AN ECONOMIC AND LAND USE MODEL FOR A MULTI -COUNTY REGION Dissertation for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY ROBERT C. MILEY 1977 I, 7 I"??? a . 1" 4’1 R Y 5 I" ';.“n State Lina L, .nv crsity i.’ J‘ § Hr. ”'""‘-'a- This is to certify that the I thesis entitled AN ECONOMIC AND LAND'LI’SE MODELING SYSTEM FOR A MULTI-COUNTY REGION presented by Robert CIark Miley has been accepted towards fulfillment of the requirements for Doctor of Philosophy degree in Resource Development Major profes I Date August 12, 1977 0-7639 IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII 17(3/ H J 5 ABSTRACT AN ECONOMIC AND LAND USE MODEL FOR A MULTI-COUNTY REGION By Robert C. Miley This study was designed to provide an economic and land-use model for the assessment of future impacts of recreation demand upon forest resources in the North Central Region. Its primary objective was to provide a framework for investigation into the economic structure of a given study area, to assess land-use trends and to aid the prediction of future impacts upon forest resources. Methodologically, the study began with a review of economic history of a select study area comprised of the eighteen northern- most counties of the lower peninsula of Michigan. The study area's physical history. climate, and landform features provided a back- ground against which to assess its current economic and land use trends. Two types of model were developed in the study. First, an input-output analysis of the economy of the study area was formed and used to develop an economic trend analysis for several key years. The input-output model was then expanded into a linear programming model into which units of land were entered as productive factors. Robert C. Miley The study outlines the structure of the model and discusses procedures of model formation to help achieve optimum model efficiency. Methods of data acquisition and design considerations for a land-use data system are described in the study. Using specific methods of evaluation of indexes based on water area, local topo- graphic relief, and transportation accessibility, the study assesses the spatial locations of summer and winter season recreation demand in the study area for comparison with the location of its forest inventory. Limitations of the model and suggestions on the efficient combination of the model with the land-use data base are discussed. AN ECONOMIC AND LAND USE MODEL FOR A MULTI-COUNTY REGION By Robert C. Miley A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Resource Development 1977 ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to his major professor and dissertation director, Dr. Daniel E. Chappelle, Professor of Resource Economics and Regional Science, for his guidance and help in the author's dissertation research. Special thanks are expressed to the members of the author's guidance and dissertation committee: Dr. Robert J. Marty, Professor of Resource Economics and Programming, Dr. Milton H. Steinmueller, Professor of Resource Development, and Dr. Paul Strassmann, Professor of Economics, for providing valuable review of the dissertation. Appreciation is extended to the many pe0ple who offered critical help and evaluation during the study, and especially to Dr. Peter Niehoff for his suggestions. Finally, the author wishes to extend his sincere appre- ciation to his wife for her encouragement and support for without her patience, courage and understanding the work could not have been done. ii TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES Chapter I. II. III. INTRODUCTION The Need for Land-Use Planning in the North Central Region . A Typical Rural Recreation Region Objectives of the Study Prior Modeling Efforts . Plan of Report THE STUDY REGION Introduction Geology . Soils Climate . . Inland Lakes and Streams Rivers and Floodplains . Groundwater Forests . Economic History. . . Seasonal Home Development . Other Shifts of Land Use . . Campgrounds and Multi- Use Recreation Centers Supply of Land . . . . . . Summary . REGIONAL INPUT-OUTPUT MODEL Introduction . . Input- Output Analysis iii Page vii Chapter IV. V. VI. Development of the Regional Input- Output Model . Derivation of Regional Gross Output Derivation of Regional Final Demand Comparisons and Results of the Analysis Results . . . . . . Summary . A LINEAR PROGRAMMING LAND-USE MODEL Introduction Summary . LAND USE INVENTORY . Introduction . . Land Survey Techniques . . . Site Quality Determination by Indices . Composite Index Evaluation . . Sample Bias: Monte Carlo Estimation Measure of Relative Economic Productivity Land Use Inventory and Remote Sensing Data Summary . . . . . . . . . SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Summary . Conclusions Recommendations BIBLIOGRAPHY APPENDICES . A. B. A HISTORY OF LAND RENT CLASSIFICATION TABLES USED IN DEVELOPMENT OF THE INPUT- OUTPUT MODEL. PRECISION COMPUTATIONS FOR INDICES OF REGIONAL RECREA- TION POTENTIAL iv Page 44 51 54 80 82 82 127 129 129 130 134 138 142 145 158 162 164 164 166 169 l8l 187 188 200 203 Table 10. 11. 12. 13. 14. LIST OF TABLES Sectors Included in the Consolidated Sector State Transactions Table Reduced from the 87- Sector MRIO State Tables Sectoral Final Demands as Proportions of Gross Output for each Key Year of the Economic Trend Analysis l947 Regional Transactions Table (1000' s of l963 dollars). . 1958 Regional Transactions Table (lOOO' s of 1963 dollars). . . . . l970 Regional Transactions Table (lOOO' s of l963 dollars) . . . l980 Regional Transactions Table .(lOOO' s of 1963 dollars). . . Formation of Composite Score Indexes Indicating Regional Attractiveness for Hater-Related Recreation Development by County . . County Shoreline Area Determination Summary of Example Model Run for Water-Related Recreation Potential . Rank Ordering of Counties Based on Evaluation of Potential for Water-Related Recreation Development . Normalized County Development Potential Results of Regression of County Housing Census Against Model-Derived Summer-Season Recreation Potential for the l8-County Study Region . Precision of Variables Used for Computing the l8-County Water-Related Recreation Potential . . . . Page 46 47 55 60 64 68 72 103 105 106 107 109 112 119 Table Page 15. An Appraisal of Water Sports Areas for a Michigan County. 146 16. A Determination of Indices for Warm and Cold Water Fish- ing for a Michigan County . . . . . . . . . . 147 17. Number of 2% by 21 Mile Cells in Each Relief Interval Organized by County . . . . . . . . 149 18. Local Relief Intervals . . . . . . . . . . . . 150 19. County Local Relief Index . . . . . . . . . . . 151 20. Computation of Transportation Index . . . . . . . 154 21. Normalized Indexes for Recreation Land-Use Potential . . 159 vi Figure \l 0'! U" «D 00 N o o o o o o 10. 11. 12. 13. 14. 15. 16. 17. 18. LIST OF FIGURES The Eighteen4County Study Region Schematic Diagram of the Regional Economy Diagram of the Model Developed in the Study Major Physiographic Areas of the Study Region . Lakes and Rivers of the Region . Public Forest Lands in the Study Region . Final Demand as Proportion of Total Gross Output for 4 Sectors for All Key Years of the Trend Analysis . Trends of Construction Sector Gross Output . Trends of Agricultural Sector Gross Output . Trends of Lumber and Paper Products Sector Gross Output. 1980 Projected Total Sectoral Gross Outputs with Pro- jected Tourism Expenditures Shown for Comparison Major Sections of L.P. Model in Tableau Form Counties Rank-Ordered by Water-Related ReCreation Potential . . . . Gradient Profile of Water-Related Recreation Potential for the Study Region . . . . Total Variance of Model Results as a Function of Data Precision and Model Specification Error Gradient Profile of Precision for Figure 14 Potential Rent Change Step Function Lagged Response of Land Units Shifting to a New Use Category Following Imposition of Rent Change of Figure 17 . vii Page 15 16 21 23 26 56 57 58 59 77 90 108 110 114 121 124 124 Figure Page 19. Example of Piece-wise Linear Regression Model of the Type Used to Estimate Site Productivity . . . . . 141 20. Critical Sample Size N as a Function of Sample Bias for Given Population Variances 02 in a Simulation of Field Observations of Ordinal Ranked Values . . . 144 21. Gradient Profile of Local Topographic Relief . . . . 152 22. Regional Highway Transportation System . . . . . . 155 23. Distribution of Transportation Multipliers by County . 156 24. Gradient Profile of Transportation Accessibility . . 157 25. Gradient Profile of Composite winter-Sports Activity Index . . . . . . . . . . . . . . . . 160 26. Three-Region Accounts Table for Greater Resolution of Export Activity . . . . . . . . . . . . . 174 27. Suggested Expansion of Input-Output Model for Detailed Export-Import Analysis of the Regional Economy . . 176 viii CHAPTER I INTRODUCTION Rapid increases in intensity of use in many areas of the Midwest have resulted in scarcities of prime land resources. In- creasing demands for land for growth of cities and development of highway transportation systems have resulted in rapid rates of land use conversion. Inroads into the remaining supply of agricultural and forest land create concern among resource planners who extrapo- late present trends forward and predict future non-availability of land for these important major uses. Consequently, land use planning occupies an increasingly vital role in regional development. Land use modeling forms an important linkage in the planning process, permitting simulation of effects of land use policy, observation of the temporal and spatial distributions of projected land use changes, and exam- ination of likely resulting changes in regional economic struc- ture. This study outlines a combined economic and land-use model designed for studies of rural regions. The Need for Land-Use Planninggin the North Central Region Approximately 30 percent of the U. S. Population resides in the states of Michigan, Ohio, Indiana, Illinois, Wisconsin, Iowa, and Minnesota, a contiguous group of states which together form the North Central Region. Compared to other regions of the United States the proportion of publicly owned forest and recreation resources to population in the Region is low, resulting in intensive utilization of resources with potentially serious problems of over— use. The 1976 Annual Report of the research program "Guidelines for More Effective Regional Development of Forest and Recreation Resources in the North Central United States"1 outlines several objectives: (1) to "systematically investigate major forces that may substantially affect the use of forest and recreation resources in the north central UnitedStates during the next decade;" (2) to "Evaluate alternative methods, both substantive and organizational, of protecting and managing these resources to increase their capacity to absorb these impacts without major loss of attractiveness or productivity;" (3) to "Evaluate public attitudes toward systematic planning of land use, and to test the effectiveness of improved information in making these attitudes more favorable.“ .IReport to the Rockefeller Foundation, Cooperative Regional Research Program "Guidelines for More Effective Development of Forest and Recreation Resources in the North Central United States," 1976. As most of the Region's population is concentrated in urban centers removed from forest and recreation resources, the problem is one of externally imposed periodic demand from large urban popu- lations. Outdoor recreation will almost certainly continue to grow as a use of land and water resources in rural areas. Every study or plan considering this activity has projected a greater future volume. More people, higher real income, shorter work weeks, more paid vacations, and better travel facilities will all lead to a greater attendance at outdoor recreation areas. To this direct impact must be included the indirect impacts of transportagion and service to the users of the various kinds of areas. In the Midwest, an important component of the demand for recreation land is demand for seasonal home development. A growing proportion of dwellings in outlying areas of the Region are owned by persons who normally reside in the large metropolitan centers. The high mobility of the latter population with its increasing demand for recreation imposes increasing pressures for conversion of land from forest and agricultural uses to developed recreation uses. Through operation of the private land market much of this conversion takes place in commercial forest areas where primary existing land use is timber production. In addition public lands are often adversely affected by recreational over-use. When public forest lands are near recreation centers or water areas of suffi- cient size the development pressure upon them increases. Moreover, rapid use conversion of private lands in the vicinity of public wildlife habitat may create problems of overtaxation of delicate 2Marion Clawson, Suburban Land Conversion in the United States (Resources for the Future, Inc., 1970), p. 53. ecosystems and may possibly result in serious impacts upon fragile forest soil resources. Long-range controlled planning may be needed in these regions to permit economic development while avoiding undesirable consequences of rapid land-use conversion. Transition from private sector to centralized, public sector planning requires models which recognize both private and public planning activity. While this does not of itself create conflicting requirements in terms of the modeling effort, most prior modeling attempts have tended to concentrate upon one activity and to exclude the other.3 The present effort in model design attempts to combine private sector and public sector decision processes into a single comprehensive land-use planning model whose predictive ability resides in identification of land resource areas most likely to respond to economic change. A Typical Rural Recreation Region The 18 contiguous counties which form the northern portion of the Michigan lower peninsula are typical of counties in rural areas experiencing recreation-induced land—use pressures (Figure 1). Primarily forest land interspersed with numerous lakes and rivers, .this sub-state region has a long history of agricultural, com- mercial forest, and recreation uses. A concentrated band of metropolitan growth to the south is the origin for a large incidence 3See Michael A. Goldberg, Urban Land Economics: Quantitative Approaches to Land Management: A Survey, Critique, and Exposition offi Recent Work (Vancouver, British Columbia, 1973). .cowmmm xvapm zpcsoulcompcmwm ugh11._ mcamwm H3343. Mamet; 1 $5.21; J 383$ . 