. 1 d. demand from sector i per unit of consumption of group g. e. demand from sector i per unit of national exports. m national import demand per unit activity. r total revenue per unit activity. v ,g value added per unit of output of public service. s, accruing to income group g. v. ,v value added per unit of output of private sector i, accruing to 'g income group g. c 9private consumption per unit of income. r direct taxes per unit of income. kj Eh. capital requirement per unit activity. f rate of physical wear and tear per year. 1 9lqs labor requirement of quality q per unit of public service. lqi5 qi labor requirement of quality q per unit of private activity i. pqg labor supply of quality q per inhabitant in group g. u rate of population growth per year. t land requirement per unit activity. cs ,money outlay per unit of output in public service s. cQs money outlay per unit of public investment for service s, constructed with modern technology. Ps charges for services s to industry i. - 10 - Table 3 (Continued) w share of total public revenue allotted to service s. S esi requirement of service s per unit of private activity i. ns requirement of service s per inhabitant. t land requirement per unit of solid waste disposal. es, eB output of pollutants (residuals) per unit activity. CK marginal welfare (weight) per unit of investment. Wag marginal welfare (weight) per unit of consumption of group g. - 11 - Table 4: CONSTRAINTS AND OBJECTIVE FUNCTION OF THE PROPOSED MODEL National Constraints: r rr Zr +D4 Z ,jr r + (W .Yfh~~ + Zzh. X + -ZA DQ, + is sa~i j si rs is s rs is sDs s s ij + (national intermediate demand) + Ed. Cr + Zb.DK + e. EX < FX + rglgg r 1 1 - ri r 1 (national final demand). < (national output) ,r + ZZ-r + ..mr DQr + ,-rBr + r + r (2) HZiiX +Zm X m DQ +Em D + m..+E .X. + rs Ss rs ss rs Qs s rs Qs s ra j rj j j (national import of igtermdiate products) + Em Cr + Em, De EX + QM + KM rg gg r (national imports of = (national export) + final products) (net capital import) S r + -r r (3) EZr.X. + EZ.X. + HrY = R - B rj jj rj i i rg g g (indirect + direct taxes) = (revenue) - (net internal borrowing) (4) EM +EM = 0 r (migration balance) * * m MI 5 IMM i 1 Mtt b M Im on - 12 Table 4 (Continued) Regional Constraints: (5) Eh. Xr + Fh. Xr + E. DQ + Eg. EQ + a + Za r + S is s s is s s is s s is s i ij j s ijj + (regional intermediate demand) + H = 1.0 for i 6,7 Ed. Cr + b.DKr + e.EX < U (Xr r+ 2.0 for g Igg 1 (x1 1 9i 3, 9 (regional final demand) -C (regional output) (6) Xvr Xr Rr + Ev. Xr . r s sg s s-sg s 1 I is I 1 1 .g g = 1,2 (value added) = (income) r = . (7) c(1-r )Yr Cr g = 1,2 (marginal consumption o = (consumption) disposable income) (8) k Xr + < (1-f)(t-) + DQr s 1,.009 SS S s S s r (capital < (capital stock - depreciation) + required) (investment) (9) Ek.X' + EK.7' < (1-ff)Kr(t-1) + DKr (capital required) < (capital stock-depreciation) + (investment) (9a) DKr > .05Kr(t-1) r = 9 (flexibility constraint for investment) 긔 14 Table 4 (Continixed) (17) Y r > .25Y r 1 2 r (inter-group equity standard) (18) ,r-/,l>r > .3Y* 9"*g g g - r (inter-regional equity standard) (19) Et D + Et D + DK' + DT r < DT r s s s Z s s tk G - B (land requirement) < (available land) (20) t + t T r(t_,) + DT r s 8 G s G s G G (disposal land t_- (disposal land used) r F 1!00..g requirement 0 r (21) eA + e < B DjU B s B s S 9 r 13'foeq (water quality standard) Objective Function: r (22) W DKr + W C 4. MAX Kr C9 9 (weighted (weighted investment) consumption) ............... -15- expansion of public services on total demand. By the summation over r it is assumed that goods are perfectly mobile between regions, except for services (sector 6) and construction (sector 7). Likewise, there is only one national constraint for foreign trade (2), stating that total imports must always equal total exports plus private (KM) and public (QM) capital imports. Since export possibilities are usually limited, imports are constrained by foreign lending. Equation (3) defines the public revenue from direct and indirect taxation and domestIn borrowing; equation (4) is the interregional migration balance. Regional Constraints: For each region, the demand for the output of the service and construction sectors is specified in equation (5). Income and consumption of the two groups of recipients are described by (6) and (7), the latter specifying also the withdrawal effects of direct taxes and user charges. Ineqtalities (8) and (9) describe the expansion of production possibilities by public and private investment. The use of inequalities permits the possibility that production can be expanded without simultaneous investment, if excess capacity exists. Equations (10)-(13) deal with employment and population. Labor supply of the four different categories is assumed to be price-inelastic. It can be used in the sectors by applying capital intensive or labor intensive technologies, but if commodity demand cannot be raised high enough or if available technological options cannot absorb sufficient amounts of labor of a certain qualification q, the persons who cannot find employment - 16 - according to their level of training have either to be "sub-employed" in the next lower qualification (S ) or this resource will be wasted q+l, in the form of unemployed labor (U). Population growth is described by equation (11), the lower and upper bounds (12) specify flexibility constraints for migration, whereas equation (13) states that members of the first group do, by definition, comprise 40 percent of the total number of inhabitants. Budget constraints for each public service and each region are spedified by (14) and (15). The left side of (14) contains the monetary costs of operation for each service. These costs have to be met by charges to industrial users (p .X.) and private households (CH), or by allotting S1 1 certain fractions (w s) of total revenue for this purpose. For health, administration, and education these charges are zero. Investments costs (left size of (15)) can be met either out.of general revenue (w Qs) or by net borrowing. The dual variable corresponding to this constraint will show, how one additional unit of borrowing in service (s) in region,l(r) would increase welfare as measured by the objective function. Constraint (16) concerns the requirements of public services. It states the objective that public services should at least meet certain normatively determined standards (which would be reconsidered in the light of sensitivity analyses). Dual activity levels corresponding to this constraint show, how much society would have to sacrifice in terms of the objective function, if one unit of the respective service were to be provided additionally, and how much could be gained if one unit less would be required. Those are the social opportunity costs of public services. - 17 - Constraints (17) and (18) state equity targets (in addition to the weights in the objective function). Income of the poorest 40 percent of population should at least be raised to, say, 20 percent of total income within each region (17), and regional per capita income should at least be, say, 30 percent of the national average (18). The dual variables corresponding to the equity targets will show their opportunity costs in terms of the objective function and may lead to a revision of the initial targets if their "costs" (output foregone) seem too high. Constraints (19)-(21) are related to the technological choices to be made in the solid waste sector and in the sewerage sector. Where land is scarce, the disposal activities compete with other activities for the use of this scarce resource. Inequality (19) therefore specifies the available land and the competing land use activities, whereas (20) describes the land requirements of different disposal technologies, so that a land saving disposal technique can be chosen if warranted by scarcity of disposal sites. Inequality (21) serves essentially the same purpose for sewage disposal: If tight water quality standards (BODMX) are chosen, that sewage disposal technology has to be used which reduces the BOD content of liquid waste in order to allow as much production of goods and services as possible. The dual activities corresponding to contraint (21) show the price of maintaining the chosen water quality in different regions. Objective Function: Whereas targets (16)-(18) and (21) are considered as not substitutable through accomplishments in other fields, the objective function (22) deals with target variables that are mutually substitutable. Weights w describe their marginal rates of substitution. - 18 - Consumption of the poor will be given a higher weight than that of the rich, and investment will be weighted differently from consumption, in accordance 1/ to the theory of "social" shadow pricing.