13.34% - 35.3.28 13.2mm _ _ _ _ L _ = 3...... . :8... u... 3.. 1 9... .52....a25 2... 5.5.5... .25.... .385... S. 9...... . Am a... .2..=ox .o. 9...... .uxo. .o... e... hazmzomwi... 5 2‘! 38.2.... 33.... u .. .a ... .5352... 23......»8 8.2.5.2... .43 k .352... 8 22.5.8 . 2.5.5.. 8. 9...... TI . . 3:35.. Juno. & .— Deb‘s—C 4.35352: 1 no. 2.3.3 I. .83.. _ 5.... a... ....o. . .8. 8. 9...... 2.2.3... _ 32...... a... 2...... 8......w..... 9.308 u lo;ga:m88 :¢._=t2 ward as o . www.0wthumc .28.... .33.. 1.8.8. 3:53.88 . .IL ml. _ 7.. fig Sb Ozcaua g gu‘ 23.8 . .383... .o _ 2.28 _ rugg- _ £53.25... 8.5255... 3.. 28.8. H _ 8.3. 8350 3.5.5.05 l6 Exogenous Economic Demands Regional Objective \l Regional Formation of Economic Land Availability Structure Constraints L.P. Model Demand for Regional Land Shifts of Regional Land-use Shadow Prices of Resources Figure 3.--Block Diagram of the Model Developed in the Study. Plan of Report Chapter II contains a description of the eighteen-county Study Region and its background of economic development with identification of current land-use trends. Chapter III develops a basic input-output model of the regional economy in order to identify primary land-using sectors and their relative ranking and to identify regional economic infrastructure. Chapter IV expands the input-output model of Chapter III to a comparative static linear programming model providing 17 information concerning likely land-use distribution responses to changes in economic demand. Alternative methods for inter-period analyses between comparative static solutions are discussed. Procedures to estimate precision of model results are developed. Chapter V presents site and area classification and description methods using site indices for development of land productivity data. Chapter VI presents a summary and recommendations concerning future model development. CHAPTER II THE STUDY REGION Introduction The l8 northernmost counties of the lower peninsula of Michigan comprise the sub-state area that is the focus of the study. Primarily forestland interspersed with numerous lakes and rivers, the Study Region is one of the most popular recreation areas in the Midwest. Climatic factors place it within the general range of agricultural suitability, but sandy soils make the Region more viable for commercial timber production, and the latter activity forms the dominant land use. These extensive areas of forest contribute to the Region's high recreation potential. Increasing demands for intensively used recreation areas and for second home developments have, however, placed serious burdens upon regional land resources and particularly upon forest resources. This chapter contains a general description of the Region's topography, climate, geology, history and economy against which to assess current land use trends. Geology The Region's location at the top of the Michigan Lower Peninsula places it above extensive bedrock deposits of Devonian and Mississippian origin, some of which appear in prominent l8 l9 outcroppings along the Region's coastline. These deposits consist of old, dissected, prevglacial limestone which slopes gradually downward from the coastal areas toward the center of the Region to form a large bowl shaped depression.1 Older bedrock deposits forming this basin became overlain in successive periods by depositions of sedimentary rock from the warm inland seas which once covered the Michigan basin. The new layered limestone and sandstone deposits form the foundation over which large quantities of glacial drift were subsequently deposited during later stages of glacial retreat. The topography of the Region today thus reflects the location of glacial sand and till which form outwash plains, moraines, and other surficial features associated with the recession of the last ice cover, an event which began about 20,000 years ago.2 The topography of the western portion of the Region is characterized by sandy hill~lands of glacial origin sloping toward the Lake Michigan coast. Upon this coast extensive dune formations are found. .The combination of rolling hills and sand beaches provides one of the major tourist attractions to the Region. The topography of the eastern portion of the Region is characterized by a gradual slope downward from the central highlands toward the relatively flat, lake-washed plains of the Huron coastline.3 1John A. Dorr, Jr. and Donald F. Eschman, Ceology of Michigan (University of Michigan, 1970), p. 26. 21bid., p. 159. 31bid., p. l76. 20 Figure 4 shows the distribution of major physiographic land—forms in the Region. 5.911;. Soils of the Region vary widely with respect to both texture and drainage. Sharp demarcations of soil types occur throughout the area, reflecting the glacial origin of the topography. Depths of glacial till reaching several hundred feet occur in the western portion, while in the low-lying plains of the eastern portions shallow depths of glacial materials are found with close surface proximity of the underlying limestone bedrock. Soils vary from extensive peat and muck soils found within low-lying wetlands to sandy forest soils typical of the hilly uplands. Soils can be found composed of fine sands, usually occurring in glacial outwash areas, or composed of relatively course gravel deposits associated with glacial moraines. Soils vary with respect to drainage characteristics and capacity to support agricultural production. Many areas of the Region are unsuited to extensive agricultural cultivation. Climate Moderate summer temperatures are an important factor in the sumner season recreational attractiveness of the Region. Winter temperatures usually permit permanent snow retention throughout the winter season, and adequate precipitation contributes to the suitability of the Region for winter sports activities. The climate is moderated further by the proximity of the Great Lakes which provide 21 HURON LAKE-BORDER PLAIN PRESQUE ISLE ROLLING PLAIN EMMET-ALCONA HILL LAND CHARLEVOIX ORIENTED HILL LAND MANISTEE-GRAYLING PLAIN KALKASKA HILL-LAND CADILLAC HILLY UPLAND HOUGHTON-HIGGINS UPLAND PLAIN MANISTEE HILL-LAND CHARLEVOIX ORIENTED HILL-LAND AU SABLE PLAIN MICHIGAN LAKE-BORDER PLAIN I—XL—ICD'UITIUOCDD Figure 4.--Major Physiographic Areas of the Study Region. 22 an evenness to yearly temperatures, contributing coolness in the summer and absorbing somewhat the extreme cold of the winter season. Inland Lakes and Streams The Region possesses over 4,000 inland bodies of water lZl of which are lakes of over 200 surface areas in size.4 There exist over 4,200 miles of streams and 500 miles of Great Lakes shoreline5 (Figure 5). Scenic features are prominent factors in the natural attractiveness of the Region, as many lakes and streams possess clear waters capable of supporting a variety of game fish. Specialty streams for trout and salmon exist along with lakes containing many species of fresh water game fish. In addition, the Region contains many wetland areas in which important wildlife species live in their native habitat. Deer, snowshoe and cottontail rabbit, raccoon, squirrel, pheasant, ruffed grouse, duck, goose, bobwhite, and songbirds including the unique Kirtland's Warbler are among important species to be found. Elk are preserved in a special wilderness area in the Region's central highlands. Rivers and Floodplains The Region is noted for the scenic beauty of its rivers. Beginning in the highlands of the central portion, the Region's , 4Humphrys and Colby, Water Bulletin, Nos. 15 and T6, Depart— ment of Resource Development (Michigan State University, 1962). 5Michigan Department of Natural Resources, Water Resource Commission, County Water Resource Data Sheets (Lansing, Michigan, l959). 23 .cowmmm ms“ mo mcm>wm ucm mmxm4--.m mgzmwm 24 Rivers flow toward the eastern, northern, and western shores, creating scenic lakes along many less swiftly moving portions. Major river systems include the Boardman, Betsie, Jordan and Big and Little Manistee Rivers which flow into Lake Michigan, and the Cheboygan, Indian, Thunder Bay, and Au Sable rivers which flow into Lake Huron. Of these, the Big and Little Manistee and the Au Sable are renouned for extensive scenic reaches and offer excellent canoe travel possibilities. Other regional river systems are important breeding areas for trout and salmon and provide extensive stocking for the Great Lakes' commercial and sport fisheries. Groundwater The Region possesses widely varying supplies of groundwater due to variance in soils and subsurface water—bearing strata characteristic of glacial depositions of the area.6 Generally, in regional gravels and sands, water is abundant. Near the Lake Huron shoreline groundwater supplies are restrictive due to impervious clays found in the shoreline plains. For most ground- water supplies, recharge areas are nearby and regional groundwater constitutes one of the most important of the Region's natural Y‘ESOUY‘CES . Forests The Region possesses over four million acres of forest land in private and public holdings, 99 percent of which is land in 6Dorr and Eschman, op. cit., p. l85. 25 7 Approximately 68 percent of regional commercial timber production. land is forested. Many of these lands are managed for multiple uses, permitting wildlife preservation and recreation in addition to commercial timber production. Important commercial softwood species include pine, spruce, balsam fir, hemlock, and northern white-cedar; principal hardwood species include oak, yellow birch, hard and soft maple, beech, ash, paper birch, aspen and elm. Timber output is increasing as growing stocks reach maturity. Many stands were established during extensive state replanting operations which occurred in the early part of the century. Total regional annual allowable cut is presently approximately 76,000,000 cubic feet of timber. In 1972 actual timber harvests for all classes 9 The and species totalled approximately 43,000,000 cubic feet. Region's timber industry forms an important component of the regional export base, supplying a large volume of wood and paper products to other areas of the Midwest. Figure 6 shows the distribution of State and Federal forest land in the Region. Economic History Although the indigent Indian population formed systems of barter and exchange throughout the Region, fur trapping was the sole 7U. S. D. A. Forest Service, Forest Areas in Michigan Counties (St. Paul, Minnesota, l966), p. 5. 8U. S. D. A. Forest Service, The Growing Timber Resource of Michigan, 1966 (St. Paul, Minnesota, l970), p. 4. 9U. S. D. A. Forest Service, loc cit. ...... c .u.....--.. 0-... I' o Figure 6.-Public Forest Lands in the Study Region. 27 export activity dating from the time of earliest French explorations in the 17005 until the mid 18005. At that time lumbering became the dominant activity and until well into the 19005 flourished as the Region's economic mainstay. The Region's rivers and streams were used for transportation for extracted timber and permitted the floating of logs from forest logging areas to sawmill towns near the rivers' terminations at the Great Lakes. .The lakes themselves provided transportation of timber to points to the south. Lumbering had become big business in Michigan by 1880. In the years following the Civil War, large in-migrations of people occurred as the pine forests provided raw materials for the flourishing mills. Lumbering reached its peak around 1900, and at that time regional population reached its highest level in history. As the land became cleared, pioneers moved northward to homestead. They found soils which were fragile and unable to systain the demands of cropping. Soils were seldom good for more than a short period before fragile top layers gave way and farming had to be discontinued. The rich lands to the south did not find their counterparts in these northern areas, and many homesteads were 10 Nevertheless, certain abandoned soon after their founding. specialty crops such as berries, cherries, and potatoes, and low intensity uses of land such as pasture and hay production became important longnterm agricultural activities. 10F. E. Lewis, Michigan Yesterday and Today(Hillsda1e, Michigan, 1967), pp. 292—300. 28 By 1910 lumbering had declined as the principal industry of the Region and many towns which had sprung up in the wilderness to serve as transportation centers or sawmill towns were left without an economic base. Some of these towns became centers of trade for agricultural enterprises which were gaining a foothold in the Region; others faded from existence. Towns fortunate enough to be situated on the shores of the Great Lakes or at important transshipment points between railroads and rivers were able to maintain their existence and many are important centers of trade in the Region today. In 1902 the State Forestry Commission was chartered by the State Legislature, its major mission to re-forest the cut-over areas created by the extensive lumbering operations of previous deCades. Reforestation was a vast undertaking. The first replantings were performed in the Higgins Lake area around 1902. Thereafter, successive forest preserves were established. These eventually covered most of the stateowned lands of the Region with new fbrest growth. Private land holders cooperated in the effort through private plantings, and by the 19505 major forests once again covered the Region. The Region has attracted tourism and recreation since the Civil War. Tourism increased steadily after 1945, and construction of modern highways has made the Region accessible for increasing numbers of visitors. Efforts to take advantage of this potential for recreation have resulted in many public and private recreation services. The availability of high quality public forest lands and 29 the many lakes and streams of the Region combine to sustain a high demand for recreation development. Seasonal Home Development The natural attractiveness of the Region has also created 1] Many lake areas a strong demand for second or seasonal homes. experience intensive shoreline development. Rapid development of former agricultural and forest lands currently is taking place as private developers seek to satisfy a growing second-home market. As demands upon regional recreation resources continue, extensive planned developments of seasonal homes are more likely to occur. There are many areas within the Region where large numbers of second homes presently exist. In many of the Region's lake areas, new seasonal home developments have limited easement access to lake frontage for boating and other purposes. It seems likely that some lake areas will experience major re—development to accommodate greater numbers of seasonal dwellings.‘2 A survey conducted by the Northeast Michigan Regional Planning and Development Commission indicated that as many as 35 nRecent Surveys indicate that seasonal dwellings may contri- bute as much as 43 percent to the total regional stock of single- family dwelling units. Northeast Michigan Regional Planning and Development Commission, Regional Planning Handbook (Rogers City, Michigan, 1973), p. 29. ‘_E IZPrivate developers have not confined purchases of land to marginal or sub-marginal agricultural areas; prime agricultural land in the Mission Peninsula, for example, is currently under high pressure to shift to recreation use. In other parts of the Region land speculation has resulted in the purchase of land as much as two miles distant from a prime lake access. Such pressures if unv regulated will cause ultimate loss of regional scenic and recrea- tional resource potential. 30 to 50 percent of regional dwellings are seasonal with some regional 13 July areas reporting as much as 90 percent seasonal occupancy. and August retail sales are typically double those of January and February, an indicator of regional population changes due to summer season transient population.‘4 Future shifts in land use are likely to follow the current trend from agriculture use to second home development. Many present seasonal homes are single units located upon small tracts of former agricultural land situated in the Region's outlying areas. The purchase of asmall tract of remote agricultural land or agricultural land which has reverted at least partially to forest is not uncommon for new single home development. Other seasonal home develOpment occurs near cities and accounts in part for the rapid expansion of urban areas. Pronounced linear expansion outwards from city centers takes place along major regional transportation routes, especially notable in areas of high recreation density. Other Shifts of Land Use The diminishing amount of agricultural land within the Region is explained in part by urbanization and increases in recreational usage, but other shifts from agriculture occur through a process of reversion to forest production either by deliberate re-stocking or by natural re-seeding of idle lands. 13Northeast Michigan Regional Planning and Development Commission, op. cit., p. 13. ”mm, p. 21. 31 Some of this land might be reclaimable for agricultural uses should agricultural prices rise high enough to justify return to production. However, many of the soils of the Region are of marginal fertility and require large maintenance to produce adequately. Lands shifting out of agricultural production are likely to be marginal lands which may not move back into production easily or cheaply. Shifts to predominant recreation usageof privately-held timber land is also widespread. There is an historic attraction to the region by private wildland groups and hunting clubs. In other areas, forest land suitable for skiing and winter sports is often partially cleared of timber in the recreation development process. Still other forested lands may become partially cleared of timber for developments of second homes, especially when sites are near other recreational attractions. Campgrounds and Multi-Use Recreation Centers Private campgrounds typically experience a condition of overcrowding even in areas away from major attractions during the summer season. The installation of winter recreation activities has extended the popular times for recreation into the winter season and more private recreation enterprises are appearing in areas where winter activities as well as summer activities can be developed. The density of private campgrounds along the Great Lakes shorelines has steadily increased, and these sites are normally among the most populated during peak summer season demand. 32 Recent studies suggest that regional campground activity may be reaching a peak in terms of annual numbers of newly installed units,15 but there are indications that more complex recreation centers may become increasingly popular as the need appears for new facilities to meet increasing demand. Multi-use centers offer a range of recreational activities and possibly seasonal-home condominium type developments as well. Large recreation complexes constructed along these general lines can become important future focal points of regional recreation attraction. Sppply of Land Transportation While adequate highways exist in the western part of the Region, there are many areas in the eastern part that are less well- serviced by highways. Rail service is an important component of the transportation system. Greater passenger transportation to regional centers of recreational attraction may become more significant as costs of fuel for transportation increases. At present rail service on many lines is being abandoned or proposals for termination are being considered. . Important service for basic industrial activity is thus curtailed. The Michigan Department of State Highways and Transportation recently issued a report outlining the requirements for railroad continuance, 15Eugene F. Dice, Supply-Demand in Michigan Campgrounds (Michigan State University, l975)1 33 consolidation, and abandonment.16 The report states that 61 percent Of total track miles within the 10 counties of the western portion of the Region (those comprising State Planning Region 10) are cur- rently classified as potentially abandonable. The report makes clear that the impact of rail abandonment in the Region will have to be offset by increased highway use if continued industrial growth is to occur. Air transport facilities are available at major points throughout the Region. An important service for tourism, small regional air carrier services operate supplemental routes to the 17 Rapid access to large metropolitan larger common carrier routes. centers outside the Region is available on a demand basis. Pro- vision of adequate airports and facilities forms a major public input to regional growth. Agriculture and Timber Land In the face of steadily diminishing numbers of farms and diminishing amounts of land agricultural production still contributes importantly to the economy of the Region. Several specialty crops are grown. The Traverse City area produces over 60 percent of the Nation's sweet cherries. Potato production, especially in the 16Michigan Department of State Highways and Transportation, Michigan Railroad Needs; A Planning Report (Lansing, Michigan, 1975). 17Small air carrier services may however be unstable providers of transportation due to fluctuating demands and typically small profit margins affecting these operations. 34 eastern portions of the Region, is an important economic use of land. Small grains and livestock are also important to regional agriculture, and the Region's hay producing lands supply hay for export to other State areas. Total land area presently in regional agricultural use is 1,023,000 acres, or 16.1% of total regional area. This compares with 32.7% of State area in agriculture use. Total acres in forests are 4,167,000 for 65.5% of regional area.18 The availability of agricultural land is constrained, as forest land must be cleared and converted to provide new agricultural areas. Forest land area may be increasing somewhat as marginal agricultural lands revert to forest use. Housing Land Housing availability in the Region is complicated by the large incidence of second or seasonal homes. The 1970 Census of 19 shows a rising trend in numbers of regional housing Housing units but other sources estimate that as much as 40 percent of available housing currently exists in the second or seasonal home category.20 Land for new housing is usually made available from prior agricultural or timber lands. Areas of heaviest conversion to 18Michigan Department of Commerce, County and Regional Facts (Lansing, Michigan, 1972). I9 U. 5., Bureau of Census, 1970 Census of Housing, 20Cf. footnote on pg. 29. 35 housing occur near the Region's growth centers. However, de- centralization of housing developments seems to be one of the factors characterizing present regional housing trends. In outlying areas platting of former agricultural land and timber land for home development is occurring.21 While region- wide there is adequate supply of land for new housing, housing land supply in the most desirable areas is increasingly scarce and competes with other potential uses. Taxes affect costs of land-use conversion and profitability of operation. Currently, little variation in tax structures exists in the Region at the county level. Township taxes, however, have recently exhibited instability. Township tax rates may be a highly important factor in decisions to locate new housing development, and tax rate uncertainties are a factor operating against housing development.22 Zoning structures also are important determinants of land use. Within the present framework of enabling legislation, zoning may be used effectively by local units of government to control certain types of development. However, the full power inherent in zoning has not been fully implemented by local governmental units in the Region. Effective zoning control over land use requires strong applications of the zoning regulatory mechanism. The type 2IFor projections of regional second home demand into the 1980's see Thomas Marcin, Projections of Demand for Housing_by Type_ of Unit and Region (St. Paul, Minnesota, 1972). 22Allan A. Schmid, Converting Land from Rural to Urban Uses (Washington, D.C., 1968). 36 of zoning applied by local governments and its extent can be an important determining factor affecting the future allocation of land. Educational efforts by county extension agents and other organizations are currently attempting to make zoning options visible to local residents and land owners. These efforts will have a marked impact upon future land availability. Recreation Land An exact determination of the recreation land supply presently available is complicated by the various types of recreation use as well as the multiple-use character of much existing recreation land.. The Upper Great Lakes Regional Planning Study projected to 1980 the expected user-day demands upon recreation services by 12 recreation categories. These demand projections show the relative importance of each category of recreation to the Region.23 The relationship, however, of categories of recreation to corresponding demand for land is not completely specified by current data. The amount of land used for public and private campgrounds, for example, is known; but the total amount of land used for recreation in all categories is not known. The regional supply of land for recreation use is not assumed to be scarce. Exceptions are those areas around inland lakes which have high recreation potential and along Great Lakes 23University of Wisconsin Extension, Upper Great Lakes Regional Recreation Planning Stuqy_(Madison, Wisconsin, 1970)I . I 1“ XIII Ill! IIIA‘ 37 shorelines where competing pressures for second home developments are high. Summary The chapter has outlined the physical and socio-economic background of the 18-county Study Region. The Region possesses highly desirable climatic factors for recreation activity. Surficial features of hill-lands interspersed with numerous lakes and rivers characterize the Region's topography and forests comprise the dominant land use, contributing to the Region's recreation attractive- ness. Significant land use pressures currently exist in the Study Region due to demands for industrial growth, permanent and seasonal home development, urban expansion, and recreation development. Regional land for these uses is scarce near areas of high scenic and recreation attraction and there is competition for land among several major alternative uses near centers of population and trade. CHAPTER III A REGIONAL INPUT-OUTPUT MODEL Introduction Considerable growth and diversification have occurred in the Study Region since the period of the 1940's. As the research intent focuses upon the relationship between macro-economic regional infrastructures and the micro-economic aspects of regional land use, the detail required by the macro-economic portion of the study justified construction of an input-output model. This chapter contains a description of input-output methods and a description of a l4-sector model developed for the study from secondary sources in which regional employment figures are used to derive final demands and total gross outputs. Economic sectors which are primary users of the land resource base are given model emphasis.1 ’The input-output model formation begins with an 87-sector state accounts table from the U. S. Multiregional Input-Output Model (MRIO) originally developed as part of the Harvard Economic Project.2 A series of key years from 1947 to 1980 are analyzed with the model to assess major trends in regional growth. IIn an input-output model, sectoral aggregation procedures are based on similarities of the physical inputs and outputs and similarity of the product mix of the various industries. For a reference to the methods employed, see Harry N. Richardson, Inputv Output and Regional ECOnOmics (London, 1972), p. 92, 2Karen R. Polenske, et al., A Guide for Users of the u.s. Multi-regional Input-Output Model (Washington. D.C., 1973)} 38 39 Input-Output Analysis Input-output methods are frequently used components of regional economic studies as they provide detailed information about relationships which exist among a region's economic sectors. A major purpose of a regional input-output model is to track various components of regional growth, to identify specific growth sectors, and to help the economist determine existing levels of import and export activity to and from the region. In an input-output model, the structure of the regional economy is defined through a quantified network of interdependent relationships. Interdependencies existing between industries purchasing and selling goods and services among one another and hndustries selling to market final demand are represented explicitly. ‘ Determinations of the influence of these relationships upon other regional economic entities, as for example the effect of an increase of a sectoral final demand upon total regional gross output, can be deduced through the use of mathematical procedures. To construct an input-output model we adopt an accounting framework. Let inter-industry transactions represent the flow of commodities, in dollars of trade value, between industries. These transactions represent intermediate industrial sales and purchases of goods and services which enable the creation of final commodities 3 to meet consumer demand. In terms of a general equilibrium model, the system appears as follows. 