- For w = C2 = WK = Q the objective function corresponds to the form which is used to determine "efficiency" shadow prices. The mechanism of deriving the optimal"spatial allocation is quite straightforward: Maximizing the objective function (22) subject to the resource constraints and the normative targets means that production has to be maximized. This can only be achieved if those locations and those technical processes are chosen, in which the limited resources (e.g. land, labor, capital, foreign exchange, environmental carrying capacity, public finance) are used as efficiently as possible. The model therefore directs new vintages of capital and migration to their optimal locations and into the optimal processes of production. Production and-migration are connected through labor demand (Qn the input side) and through the demand for goods and services (on the output side). Growth, therefore, in the present model takes place not simply where labor is located (or can be provided by migration), but also where public services for the population can be provided with the least resource requirements. In this sense, the social opportunity costs of migration are minimized by the solution of the model, and the results show which pattern of migration would be most advisable in the light of the fundamental objectives. The same holds true for the regional pattern of investment. The model thus shows if a decentralization policy or a centralization policy should be pursued in the country in question, and where the concentration or deconcentration of economic activity should take place. 1/ See Lyn Squire and Herman G. van der Tak, Economic Analysis of Projects, (Baltimore: John Hopkins University Press, 1975). - 19 - III. EXPLORATORY APPLICATION AND RESULTS FOR COLOMBIA, 1975 In the initial application of the multi-regional linear programming model described in the preceding section it was necessary to assume that input requirements (technologies) for the basic input-output relations of the model do not vary across regions, due to the absence of region-specific input-output coefficients. The national input-output coefficients were drawn from the Colombian simulation model SERES,-/ whose basic structure conforms with the structure of the model presented above. As a further limitation, data were available only for two public services (telephones, and solid waste disposal) and for four regions (Bogota, Cali, Bucaramanga, and Rest of the Country).- Finally, the land and water quality constraints contained in (in)equalities (19) to (21) above were not included in this application of the model. For this reduced model two alternative optimal solutions were calculated, which show the consequences of two different sets of weights in the objective function: the first "uses "efficiency weights" (i.e., WC1 :WC2 : WK =1:1:1), the second applies "social weights" (WCl : WC2 W K 3 : 1 : 2).3!/ The results discussed below are merely indicative, and by no means exhaustive methodological examples of the type of information which may be gained for the development of a consistent regional policy, based on a more 1/ Corporaci6n Centro Regional de Poblaci6n, "Modelo SERES, Estructura y Usos". Monograffas de la Corporaci6n Centro Regional de Poblaci6n, Vol. 3; Bogotg, 1974. 2/ Data collection on telephone services was carried out by the Colombian National Planning Department. Data for solid waste disposal were provided by Alfredo Sfeir-Younis. 3/ The second set of weights was chosen arbitrarily; but it reflects a plausible set of normative judgements. - 20 - complete and extensive application of the basic model framework. The policy implications which are drawn from these results are therefore also only of heuristic value, rather than definitive conclusions. 1. Public Services and Public Investment The main objective of the analysis was the estimation of the social opportunity costs of public services. For the two services included, those results are shown in Table 5. The numbers show by how . much the value of the objective function (22) would be reduced if ceteris paribus one additiqnal unit of service s had to be provided. This cost arises because if more public services are required, resources have to be used which could otherwise be used for private consumption and/or investment. No matter which weighting system is applied, the output foregone is highest in region 2 (Cali), whereas these costs are relatively low in region 1 (Bogota). The marked cost differences for telephones between Bogota and Bucaramanga (regions 1 and 3) on the one hand, and Cali and the Rest of the Country (regions 2 and 4) on the other hand, are due to the fact that in regions 1 and 2 there exist excess capacities of capital, so that (for a limited range) no investment costs are incurred if the services is to be expanded. For a final judgment, of course, all the other services would have to be evaluated in the same form and this may reveal compensatory effects in other services. Tables 6 and 