3The reader not familiar with the mathematics of input‘output will find materials in Richardson, op. cit., Chapter 2, to be helpful. 40 Let xi represent the amount of production of industry 1_ J which is sold to industry j, Let FDi represent final demand existing in industry i_or the amount available, after inter- industry sales from industry i to other industries, for final consumption. Then GOi represents the total output of industry j, the sum of all sales to other industries plus final demand: Total Industry Final Gross Purchasing, l 2 Demand Output Industry Producing 1 x11 x12 FD1 GO1 2 x2] x22 FD2 G02 Reading across the rows (sales) the total amount produced by an industry j_is consumed by other industries' purchases plus final demand F01, the amount available to final consumption. The amount sold by industmy i_to another industry j_can be found by selecting the jth row and reading across to column 1; similarly the amount purchased by an imdusmry j_from another industry i_can be found by selecting the appropriate column and reading down until the selling industry i_is found. The inter-industry transactions can be represented as a set of ratios called technical coefficients or direct requirements formed by dividing each intervindustry transaction x.. by the total 13 column sum gross input, or GI. 41 GI ij II J. M X I X l =__l .. The aij are computed as aij GI for each xi of the original J 3 table. Since the input-output table is a balanced accounting frame- work in which GI = G0, the final form of the table can be represented in matrix notation: A x GO + F0 = GO where A_is the matrix composed of all aij’ §Q_the matrix of total gross outputs, and [Q the matrix of final demands. Rearranging and placing in factored form, FD = (I - A) x GO where I is the identify matrix and (I — A) is called the Leontief matrix. The elements of the Leontief matrix represent the proportions of amounts consumed by all regional industries in the manufacture of regional commodities to their respective industry outputs G9, In an expanded tableau4 form the table appears: GO1 GO2 4The tableau form is used in linear programming to describe the structure of linear systems such as the above. For a full dis— cussion, refer to David Gale, The Theory of Linear Economic Madels (New York, 1960), pp. 97.104. 42 and for a three sector model: GO.l G02 603 I (I'aii) 'aiz 'ai3 = F0] 2 -a21 (l—azz) -a23 = FD2 3 -a3] -a32 (l-a33) = F03 The extension to p_sectors is straightforward. Since FD (I - A) x G0 we can write (I - A)" x FD GO Assuming a projected vector of final demands {D}, it is possible to derive a new vector of gross outputs for an economy G9]. From G9] it is possible to derive a new matrix 5} of projected transactions necessary to produce the new gross output G9}. [X'] = A x 60' Thus, it is possible to obtain a set of projected regional transactions corresponding to projected final demands, and thereby trace future sales and purchasing impacts upon individual industries which originate from changes in regional demand: These analytical procedures provide insight into the network of interacting transactions between a region's industries and the outside world. When used as a forecasting tool, the general input- output model rests on several assumptions. The first is that of 43 constant technology. Like a photographic snapshot, input-output models represent an economy at a single point in time; the model is static in that changes which normally occur through changing technology or through changing patterns of demands for goods and services are not represented in the input-output framework.5 As in a still photograph, the motion of technological change cannot be observed with an input-output model. Such change must be inferred from other data sources, or by a time-series of accurately derived input-output tables. A second and related assumption is that time-rates of resource use do not change; in an input-output model, substitution of one resource for another due to scarcity is assumed not to occur, since all resources are infinitely available. These assumptions imply unchanging relationships between industries buying and selling to one another in the economic trade~mix. We say that trade coeffi- cients in the input-output model are constant with constant relative proportions between production factors. No substitution of one factor for another can occur. Because of these assumptions, technology existing in a former period transfers into the forecast period when the model is used as a forecasting tool.6 51bid., p. 168. 6The unrealistic nature of these assumptions for long economic intervals usually limits the usefulness of model forecasts for any but very short time periods. For example, the model cannot effectively illustrate changes which occur in an economy's structural relation‘ ships due to modified patterns of consumer preferences or changes in institutional factors related to matters of public policy. This limitation of "constant technology" in the input-output model is not assumed to be a severe one, however, when a relatively stable economy exists in which technological changes occur slowly over time. If 44 Within the limits set by these major assumptions input-output analysis is often used to forecast and evaluate impacts of projected changes in economic activity by tracing through an economy's short-run infrastructure. Development of the Regional Ipput-Output Model In the present study several steps were employed to develop the regional input-output model from original state MRIO accounts tables. In order, the steps are as follows: Step 1. Consolidation of the 87-sector MRIO state accounts table into a 14-sector table. Step 2. Computation of base year final demands for the state.7 Step 3. Computation of 1947, 1958, 1963, and 1970 state final demands from Michigan State Department of Commerce data. Step 4. Development of a 1980 forecast of final demand using Office of Budget Economic Research Service (DBERS) projec— tions.8 Step 5. Reduction of state final demands to regional final demands for key years 1947, 1958, 1963, 1970, and 1980. forward projections are made in the short run before any significant structural changes occur, the input-output model can be assumed reasonably accurate in its predictions. For cases where significant technological changes do occur, a modified set of direct requirements must be used and the model structure changed period by period in a comparative static manner. 7The Base year chosen was 1963, corresponding to the latest year for which MRIO data is currently available. See Polenske, et al., op. cit. 8U. S., Department of Commerce, DBERS Projections. 45 Step 6. Computation of regional gross outputs. Step 7. Expansion of gross outputs to regional transactions tables for each key year of the trend analysis. The 14 sectors of the consolidated model9 were selected on the basis of the Standard Industrial Classification Index, and are listed in Table l. Consolidation of the original 87-sector state model to a 14-sector model was accomplished by a column and row n n .. = Z 2 x.. where Xi' 1 .. is an aggregated transactions 3 i=m j=m J summation Xi. J table entry of the l4-sector model in position i'j' of the l4-sector table; xij are transactions table entries of the original 87-sector model m,n are summation limits, equivalent for both rows and columns. With these minor adjustments the aggregation procedure permitted con§olidation of the 87-sector state accounts table into a l4-sector table.10 The consolidated l4-sector state transactions table was then summarized to obtain total sales, total purchases, final and 9In certain cases, individual rows and columns of the 87- sector model were deleted in formation of the consolidated table, an adjustment intended to reduce the predominant influence of the state's automotive industry upon the technical coefficients of the regional manufacturing sector. 10In the original 87-sector MRIO model total sector pro- duction was found not to match total sector consumption. Differences . between column and row sums are partially due to import and export trade flows and minor inaccuracies in forming the state table from a national model. Adjustments applied to regional tables in the present study balance production and consumption by sector. 46 TABLE l.--Sectors Included in the Consolidated State Input—Output Model 1. Agriculture 2. Forestry and Fisheries 3. Mining 4. Construction 5. Lumber and Paper Products 6. Manufacturing 7. Transportation 8. Communication, Public Utilities 9. Wholesale, Retail Trade 10. Finance, Insurance, and Real Estate 11. Lodging Services 12. Amusement 13. Other Services 14. Government; federal, state and local ‘1 The consolidated the average value-added figure for each sector. state accounts appear in Table 2. The l4-sector state accounts were then used to develop a set of direct requirements for the region (Appendix B). Final demand tables were constructed for the years 1947, 1958, 1963, and 1970 with projections to 1980. Employment data nAverage value added (AVA) for each sector is computed by dividing total sector production (column total) into sector value- added figures. 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R1 for i = m+l, . . . , M J 3 (land resource constraints) where xj are sectoral gross outputs Cj are the objective function coefficients Di are sectoral final demands Ri are areas of land resource units Xj _3_ 0 for all j and each aij = aij (£1, £2, . . . 2") is dependent upon land charac- teristics in each of the several use categories.4 In the general case a land parcel or parcel group may contribute its productivity to several economic sectors. For exclusivity of use an additional constraint can be imposed specifying that the parcel productivity be directed to but one of a set of sectors. To establish this condition, for a single land parcel potentially able to contribute to p_economic sectors, there are p constraints of the form cu x l 1 U' | A o A O a. X - r0 (j'+p) be k 1 k p 1p 4Each of the aij (£1, £2, . . . , K ) may exhibit time varying behavior and is evaluated for each period 0 the comparative static analysis. Structural coefficients of the economic portion of the model (Xij) may also change period-by-period indicating changing interindustry transactions ratios and changing regional economic infrastructure. See Dorfman, Samuelson, and Solow, op. cit. 89 and one constraint of the form . .. + . .. = 1 r113 r1p (1+p) the additional constraint specifying employment of the parcel in one and only one sector (mutual exclusivity of use).5 Because of the presence of binary valued 1 variables (variables which take only the values 0 or 1) the model requires mixed integer algorithms for solution.6 Figure 12 shows the general outline of the tableau of the economic and land-use linear program model. Rent Changes Land is inherently fixed in quantity. Because fixed quantity creates supply inelasticity the cost of land is almost wholly market determined and returns to the aggregate land base as a factor of production are product price-determined. From the point of view of 5For the case of a land parcel potentially contributing productivity to two economic uses simultaneously (e.g. as in multiple use of forest lands for, say, both timber and recreation) a constraint of the form . x. - r.. b. < O 131 J1 J 11 T a. . x. - r.. b. < o 'sz J2 3 12 “ a. 'I can be used where r'- is a single column activity of the model. A value of rjn = l wiIl cause both land uses to contribute gross out- puts xj1 and Xj respectively. Note that bi and biz are not necessarily the same even though they refer o the same land parcel; in the general case effective area may be different in each economic use. 6For a reference to mixed integer algorithms see Harvey M. Wagner, Principles of Operations Research (New Jersey, 1969), pp. 445- 511. 9O OBJECTIVE FUNCTION GENERAL FINAL II/ ECONOMIC MODEL DEMAND LAND USE COEFFICIENTS ‘1: AREA 4 RESOURCE OTHER RESOURCE REQUIREMENTS ___ CONSTRAINTS Figure 12.--Major Sections of L.P. Model in Tableau Form. 91 a small land using enterprise in a market where a large number of firms determine the demand for land, all returns to land appear to be price determining rents similar to costs of other factors of production. These rent returns derive from specific qualities of location and inherent local resource availabilities. In a given locale, as a result of changes in aggregate economic demand in one or more regional sectors, rent returns from one land-use category may shift up or down compared to others. The small land owner perceiving these changed rent conditions may choose to shift the use of land into a new use-category. To derive probability generating functions for use-shift based on land rent change define a valuation criterion describing the “net benefit" perceived by a land owner whose activity is that of shifting land-use from use category j_to a category k_as follows: n kt A n " jt A m Cjkt letv.=Z——-- z———->:—————— 3k t=1 (1 + i)t t=1 (1 + i)t t=1 (1 + i)t where vjk is the economic benefit7 of a shift from use category j to use category k, n. are shadow prices per acre in use category at and nk t 7This formulation of the owner's returns may be recognized as a net present worth in which future revenues and costs are brought to present values by means of computation with the interest rate 1 in the formula l/(l + i)". Other formulations may be used, such as the net future value in which all revenues and costs are advanced to a future time period; or the composite internal rate of return criterion in which the average return rate of capital investment is computed; or the benefit/cost ratio in which capitalized returns are divided by capitalized costs. Although possibly yielding different optimal times, all such measures give the information required by the landowner to make the use-shift decision and are . essentially equivalent to one another in that each describes whether 92 j and k respectively in time period t; A is the area of the land is the cost of conversion. The variable 9 . . . 8 unit being shifted and cjkt i is the rate of interest on capital, the variable n is the number of periods over which the new economic use is to remain effective and m is the amortization period of invested capital. Since costs of conversion from use j to use k can be considered to be fixed and independent of returns to investment as well as independent of opportunity foregone by shift from any previous economic use, we may assume the landowner will normally make the decision to shift use categories when the difference between "kt and fljt capitalized over time is large enough to offset costs Cjkt similarly capitalized; it is the difference ”k. - “j. which is the primary decision variable.10 returns will be greater than costs over time for the considered investment. 8Marginal factor price appears in dollar units per unit of land area, and when used as a rent must be multiplied by the land area of the parcel. 9Interest rates may vary from period to period and may also be valued differently between costs and revenues during the same time period. The symbol j_denotes interest as a general factor, with understanding that interest may take different values in differ- ent time periods or different values between revenues and costs in the owner's net-benefit expression. 10As a practical matter, most land owners who act as decision makers add to the above net-benefit criterion an element for un- certainty which when incorporated into the decision criterion causes the rate-of—return threshold to rise. Thus, rather than a decision threshold at which net present returns NPR less present costs NPC equal 0, we have instead NPR - NPC = U, where U denotes some positive value indicative of the owner's general unwillingness to invest until assured of returns great enough to cover initial investment plus unforeseen losses. 93 The general fbrmula for net benefit can be expanded to include tenns in the series which reflect the influence of public policy. For example, a series of terms can reflect incidence of use-value or property tax assessment allowing the model to reflect these factors in the decision process. The comparative static nature of the model requires that all returns and costs in the benefit-cost stream be reduced to a single composite value for comparison with values associated with alternate uses. This procedure brings all future returns and costs to the specific economic period represented by the model. Evaluation of series expansions for net benefit computations is performed by a report generator portion of the model system subsequent to each solution pass of the general linear programming model. Assumptions of the Model Economic Rent.--In addition to the basic assumptions of . . 11 . . . linear programming there are several important economic assumptions of the model. A primary assumption is that of a constant returns to scale production function for each land unit identified in the model.12 1lThese include formulation of an objective function and a set of constraints as linear functions of variables, and the divisibility, additivity, and substitutability characteristics of these models. The reader not familiar with linear programming will find a text such as David Gale, The Theory of Linear Economic Models (New York, 1960) a help in understanding the assumptions of the general linear programming model. 12A constant returns to scale production function states that increases in factor levels result in proportionate increases in output at all levels of production. That is, there are no economies or diseconomies of scale. An extension of the assumption states that there exists no minimum number of land area units required 94 Under this assumption marginal factor price used as a rent proxy is not influenced by the areal size of a given parcel. Another assumption concerns use of the term "rent." The term rent is used in the model context to refer to total economic 13 Employed factors on each unit of returns for a unit of land. land are assumed to be labor and capital; with the assumption of perfect spatial mobility of the latter factors, and the assumption of homogeneous distribution of wage rates and costs of capital over the Region, the model describes economic rent in the von Thfinen sense as a return to the spatially fixed factor of production represented by the locational attribute.14 for a given economic use; size of parcel is thus not a determinant of use category selected and it is assumed that the size of parcel (parcel land area) does not enter the decision process in shifting to new use categories. This assumption will be held valid "on the average" with the large number n of land units employed in the model. 13In the model solution marginal factor price times associated parcel land area is a measure of economic output of a land unit. The Dual Theorem of linear programming specifies that the value of the primal objective function is equal, at the optimal solution point to the value of the dual objective function: chxj - 2111b]. the dual objective function is the sum of marginal factor prices times right-hand-side values; the portion of the sum which pertains to land-use constraints is the contribution made by the land units to the attainment of the stated regional objective. When the objective is to maximize gross regional output, individual marginal factor prices time parcel land area indicates total economic output of a land unit. For more elaboration on the subject of duality, see Wagner, op. cit., Chapter 5. 14The argument for the von ThUnen character of the rent solution is strengthened by inclusion of distance and proximity factors associated with transport costs in the determination of the model's land parcel productivity ratios. In a completely competitive 95 Ricardian aspects to the model-determined rent are evidenced in the non-uniformity of regional land areas and their varying site productivities. Generally, use-intensities will rise in areas having high productivity potentials for given use-categories and 15 The pattern diminish in areas having low productivity potentials. of spatially distributed productivity potential is likely to coincide with the distributed pattern of actual uses which ultimately result Under the assumptions of a perfect market the latter should show similar variation over space.16 market, von ThUnen's original analysis assumes constant money wages over space implying a fall in real wage (purchasing power) with increasing returns to the varying location factor; on these grounds von Thfinen's wage arguments have been criticised as logically inconsistent with assumptions of a static state. Our assumption of homogeneous wage rates and capital costs over the Region is similar to a von Thfinen analysis in the classical sense in that it ignores changes in spatial distributions of capital to labor ratios. Recognition of distributions of wage rates and capital- labor intensities properly belong to a market analysis, but are omitted in this model as in von Thfinen's model, for simplicity. Logical consistency is obtained on the basis of the comparative static nature of the model. Structural changes can be introduced in inter-period analyses to reflect local differences in capital- labor intensities. A discussion is contained in Peter Hall, ed., Von Thfinen's Isolated State (London, 1966), p. xxii. 15Identification of the intensive and extensive margins of use are implied in the choice of use-intensity for a given site location. For a full discussion of intensive and extensive margins and their implications to the location decision see Raleigh Barlowe, Land Resource Economics (New Jersey, 1972), pp. 120-155. 16The rent factor in location decisions is iven thorough treatment in William Alonso, Location and Land Use ICambridge, Massachusetts, 1964). 96 A final assumption is that the difference (iikT - an) remains stable over the economic period covered by a model solution.17 An economic period of sufficient length to accommodate the planning horizons of most business firms investing in land-use changes should 18 Since the purpose of the model is to aid the prediction be used. of major, long-term land-use changes, a period of sufficient length to accomplish the latter prediction should also allow the model's use in identification of areas likely to receive impact due to recreation activity in public recreation areas. The rationale behind this assumption is that investment in public sector recre— ation should respond to the same demand conditions as exist for private recreation. Model periods of sufficient length to accom- modate the private sector planning horizon should also be adequate to allow identification of public areas likely to receive signifi- cant recreation impact. 17While a constant rent differential may not appear realistic and in accord with adjustments which occur due to entry and exit of firms from the market, the model intent is to show the potential differences at the beginning of a period. Subsequent analyses of the inter-period will accommodate rent-modification due to entry and exit of firms. 18Five years is suggested as the most useful interval between subsequent program analyses since changes indicative of major economic trends are likely to occur only in a period of this length or longer, given the measurement accuracies of most economic trend data. Short period trend data indicating minor business cycles are likely to become distinct in a five-year period, and the direction and rate of major long-term trends can be computed by comparison of series of five-year periods. 97 The Regional Objective Function The regional objective function should be formulated to optimize total regional economic well-being, and this goal almost certainly requires that all economic sectors be represented in the relative proportions of their contributions to the regional economy. As in previous studies, an average value added ratio (AVA) is derived for the 14-sector regional input-output accounts and is assumed to represent the aggregate regional goal in the absence of over-riding central policy directives. The model is sensitive to proper choice of objective function coefficients as marginal factor prices depend directly on the objective function through the relationship where n is the vector of derived marginal factor prices, c0 the basic (optimal) vector of objective function ciefficients and B"1 the inverse basis. When the model is used to aid the prediction of private sector land-use changes it is necessary that the objective function accurately reflect regional economic goals. The regional planner must be aware of the constraint network operative on regional options to modify a particular sector, and also recognize social goals which may preclude or preempt the objective function emphasis of certain critical sectors. The choice of objective function is one which demands careful study and systematic examination of alternatives in the design of the model. 98 Spatial Shift Likelihood With these assumptions it is possible to define a shift likelihood using model-derived marginal factor price as a principal explanatory variable. A matrix of spatial likelihood of land use shift was defined as follows: Let a probability of land use shift p = f(vjk) where vjk is a net-benefit criterion evaluated as before, specifying returns to private investment for a given shift of use of a single parcel of land. Let the general probability of shift of a single parcel from use category j to category k be pjk' Then ij represents the spatially ordered matrix of probabilities of use shift from category j to category k for all land parcels in the region. The matrix P11 P12 ' ' ' p1k P = 92] P22 . . . sz PK1 sz . . . PKK K K 2 z p. - 1 .1ka can be used to represent land use shift probabilities for K land use categories.19 19A large number p_of land parcels or parcel groups is required to determine probability estimates in the model. The required number increases with increases in the number of use categories K. Aggregate land areas can reduce the discriminatory power of the model and may necessitate a reduction in the number 99 Solution Sequence and Calibration The model has economic constraints of the form 2 a.. x. > D. 13 j —- l and land-use constraints of the form 2 a..x. < R. ij J —- l Constraints of opposing sense create possibilities of infeasible solutions which may be troublesome especially during multiple runs (sensitivity testing or recursive solution sequences used for calibration). Some researchers use a bounded right-hand- side vector in the economic portion and reverse the sense of the economic constraints to insure initial feasible solutions.20 The approach adopted in the present study is somewhat different. Calibrated input-output data from an input-output model solution is used to form economic technical coefficients aij and economic demands Di of a Stage 1 solution pass with only economic data present. A Stage 1 solution pass with only economic constraints present allows calibration adjustment of objective function coeffi- cients so that all sectors appear active in the linear programming solution at the specified levels of final demand. Adding land-use of use categories K into which land use may be classified. The critical relationship between n and K depends somewhat upon the level of significance of the tests of hypotheses employed in analyses of model results; it also depends to a considerable extent upon accuraey and precision of data used in the model. 