7 show the corresponding activity levels for public services and public investment. For telephones, no technology choice is available and therefore the activity levels are the same for both solutions. For solid waste, however, a choice can be made between more or less labor 二 Table 6: TELEPHONES AND WASTE DISPOSAL (Service Levels) WITH EFFICIENCY WEIGHTS WITH SOCIAL WEIGHTS Region Telephones Waste Disposal (t/day) Telephones Waste Disposal (t/day) (Lines) Process 1 Process 2 Process 3 (Lines) Process 1 Process 2 Process 3 1 402,9-00 2,100 450 402,900 2,550 2 96,goo 70 69o 96,goo 76o 3 20,8oo 10 14o 20,8oo 150 4 858,6oo 5,100 2,370 858,6oo 7,470 Total 1,379,200 7,28o 3,630 1,379,200 10,930 Table 7: TELEPHONES AND WASTE DISPOSAL (Investment) WITH EFFICIENCY WEIGHTS =WITH SOCIAL WEIGHTS Waste Disposal (t/day) Waste Disposal (t/day) Telephones Collection Deposit Telephones Collection Deoosit Region (Lines) Capacity Capacity (Lines) Capacity Capacity 1 1,200 580 2 22,4oo 280 22,4oo 260 3 50 4o 4 348,300 3,380 348,300 1,850 Total 370,700 .,910 370,700 2,760 4,9102,76 -24- intensive technologies leading to a greater or lesser reduction of the amount of waste to be deposited. Process 1 is a "modern" technology by which garbage is collected and deposited; in process 2 the amount of waste is reduced by 30 percent before collection by "pickers" who collect recyclable materials from households, etc.,; in process 3 the amount to be deposited is further reduced by pickers who work at the disposal site and collect another 20 percent of recyclable materials. As the tables show, according to the present preliminary formu- lation of the model the application of process 3 is not advisable in Colombia, because there are other, more productive, activities which could be expanded with the use of any resources saved in public service provision. The reader should be cautioned, however, to keep in mind that the present model does not allow for the resources saved by the recycling of materials and that scarcity of suitable land for disposal sites (constraints (19) and (20) above) may enforce a choice of the most space-saving process 3 at least in some regions. Table 6 shows also the technology switch to a more labor intensive process which becomes optimal when changing from efficiency weights to social weights. The investment consequences are demonstrated in Table 7. As was already mentioned in connection with the dual variables, excess capacity existed in telephone lines in two regions in 1975, so that no investment is necessary. The same holds true for waste collection capacity in all four regions. 2. Decentralization Tables 8 and 9 show the optimal distribution of population, migration, private investment, and production. They show the direction in which private activities (along with public investment) should be guided. The example - 25 - shows that the dichotomy "centralization" versus "decentralization" is too simple for the case of Colombia (as probably for any other country). First, it should be noted that capital and labor should move in opposite directions: New investment in the three cities should be kept as low as possible and should be mainly directed into the rest of the country (region 4). This result is very similar to the findings for industrialized countries, e.g. Germany. If productivity differentials were introduced into the present framework, one would expect that this effect would be even more accentuated, since capital productivity can be supposed to be inversely related to capital intensity. Table 10 shows how much could ceteris paribus be gained if investment could be reduced by one unit below the lower bound set in the model (constraint 9a). Second, even if one looks at labor alone, the results do not permit a simple answer in terms of the above dichotomy: Comparing the three cities with the rest of the country, the optimal distribution would be clearly one of higher centralization of population. Between cities, however, the largest as well as the smallest (of those included in the model) should be permitted more in-migration, whereas the second largest should receive less than the amounts specified by inequalities. Table 10 shows the costs of limiting migration to Bogota and Bucaramanga (regions 1 and 3) and of insisting on migration to Cali (region 2). The optimal distribution of private production which corresponds to the above factor flows may be seen from Table 9. Both weighting systems 1/ This confirms the conclusions of Linn, op. cit, pp. 71 ff. which are based on a more comprehensive, but qualitative evaluation of this dichotomy in Colombia. Table 8: OPTIMAL DISTRIBUTION OF POPULATION AFD MIGRATION (in millions) Private Region Population -Migration Investmént 3.47 .o84o (UL) 9.75 (.L) 2 1.24 .0200 (LL) 1.75 (LL) 3 .4o .0048 (UL) .75 (LL) 4 18.67 -.1488 29.33 EM - .0400 - 1/ Alternative weights do not change the optimal distribution. Table 9: OPTIMAL DISTRIBUTION OF PRIVATE PRODUCTIGN (in billion Colombian Pesos) WITH EFFICIENCY WEIGHTS WITH SOCIAL WEIGHTS Sector 1 2 3 4 5 6 7 1 2 3 .4 5 6 7 tegion 1 5.5 . ioi.8 1.8 . 