20See Lofting and McGauhey, op. cit. 100 resource constraints then expands the model and permits a Stage 2 "full" model solution. This procedure reduces chances of infeasibility during calibration adjustments in the objective function. With resource constraints specified correctly, the majority should be binding upon their respective sectoral gross outputs. When using the AVA objective function Stage 2 sector outputs may drop slightly in the resource constrained sectors and increase slightly in one or more of the "free" (resource unconstrained) sectors. Differences between Stage 1 and Stage 2 sectoral gross outputs can be used to determine the point of attainment of a final, balanced model solution.21 Example Program Run Summer Season Recreation Development Water related recreation is a central feature of summer visitor activity in the Region. Most regional second or seasonal homes are summer-season lake-front cottages offering a variety of water-based recreation activity. As a preliminary exercise of the model it was desired to know the regional distribution of potential 2llhe differences may be of any specified magnitude consistent with the levels of sectoral output and the precision levels attainable with the economic data. A general criterion can be formulated max I le — sz I §_e 3 where x'i, x-Z are output levels of sector j in Stages 1 and 2 respectively, and e is a stated error criterion, derived as a function of precision of the economic data. A value of 2 percent in each zector or ej = .02 (le) might be used if justified by the economic ata. 101 for summer season recreation development. Estimates of county rankings of potential for water-related development were obtained from an earlier USDA Soil Conservation Service study of Michigan counties. The SCS survey considered factors such as county scenic potential and transportation access as well as county nearness to population centers. From the SCS study preliminary estimates of three general recreation types related to water use were obtained (Table £3 ). These were combined into a composite score to indicate relative county potential for water-based summer recreation development. The tendency to locate seasonal homes in proximity to shore- line areas led to right-hand-side resource constraint values based on regional county shoreline extent.22 For the example model run shoreline extent was obtained and evaluated from supplemental data 23 sources. The SCS Recreation Data From the SCS study three general water recreation types were selected: warm water fishing, cold water fishing, and general water 22Length of shoreline, rather than extent of water area, is a logical choice for the constraint upon water-related land use as the length of shoreline, when multiplied by average depth of property in riparian ownership should give an approximate amount of land available for water-related recreational development in each county area. 23Data were obtained from the Michigan Department of Natural Resources, Water Resources Commission, County Water Resource Data Sheets, 1959. Also Humphrys and Colby, Water Bulletin, Nos. 15 and 16, Department of Resource Development, Michigan State University, 1962. 102 sports which includes swimming and boating. With suitable weighting these were combined into a composite score for each county of the Region (Table 8).24 The final composite county scores were numeri- cally inverted and multiplied by a scale constant to form the model structural coefficients. Each structural coefficient is of the form where CS is the composite county score and k = 100 is an index adjustment factor, constant for all regional counties.25 Values of the index j_range from 1 to 18 (for each county of the Region) and the value of ins set at 12 to allow all recreation development to fall within a single economic sector.26 24A weighting factor was applied to account for differences in projected 1980 user-days for general water sports activity over those for fishing. Data obtained from the Upper Great Lakes Regional Recreation Planning Study indicates a projected value of 22332038 user-days for swimming and boating and 6287397 user-days for fishing. The ratio of the two values is 3.55. As more water areas are suitable for fishing activity than for boating and swimming an additional use correcting factor of about 5/6 was applied, making the total weighting factor equal to 3.55 x .82 = 2.91. 25The scale factor was adjusted to allow L.P. model solution values to appear on a scale of 0-10 to provide a direct index ranking. 26As the SCS data do not provide estimates of actual economic potential but only relative county rankings, and since seasonal home development of shoreline areas is a dominant regional land use, no alternate economic uses of shoreline areas were considered in the preliminary model run. Sector 12 of the l4-sector regional economic model was selected as the economic sector closest to water-related recreation development. Since alternate economic uses (uses in other sectors) of shoreline land is not considered in the example, the selection of economic sector will not alter the relative county rankings obtained. 103 ucnemo :oNNnmcumm icowcw anuicmma ome wo mucaom mNNaanou anoh wheNmN .cNaaouth .comNunzv aumnumcom ucn Nw>cam "N Ncna nxuapm chcana cowunmcumm Nnconmm mmxna pnmcu cmam: ms» mN coNNne .Nev ucn NNN N: 2538 we 53255.». n aw 5 $38 wo mcoum .ammuun mahgan cow couunw NncaNNNuun an Na umpumccou .NuN>NNun mawzaww wo axnu 1cmma ome ow NuN>NNun :oNNnmcumc Nncmcmm umNnchicman wo amnuicmma ome umNumnoca wo coNucoaoca m5 .Nm.N coNonw mcwucmwmz m5 3 umNNaNuNaE 83? E 5:38 mcn mmaNn> Nev SENS N NNN aaN ma mN Nm acawxma maN aNN aa aN mm mocaammNz ama amN am am NN ooamNeaz Nae aNN am am Na aaeaNoaa aaa aaN aa mN mN acnexNax mNa amN an mm mm mmco>acN aeaca Nmm amN mN aa Na aoEEa Nam amN mN aa Na xNa>oNcnaa mma mmN mm mm Nm mNNeom ama amN NaN aa aN eNcaaa Nmm amN am am Na mNmN oaamoca aaa amN Na aN NN aaomaa NNN maN am mN Nm aaaoma aNN NNN mm aN aN Neeocagaaaz NaN maN am am Na acawznca Nam NNN aN Na NN aammaaaaa mam aaN aN ma mN eeoaNa Nam mmN Na mN Na neaoN< Nma Nam NNN NNN NNN Nueaaa mcoum apcoam cmpnz aucoam NNN>NN6< NuN>NNu< mNNaanou N Nncmcmo cmunz mcwnmhm mcwsmwa Nnuoh umusmNmz Nncmcmw cmunz uNou cmpnz Ecnz .Nucaou Na mamsaon>mo cowpnmcumm umpnNmm -cmunz cow mamcm>Nuuncuu< Nnconmm chNnuNuaN amxmucN mcoum muNmoaeou wo :oNNnecomii.m ummeh 104 County Shoreline Extent Extent of county shoreline was estimated from secondary data. For each county these data specify area of inland water and the number of inland lakes over 200 acres in size. The computations proceed as follows: Let A = total county water area in acres, n = number of county lakes over 200 acres in surface extent, L = estimated length of shoreline. With the further assumption that such lakes are circular areas, L = 2 VEA x n x k x n where_k is a constant of conversion of acres to square miles and n is the ratio of circumference to diameter for a circular area. For example, a county having 150 individual water bodies of over 200 surface acres and a total surficial water area of 30,277 acres can be assumed to have approximately 2 / 30.277 x 150 x .0015625 x 3.14159 r II 298.6 miles of shoreline To this figure is added existing miles of Great Lakes shore- line for an estimate of total county shoreline. The results of these computations for the 18 counties appear in Table 9. Results Marginal factor prices and evaluations obtained from the model solution are presented in Table 10 which shows evaluations of 105 N.NaN a N.NaN a.aaNa a amN acawxaa a.NNN a a.NNN a.mama a.maN oaxaanmNz a.aaN a.aN a.aNN a.aaNa a.Nam oaamNenz N.mNm a.am N.NNN a.NamNN a.NaN aaeaNaoa N.NaN a N.NaN a.Nmmm a.Nmm acaacNec a.mNa a.Nm a.mNa a.aamNN a.maN amcaNacN aeaca a.mam a.ma a.mmN a.NNaaN a.NNN Nessa N.amN a.NN N.NNN a.mNamN a.Nm xNasaNcaaa m.aaN a.aN m.NaN a.aamNN a.aaN aNneam a.mNa a.mN a.amN a.NNNan a.amN eNcNaa a.amm a.NN a.aaN a.aammN a.mNN aNmN aaameca a.NmN a a.NmN a.NaNN a.aNm amamaa a.mmN a a.maN a.aaam a.amN aaaoaa N.NaN a N.NaN a.aaNNN a.aaN NoemcaEaeoz N.am a N.am a.aamN a.NaN acawzaca a.mNa a.am a.mam a.aamNm a.aam eamcamaaa a.aaN a.Nm a.NmN a.mNmmN a.Na aeaaNa N.maN a.mN N.aaN a.amamN a.mmN neaoNa va Nev NNN NNN NNN caeaaa NaeNNza aeNNacaam NaaNNz .amv Nmacoav maxaa ca mchmcogm mmxna unmcw nmc< mxnm nmc< mxnm cmaEsz caeaaa NaaaN ca maNNz .coNNncNEcmpmo nmc< chchocm mucaooii.m mmmeh 106 .m oEaNoo x N :EaNou a .m aNaaN we a eEaNaaN .a eNaaN we a eeaNaaN NmN.N N.aaN mmNaa. NNN acawxaz mmN.a a.NNN Nmaaa. maN oexaammNz amm.N a.aaN NmaNa. amm eoNnNaaz NNN.N N.mNm amNNa. Nae aaeaNooa aaa.N N.NaN aamaa. aaa exmaanc aam.a a.mNa NaNNa. mNa omco>acc aeaca mma.N a.aam mmmaa. Nmm aoeeN NNN.N N.maN mmmaa. Nmm xNa>aNceea aaN.m m.aaN NaNNa. mma oNnaom mam.m a.mNa mNNNa. ama eNcaaa maa.a a.amm aNaNa. Nmm onN oaamoca aam.N a.NmN NmaNa. aaa aaomaa aaa.N m.mmN amNaa. NNN eaaona NmN.N N.NaN Nmaaa. aNm NoeocaeNeaz amm.a N.am mmaaa. NaN acawznca ama.a a.mNa mmaNa. Nam enamaaoaa NNN.N a.aaN mamaa. mam aeoaN< Naa.N N.maN aNmaa. Nam naaoN< Nev Nam NNN NNN Naeaaa mxmucN NnNNomuoa moNca coNonN NoNncNaooo NomNonwmoo aeoEaaNecoa Naechnz oamm-aeaa-aemNa NN>NNoaaaca umNnNmmicman II . Nnhuomuoa :oNNnmcomm umunNmaicmpnz cow cam Nmuoz mNoEnxm wo cacao—Naming momozumozhzoz 109 TABLE 12.--Normalized County Development Potential. County Area Normalized Index Alcona 444160 74.8 Alpena 377600 60.9 Cheboygan 510720 168.8 Crawford 362240 21.9 Montmorency 362880 79.2 Oscoda 363520 38.4 Otsego 344320 98.4 Presque Isle 433920 lll.O Antrim 332800 158.3 Benzie 218880 193.3 Charlevoix 288640 102.8 Emmet 305208 126.7 Grand Traverse 313600 184.8 Kalkaska 366720 68.5 Leelanau 239360 213.3 Manistee 363520 95.0 Missaukee 366080 29.1 Wexford 364800 49.7 Mean 104.16 so do >Standard Deviation 56. To obtain a smooth distribution of gradients of recreation 28 potential over the Region a gradient profile algorithm was employed, resulting in the map of Figure 14. 28Refer to Donald Shepard, "A TWO-Dimensional Interpolation Function for Irregularly Spaced Data," Harvard Papers in Theoretical Geography (Cambridge, Massachusetts, 196812 humqohwfimflohwufifl0htflthrumHHo H.0¢.....Oo.oo~.ooo...... ....... moooooooooooooooooo...... ....... 110 othuqounwuuahw00~oruemewhwwvdlowflfin*W*fido*WflfidlhwfiouoH .anucmuoo cmgmN; wo anmcn munowucN mnmcn cmxcno Nuaum ms» cow NnNNomNoa :oNunmcomm aoaaNoN-coaaz ca aNNwoca aaoNuaca--.aN ocamNd 000000000000000000000 oo 0000000000000000000000 o 0000000000000000000000 00.0000. 0000000000000000000000 .0000... 0000000000000000000000 o 0000000000000000000000 o 0000000000000000000000 o 00000000000000000000000 coo-coo 000000000000000000000000 0.0.00. 000000000000000000000000 ......- 000000000000000000000000 0.0.... 000000000000000000000000 out... 0000000000000000000000000 coo-o 00000000000000000000000000 coo 0000000000000000000000000000000 000000000000000000000000000000000000000000000 00000000000000000000000000000000000000000 000000000000000000000000000000000000000 000000000000000000000000000000000000 000000000000000000000000000000000 0000000000000000000000000000 oooooooooooooooaoo..oaoo cocoa ooooooooooooooooooooo oooooo .....ooooo...o.o.o coooooooo ooo...........o. ecooooooooo oooooooooo... ooocOOOnoooo o...o...o.. 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H oonooouo onoooionnoo o oonncoco ooouooooonu H oouoooou ouuuuooono H uououoou uuuuupuo w uouuuoou Quuuu oeuuouo ~ couuu H 00000 M QOnQOOCn oouuooouuo H 00uuuoouuou o oououopuuou w caucuooonoéo oooooaooooouu ovuuNuouuowu uouuuoouuuuu acooooouooo N oooooocooo H ocuuccouu w oeuuuoou oeuooco 0 000000 H ouuuooo H 00 w ..O..-...'.-m...-.....0.--’....-M...-.-.O.NOOOOOODOOHOOOOOOOOO0 111 While the solution to the example problem cannot precisely estimate actual economic potential fin~recreation development it nevertheless indicates a distribution of relative county potential for the Region, assuming that the SCS survey maintained sufficient lack of bias to provide close estimates of county rankings. Con- straining the model with county shoreline area provides a weighting of the original SCS data. It is assumed that distribution of water— related recreation potential is close to the actual regional distri- bution of the parameter, considering employment of county-sized data aggregates. To check the results it was noted that a high percentage of dwelling units in the Region are seasonally occupied units. Data on the number of dwelling units by county were obtained from the U. S. housing census for the years 1950, l960, and 1970. Data for each census year were regressed separately against the model derived 29 The results are shown in water-related recreation potential. Table l3. The results show a high correlation of housing units by county with county water-related recreation potential. The results encourage detailed disaggregation of county areas into smaller land units to more fully assess the degree of linkage of housing units with existence and types of water areas. 29Regressions were performed using SPSS regression routines. Refer to Norman Nie, et al., Statistical Package for the Social Sciences 2nd ed. (New York, l975). 112 TABLE 13.--Results of Regression of County Housing Census Against Model-Derived Summer-Season Recreation Potential for the 18-County Study Region. Dependent Variable: Independent Variable: Number of Dwellings Recreation Potential (average for 18 (average for l8 Year counties) counties) Simple R 1950 2601 104.1 .714 (763.9) (58.6) 1960 4237 " .790 (1255) 1970 5168 " .645 (1538) Precision Analysis The precision analysis proposed in this study utilizes a deterministic solution to the linear programming model followed by a post-optimization calculation of precision. The approach avoids over— burdening the extent of the linear progranming model which may be large due to the number of individual land parcels and use-categories included. Bias errors introduced due to approximations to optimal solutions are assumed when bounded to be no greater than those which 30 are introduced in the formulation of risk-programming models. The method employs the assumption that the deterministic solution expresses the expected value of all data elements having variance.3] 30 31After William Alonso, "The Quality of Data and the Choice and Design of Predictive Models," Urban Development Models, Highway Research Board Special Report No. 97 (Washington, D.C., 1968). Wagner, op. cit., pp. 639-686. 