13.2 4.o . . 91.3 5.9 16.5 14.7 8.1 2 . 2.1 14.1 ,6 17.0 4.9 1.3 . .7 38.1 .7 .4.9 2.4 3 . . 9.3 6.0 . 2.1 .6 . . 12.5 2.1 . 1.8 .6 4 58.0 56.4 99.1 3.1 . 51.8 23.8 60.9 59.5 81.6 2.8 . 44.9 16.3 Total 63.5 58.5 224.3 11.5 17.0 72.0 29.7 60.9 60.2 223.5 11.5 16.5 66.3 2T.4 -28 - lead to very similar regional distributions of production. The sectoral distribution, however, must be expected to be different since it has to take into account the different structure of consumption resulting from changes in the income distribution which in turn depends on the weighting system applied. Agricultural demand,for instance, will be higher and service demand will be lower, if social weights are applied. 3. Technological Choice The consequences of different weights for the technology to be chosen were already mentioned in connection with the provision of public services. In order to increase consumption of the poor, technologies have to be selected which provide more income to that group. For the private sectors, the consequences of different weights are shown in Table 11. The results suggest that neither agriculture (sector 2) nor industry (sector 3) in Colombia are very good candidates for the application of labor intensive techniques. Only in the service sector (6) does the optimal process mix depend on the weights chosen. 4. Employment The production levels of Table 9 and 11 allow full employment to be achieved in all regions. Table 12 shows how the value of the objective function could be increased if ceteris paribus one additional unit of labor could be provided. 5. Equity and Income Level The targets for minimum equity between groups and between regions have been set so conservatively that they do not become binding constraints in either the efficiency or social. solution. Therefore their dual variables are zero, indicating that the constraints could be tightened by (at least) one unit, without any sacrifice in terms of the objective function. 二 Table OPTIMAL DISTRIBUTION OF PRODUCTIORr BY NODERNfr (1) AND "TRADITIONAV' PROCESSES (2) WITH EFFICIENCY WEIGHTS WITH SOCIAL WEIGHTS Sector: 2 3 6 2 3 6 Region Process: 1 -1 2 1 2 - 1 2 1 2 1 2 1o1.8 -7 12.5 91.3 14.7 2 2.1 14.1 2.3 2.6 -7 38.1 4.9 3 9.3 -5 1.6 125 1.8 4 56.4 20.8 99.1 31.0 59.5 81.6 44.9 Total 58.5 224.3 24.3 47.7 6m 223.5 66.3 Table 12: MARGINAL SOCIAL OPPORTUNITY COSTS OF LABOR (in million Colombian Pesos) WITH EFFICIENCY WEIGHTS WITH SOCIAL WEIGHTS Labor Category Labor Category Region 1 2 3 4 1 2 3 4 l.01TO .701770 .01770 .01770 .04689 .04689 .04689 .04689 2 .01770 .01770 .01770 .01770 .04689 .04689 .04689 .04689 3 .03390 .03390 .03390 .01636 .04689 .04689 ,o4689 .04689 4 .01770 .01770 .01770 .01TO .o4689 .04689 .04689 .04689 - 32 - A comparison of the two solutions. shows however, considerable differences with respect to the levels of income and consumption: If consumption of the poor is to be increased relatively to that of the rich, technologies have to be chosen which produce a relatively higher income for the poor than for the rich. Those technologies on the other hand are less productive. With the available resources therefore only a smaller total income can be produced. Tables 13 and 14 show the resulting regional and national levels of income and consumption. The figures demonstrate the importance of a full employment strategy within the realm of regional allo- cation policy. The income distribution can be influenced quite favorably even by pursuing "only" efficiency targets, since such a policy makes labor scarce and generates higher incomes for the poor. Achievement of distributional goals can be still more improved by applying social weights. However, total income in this case would be 4.5 million Colombian pesos less than the income resulting from an efficiency strategy, because the rich would lose more (as compared to the strategy with efficiency weights) than the poor would gain. The tables show further, that the regional distribution of income and consumption is by no means independent from the choice of distribution weights. Social weights lead to a heavy bias in favor of