113 With the large number and different qualities of data employed by the model a post-optimality precision analysis permits examination of the relationship between model detail (should more structural elements be employed or less?) and the precision of the data available (should more resources be expended or less be expended upon data acquisition in order to meet the levels of precision required by the planner?). Figure 15 is a graphic presentation of the relationship between level of detail included in a model, data precision, and total variance of model results. Increasing model detail can decrease error due to model specification but error introduced by the data will rise because of the required increase in number of data elements. For a given level of data precision there is an optimum point at which total model variance is minimum. Minimizing variance increases confidence in decisions concerning locations of boundaries between areas of differ- ing economic potential.32 A suggested form of precision determination follows. Let y = F(x], . . . xn) 32There must be an awareness of the existence of both un- certainty and risk in the structure of the model. Risk refers to situations in which the probability distributions of random events are known and calculable, and for which loss functions can be mathe- matically applied in the decision process. Uncertainty refers to influences of random events whose probability distributions are not known and which therefore must be estimated outside of the rigor of parametric statistics. For a treatment of the subject of risk and uncertainty see Russell Ackoff, Scientific Method (New York: 1902), pp. 46-58. 114 .coccm :o_umuF$wumam quoz new cowm_uwca name we cowpuczd a ma mppammm quoz we mucmwgm> Pmuopuu.m_ mesa?“ EUCE E Ummamocm mmEOCO> *0 .50—E32 _mnoE Nb 1/ \J are m \ mucoco> 30b 115 when y is a linear combination of x], . . . , xn the error in y is given by33 3y 2 8y 3y 02=Z -—- o2 + Z— -— ox ox r (l) y axi x1 3x1 axj i j ij where oy, Ox- are standard deviations of y and x respectively, and 1 rij is the simple correlation coefficient between factors xi and xj. As an example of application of this formula _ -1 let n - c0 8 represent the vector of marginal factor prices deduced from the solution34 of the linear programming model, where c0 is the solution vector of objective function coefficients corresponding to the solution -1 basis 8. Then total productivity Q = nb = c0 8 b where b is the vector of land area constraints. Differentiating Q with respect to bi and aij: ad -1 3b 33" sec _] ———-= c B -——-+ c -——- b + ———-B b 0 0 Bbi Bbi Bbi abi II D a- 33A derivation of this formula appears in E. B. Wilson, Jr., An Introduction to Scientific Research (New York, 1952), pp. 272-274. 34Refer to Saul I. Gass, Linear Programming Methods and Applications (New York, 1975), pp. 96-117. 116 and ll 0 so that Equation (1) becomes 2_ .2 2 .2 2 . .2 2 00" (0a) 0a.j+(Qb> Ob+Qa Qb Ua.."b rab (2) 1 1J when no correlation exists between aij and bi’ rab equals zero and the final term of equation (2) vanishes.35 In determining precision levels of model structural elements, postoptimality analysis should also be employed to determine ranges for the aij’ bi and cj within which the current feasible solution is retained. If the expected variances of the model elements fall within these bounds, no change of basis is required in the optimized model and the current solution can be assumed to represent the expected values of each of the model dependent variables. 35All precision analysis Sevelopeg for the model is based on a quadratic loss function (e 3 e) where 6 is a population parameter and 6 is a point estimate of a. Variance in e is estimated to be 2 2(9 ' é)2 0' = -——- n n = number of observations. 117 The criteria specifying the ranges within which the current feasible basis is retained are:36 For c: -(zJ - c ) . -(zj - c ) max _<_Ack 3mm (3) xkj<0 ka xkj>0 xkj for all j not in the basis. For b: -xi 'Xi max §_A b].: min (4) bil>0 bil b1] for all i in the basis. For a: zj - cj . < Aa1k if D is positive, D __ z. - c J J . Aa < ———————— 1f D is negative, (5) 1k —- D where D 15 Ebilcixkj - (zj - Cj) bkl and for 3 not in the ba51s, 36The dependent variables are the value of the objective function 2, the levels of each of the sectoral outputs x, and the values of the marginal factor prices n. The c, a, and b are assumed stochastic. For further clarification of the variables of the linear programming problem, refer to Gass, 0p. cit., pp. 161-168. 118 If parametric routines are not available in the linear pro- gramming system used, the ranges of each of the parameters a, c, and b can be determined from the above. An example precision determination for the data presented in Figure 14 was prepared. The model computed value for summer-season water-related recreation potential is P=nxA where P is the vector of recreation potentials from the model, 3 is the vector of marginal factor prices, A_is the corresponding vector of values of county shoreline area (cf. Table 14). Assume the standard deviations of the data to be as fOllows: 2 A1 18 0A = 0.1 x Ar—- where Aav = i=1 Ai av and o = 0.2 x n '11 3P 3P _ Using formula (1), 5;—= A and 5A" and assuming the correlation coefficient r1TA = 0. oP =\/(n x OA)2 + (A x 0,")2 119 mompmm. empomo. ooooou.¢w_ ouvpoo. ommmoo. ogowxwz @— Nmnmmp. mmnpmo. oooooo.pmp vpmpoo. oxmmoo. mmxsmmmwz up womurm. eupmmo. ooooom.¢¢~ «PFNoo. osmopo. mmpmwcmz 0P cameos. .Nwmvp. ooooop.mpm ommmoo. omm_Fo. :mcmpmwg m— Nmoomm. empooo. ooooom.~om oowpoo. oommoo. mxmmxpmx e— upowom. mmmmom. ooooom.mnm emmmoo. omoppo. omgm>mgp ucmcu mp cqomum. mammmp. ooooom.mom mompoo. ommmoo. meEm NP omvvqc. mpoomo. ooooom.mmm oompoo. ommmoo. xwo>mpgmgu ~_ «momma. mpm¢o_. ooooom.mom emmmoo. onwppo. «FNcmm oF mmmwmm. whoemF. ooooom.m~m mnemoo. ompmpo. 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Q WHO-n . “Ht-.1 want-0H 9 HWNWW OHHHMu-amrd o HHHH :HHNH O HMHanHHt-CH o WHHOHHH.‘ O O‘NHHKHO-O-fl 122 where Xt and X are adjacent discrete values of the time-ordered t-l variable X. In application the z-transform is useful in describing a general distributed lag process resulting from an induced change in one of several system inputs.37 The general distributed lag process can be written Yt = poxt + plXt-l + pzxt-z + ' ° ° + ant-a (6) termed a process of order a, The general autoregressive process of order g_can similarly be written Yt = qlYt-l + q2Yt-2 + ‘ ' ' + qut-b A general function can be obtained by combining the two processes qlYt-l + ' ' ° + qut-b = poxt I P1 t-l I ° ° ' paxt-a (7) or in terms of the z-transform: Q (2") v = P (2'1) xt -1 _ where Q (2 ) - q]z + qzz + . . . + qbz -1 p0 + p12 + . . . + paz 37The z-transform of the general function f(t) 15 def1"9d as F (2) = g f(nT)z'n n=0 where z is a complex variable satisfying certain distinct properties of convergence and the nT are discrete points in the time domain. 123 1 where z- is the inverse transform operator. The above equation may be written Yt = o“ (2") P (2") xt =¢Wx (& t where the polynomial P is a function of lag and determines transient response to induced change and the polynomial 0 determines system steady-state response.38 When induced change in economic demand appears as a change in use potentials between two use-categories we may characterize the changed potential obtained from the solution of the static linear model by a step function u (Figure 17).39 When the step function U_is induced, typical response in expected number of land-use shifts over time may take the form of equation (8). Figure 18 shows a typical time-ordered response of use-shifts following changed economic demand. Refer to E. I. Jury, Theory and Application of the Z-Transform Method (New York, 1964). 38The function is thus characterized by a polynomial P of order g_and a polynomial Q of order b abbreviated a TF process of order azb. See R. J. Bennet, "Process Identification for Time Series Modelling in Urban and Regional Planning." 39The z-transform method permits determination of system response to any induced change as indicated by the general form of the tenn X in equation (8). The discussion here is confined to the unit step unction U(t) for simplicity of exposition. Solutions to system response are possible when complex driving functions are imposed; it is also possible to obtain solution to the system when the driving function is distributed as a random variable [at]. Refer to Bennet, op. cit., pp. 157-160. 124 U(t) Figure 17.--Potentia1 Rent Change Step Function. H0 --- //’~\\\\i>>>—r:e Figure 18.--Lagged Response of Land Units Shifting to a New Use Category Following Imposition of Rent Change of Figure 17. 125 If the input excitation is represented by a random variable [et] distributed as N (0, oz) the process is termed an autoregressive 40 moving average process or ARMA. An ARMA process is defined - -1 -1 -1 Yt - C (x ) 0 (z ) et -1 _ -1 -2 -b where C (2 ) — c]z + czz + . . . + cbz D (2") = d + d z" + . . . d 2'3 0 1 a Although the general system response to a unit step function and to the random variable [et] is characterized by an infinite number of terms of the transform function, it is characteristic of the ARMA and TF type processes that the bulk of the response will be contained in the low order terms of the expansion. System description, therefore, occurs with a conservative number of terms in the general regression equation with retention of a high degree of precision in 41 the data fit obtained with the transform. As Bennet observes, the processes way be detectable in time series data for which a limited number of actual observations are available.42 40Box and Jenkins, Time Series Analysis, Ch. 4 provides an example of the processes of statistical forecasting described above. 41 Bennet, loc. cit., pp. 157-160. 42Ibid., p. 172. Questions of efficiency and unbiasedness “of estimators obtained with the TF and ARMA processes are relevant. It is indicative that time-series polynomials resulting from unit- step and stochastic excitations can be represented with a low number of terms in the transform expression. Bennet labels this the frugal or parsimonious characteristic of these processes. 126 The curve of Figure 18 shows a hypothetical but typical lagged response in terms of numbers of land shifts in a locale fbllowing an induced change in economic demand. Rapidly accelerating entry of new firms into the advantaged use-category results in an over-supply of firms in the market at the peak of the curve; this over-supply yields to correction effects as firms leave the market in the next portion of the response, with an eventual steady-state condition obtained in which enough firms exist to just accommodate market demand. Interperiod models describing expected land-use shifts may be created through application of a process such as the z-transform. When driven with expected capitalized returns derived from marginal factor price estimates from the linear programming model, a TF or ARMA type process can generate reasonable time-ordered probabilities of use-shifts. These probabilities in turn can be spatially distri- buted and placed in the matrix form for comparative regional analysis described previously. An important function of interperiod analysis is to provide parameter inputs to the next sequential general programming solution. Generally, in examining the question of availability, land that has experienced a recent use—shift will not shift again in the near-term, because of capital investment recovery. A master land-use file should be structured to provide indexing of all land areas which have experienced use-shifts in a recent period. Availability of these areas for future use shifts can be delayed accordingly. Input 127 to the general model should be based on periodic, programmed re- assessments of land availability. Levels of precision associated with the transform process can be determined through the usual methods employed in the formu- lation and solution of auto-regressive moving-average (ARMA) and transform function (TF) models. It is important that precision levels be explicitly computed at every stage of the model's develop- ment. Comparison of precision levels is an important guide in adjusting model specification or in directing resources toward improvement of primary data inputs. Summar The chapter describes the development of a linear programming model for predicting likely land-use shifts. The model functions either as a general land-resource allocation model or as a land-use shift prediction model. Identification of development potential derives from use of marginal factor prices of land resource units in scarce, competitive supply. Potential land use changes are predicted through model-derived changes in factor prices which in turn respond to exogenous changes in regional economic demand. The model is formulated as a comparative static model. Adjustments in model parameters in period-by-period analysis enable time-ordered control over resource allocation and investment. Both temporal and spatial distributions of shift potentials developed by the model and contribute to the model‘s use in locating and defining important sub-regional areas. 