Bogota, whereas all other regions would benefit more from the choice of efficiency weights. IV. POSSIBLE EXTENSIONS AND LIMITATIONS OF THE MODEL Future extensions of the model and its application could include the following additional features without changing significantly the structure of the model, but requiring, of course, considerable additional efforts in Table 13: DISTRIBUTION AND LEVELS OF CCNSUMPTION AND INCOME (in million Colombian Pesos) CON S UMP T 1 ON INCOME Efficiency Weights Social Weights Efficiency Weights Social Weights Region Group 1 Grou 2 Group 1 Group 2 Group 1 Group 2 Group 1 Group 2 11.0 16.4 14.9 19.9 11.0 23.4 14.9 28.4 2 6.2 T.T 4.3 6.1 6.2 11.1 4.3 8.8 3 1.9 2.6 1.6 2.2 1.9 3.T 1.6 3.1 4 63.8 47.4 66.3 39.8 63.8 67.7 66.3 56.9 Total 82.9 74.1 87.1 68.0 82.9 105.9 87.1 97.2 Table 14: DIFFERENCES OF DISTRIBUTION AND LEVELS OF CONSUMPTION AND INCOME AS A RESULT OF DIFFERENT WEIGHTS (in million Colombian Pesos) C OiN S U M P T IO N I N C 0 M E Both Both Region Group 1 Group 2 Groups Group 1 Group 2 Groups 1 +3.9 +3.5 +7.4 +3.9 +5.0 +8.9 2 -1.9 :-1.6 -3.5 -1.9 -2.3 -4.2 3 -,3 -.4 -.7 -.3 -.6 -.9 4 +2.5 -7.6 -5.1 +2.5 -10.8 -8.3 Total +4.2 -6.1 -1.9 +4.2 -8.7 -4.5 - 35 - data collection: - Expansion of the number of possible locations of private economic activities and public services by considering additional cities; - Expansion of the number of public services and further dis- aggregation of service sectors to allow for a greater choice between alternative service technologies; - explicit inclusion of inter-regional flows of commodities and their transport costs to permit study of the effects of improve- ments in the transportation network on optimal spatial distri- bution; - regional differentiation of labor input coefficients to allow for spatial productivity differentials; - replacing the assumption of a fixed sectoral coiaposition of exports by a formulation permitting changing export composition; - disaggregation of private investment by sectors, since so far it has been treated only as an aggregate variable without sectoral breakdown; - inclusion of simple migration and private investment functions showing their reactions to policy instruments to permit sensitivity analysis for reasonable ranges of parameter values; - extension of the time horizon of the study by inclusion of more time periods in a recursive manner, such that the optimal activity levels computed for any particular period are taken into account when computing optimal values for the next period. - 36 - The preceding list of extensions of the model and their application appears within the realm of a feasible future research effort given the data availability in a country such as Colombia. More remote is the possibility of extending the model in a number of other directions: - refinement of the budget constraints to allow for more detailed analysis of the hierarchy of administrative units and the institutional setting of multi-level government; - further disaggregation of the region called "Rest of the Country" to permit the study of rural-urban inter-relations (hinterland effects); - regional differentation of input requirements (technologies); - consideration of the indivisibility (lumpiness) of investment, in particular investment in urban infrastructure, by developing a mixed integer program; - endogenous treatment of prices by changing the model from real to monetary terms; - expansion of the analytical framework to a fully dynamic optimal- control model. The first three items on the immediately preceding list of possible extensions represent primarily refinements in terms of the data base, while the remaining items would imply significant advances in the complexity of the model, which for the forseeable future are probably beyond the scope of feasible research design in the context of multi-regional modelling. Given these limitations of the basic model framework presented in this paper, the policy conclusions derived from its application, even when extended in a number of respects as suggested earlier, will provide only additional guidance and information, for spatial policy rather than hard and fast prescriptions, and need to be combined with the results of partial or qualitative analyses. - 37 - The strength of the model presented above lies in that it forces the policy maker to take a comprehensive look at the impacts of spatial policy, and that it permits the testing of the sensitivity of spatial policy strategies to variations in the assumptions regarding behavioral functions, as well as regarding policy goals, targets, and constraints. I'