128 The chapter includes an example of employment of the model to determine the location of summer season recreation potential using the dominant seasonal activity of water-related recreation. Gradient profile maps show areas of likely future impact of water-related recreation development. The chapter describes a systematic procedure of precision analysis to help achieve optimal levels of model structural detail. The chapter concludes with description of a transform method with which to formulate functions to track land-use change in the inter-period between general model solutions. CHAPTER V LAND USE INVENTORY Introduction Accuracy of the land allocation model's output depends on accurate measurements of data taken in the field. Optimum design for a field survey in turn depends upon the scale, accuracy and resolution required of the data to meet specific applicational needs. The present chapter outlines procedures for data collection and organization and outlines the general requirements for a land-use inventory system capable of providing organized, spatially described information for the model. The chapter begins with a review of general techniques to form comprehensive land-use inventories, and discusses the types of data required for land-use evaluation in the Study Region. The chapter proposes indices with which to develop spatial descriptions of economic potential in the Study Region. The chapter discusses methods for combining information gathered through remote sensing with data obtained from alternate sources to produce an efficient data acquisition and data base management system. 129 130 The chapter concludes with suggestions for the design and organization of a responsive regional land-use inventory. Land Survey Techniques Among the foremost of land survey techniques is that pro- posed and developed by Philip H. Lewis1 as a response to needs for planning recreation land in Wisconsin. In the development of the method, Lewis noted that the particular cultural patterns coincide with distinct types of physiographic land-form features. He sub- sequently identified as likely areas for future development those areas which contain typical physiographic and cultural features. In Lewis' method an initial survey records perceptual quali- ties such as existing structures, highways, land cover, and scenic attributes. A soil survey helps determine the types of recreation use suited to each soil association. Lewis compares individual survey inventories by using transparent overlays, and identifies intrinsic patterns as those which occur naturally, extrinsic as those which are associated with man-made changes. Three major intrinsic patterns are significant topography of an area, existing pattern of rivers and lakes, and the distribution of wetlands. Lewis maps these major resource patterns and combines them with extrinsic cultural information. The composite pattern of resource 1Philip H. Lewis is head of the Department of Landscape Architecture at the University of Wisconsin. His work in locating outdoor recreation for the State of Wisconsin established important guidelines upon which to base land-use survey methods. An outline of his work appears in Philip H. Lewis, "Quality Corridors," Landscape Architecture Quarterly (January, 1964), pp. 100-107. 131 concentration which emerges Lewis terms the ”Environmental Corridor," within which the majority of resources, both culturally derived and of natural origin, appear. Lewis' concepts form an important contri- bution to land-use planning, in that cultural and natural feature identification of corridor sub-regions provides a basis for the analysis of other Spatially distributed resource uses. Ian McHarg2 develops an approach centering upon historic natural processes and their limiting or liberating criteria for land- use planning. McHarg emphasizes the balanced relationship between man and nature which can be achieved by pr0per land-use planning. His concept notes the dynamic quality of the interface between man and nature: We need, today, an understanding of natural process and its expression and, even more, an understanding of the morphology of man-nature, which, less deterministic, still has its own morphology, the expression of man- nature as process. McHarg applies knowledge of the natural sciences to land-use planning accomplished by a comprehensive and systematic inventory of natural features which, in turn, indicate natural processes found in the environment. McHarg identifies four major values attributable to natural process: 1. Inherent qualities or characteristics of the process itself; 2. Productivity of the process; 2Ian McHarg is chairperson of the Department of Landscape Architecture at the University of Philadelphia. 3Ian McHarg, in F. Fraser Darling and John P. Milton, eds., Future Environments of North America (New York, 1966), p. 529. 132 3. Maintenance of the ecological balance; 4. Potential hazards arising from improper use of natural processes or resources. McHarg's workplan for develOpment expands around his concept of the man-nature interface. In McHarg's concept of planning the "health" of the environment requires identification and evaluation. For example, the species of trees present in densely populated urban confines represent only the most hardy; McHarg's analysis would consider the exclusive presence of such species an obvious indicator of the health of cities. McHarg outlines the ecological determinants of a study area and applies them to its envionmental analysis. His work recognizes environment as a process of nature, subject to change and dynamically responding to man's own interaction with it. McHarg recognizes the difficulties of incorporating changing value structures into the analysis; change over time and problems of scale of natural processes are part of the problems he encounters in dealing with value.4 His method, however, allows insights into the directions by which human cultural land-use patterns interact with biological process to create a definitive environment. McHarg's analysis procedure collects study area inventory in a systematic sequence: 1. Climate 2. Historical Geology 3. Physiography 4. Hydrology 4Ibid.. pp. 526-538. 133 5. Pedology 6. Plant Associations 7. Animals ' 8. Land Use The sequential procedure permits analysis to follow the same path as an area's historical development. (For example, climate and historical geology determine physiography.) From this analytical base, McHarg's systematic development of site plans follow sound ecological principles of land use planning. Another approach to determination of use-potential evolved through the work of G. Angus Hills of the Ontario Department of Lands and Forests.5 Hills' analysis technique began initially from an analysis of soils, shifted to landform analysis, and eventually to analysis of site vegetation as an indicator of environmental condition. Hills' procedures provide a basis for selection of sites for residential or recreation use from site productivity potential, determined from comparative analyses of site classifications. Hills uses "bench marks” or extremes of physiographic formation within a study area to describe intermediate physiographic forma- tions.6 Thus, features of the areas themselves form the basis for area description. His procedure separates the present and probable 5See G. Angus Hills, "Ranking the Recreational Potential of Land Units by Gradient Analysis: A Physiographic Classification of Land for Recreational Use" (Ottawa, Canada, Department of Lands and Forests, February, 1966). 6Hills initially subdivides a land area into geographic units having common physiographic features. An ordered classifica- tion based upon ecological types determines the potential productivity of a particular area through comparative site study of similar, developed areas. 134 cultural development of the land from its true potential as identi- fied in the physiographic land survey. Site Quality Determination by Indices Often, measured variables characterizing spatial features of land areas are complex and carry little information in themselves to the planner who must form an overall measure of site quality. Indices facilitate the interpretation and integration of complex spatial data.’ Indices find use in many areas of science to describe complex system behavior. Their use in regional planning enhances the ability of the planner to form quality determinations of site environ- ment and to synthesize these quality determinations into scales of comparative ranking.7 Since indices are subjectively determined measures of site quality, a land-use model's success depends upon the planner's ability to formulate indices which describe the range of land use and cultural features in an area, and to interpret these with enough resolution and accuracy to meet the requirements of the model. Indices useful for classifying a wide range of physiographic features, cultural features and land types in the Study Region follow. Available Land. The amount of land available for transfer to new use categories. To facilitate planning, an area may be 7A. H. Voelker, Cell Indices, An Analysis Tool for the Planner (Oak Ridge, Tennessee, 1975). 135 classified by a set of indices each one for a different use category. Transportation Accessibility. This index describes the relative accessibility of the area. The number of access routes to the area and the amount of transportation land within the area influence this index as they provide indication of levels of access available to internal portions of the area. Proximity Attractiveness. Quality of surrounding areas to enhance the general attractiveness of a site for specific types of development. Both natural and cultural features should be considered in index formation. There may be several indices of this type each describing proximity attractiveness for each of several possible site uses. Proximity to P0pulation. Number of people within a Specified radius of the centroid of the area. Compatibility with Existing Land Uses. An indication of the relative compatibility of each proposed use with existing uses in the specified area and in surrounding areas. The index may consider zoning restrictions and proximity of general utilities, as well as the multiple use of forest land. Use Conversion Cost. The cost of converting an area to alternate use. General Landform Class. This index should be developed from procedures such as those used by Hills or Lewis in describing physical landform data. 136 Topographic Level. The mean elevation of an area above a datum reference. Soil Type. From soils inventory data, the soil type index should rate general area suitability for development in each of several potential use categories. Bedrock Depth. The mean depth of soils to bedrock, especially useful when the bedrock may be near enough to the surface to influence building construction. This depth may also be an important factor for determination of local drainage and general hydrological features. Depth to Subsurface Water. The mean depth to water table needed to determine specific use capabilities of the land area, again especially where buildings are to be erected on the site. Surface Water Type. This index classifies predominating types of surface water. Wetlands, small streams, ponds, reservoirs, lakes, and minor and major rivers should receive a ranked order of classification.8 Forest Type. The type of forest cover should be identified. The index should reflect percent cover, stand density and age, as well as type and characteristics of understory vegetation. There may be more than one index for an area. Residential Activity. The presence of residential develop- ment within the area; number and types of units, density, lot size, and condition of buildings comprise important data for this index. 8Development of a surface water classification index can be found in T. Murray, et al., Honey Hill: A Systems Analysis for Planning the Multiple Use of Controlled Water Areas, Vol} 1 (Washington, D.C., 1969), p. 44. 137 Site Availability. The site availability index should specify conditions determining non-availability of the land parcel for potential shifts to alternate uses. Index values denote the specific condition responsible for non-availability.9 Proximity to Major Transportation. Gravity models in one form or another form the basis for most transportation proximity measures. In the gravity model, population attraction to a point in space is determined relative to other points in space. A general gravity model takes the form Tij = K Pi Pj dij'A where Tij expresses the derivation interaction between points i and j (e.g., an origin point and a destination point) Pi and Pi represent populations of points i and j respectively, and dij represents the distance between i and j. K and A are model parameters, fitted by appropriate calibration techniques.10 To compute the index, the ratio T. J./P.1.PJ. is allowed to represent effective "nearness" or proximity of the points i and j. The general formula for the index thus becomes 9The Site Availability Index may be related to the Site Preparation Cost Index. It is possible that the costs of shifting uses are determined to be prohibitively high by an owner, a condition which could occur, for example, if the land had shifted use categories in the recent past and the owner finds it necessary to wait until investment amortization has occurred before again changing uses. 10Models of this kind are easily fitted by regression analysis when of the form log (T11/P1Pj ) = log K - Alog d1 11. A general theoretical development of the spatially interactive gravity model can be derived from statistical mechanics. Refer to an article by M. Batty, "Recent Developments in Land Use Modelling: A Review of British Research," Vol. 9, Urban Studies, 1971. 138 = -x I Kdij Gravity models can be calibrated to relate a particular recreation type or category to site attractiveness. The Michigan Department of State Highways and Transportation maintains a proximity analysis program system which can be combined with an extensive socio-economic data file, a statewide travel data file, and a statewide public and private facility file to generate mean trip times and populations served for given facility types.11 The use of this system will enable gravity model generation of attractiveness indices related to trip origin areas outside of the Region. Composite Index Evaluation In general, a composite index is of the form I = F (I1, I2 . . . Ii) where I is a composite index formed by a function F operating on the set of indices I1 . . . , I.. The specification 1 of the function F is quite general. However, in practical applica- tions F is usually a linear, first order equation of the form HMichigan Department of State Highways and Transportation, Michigan's Statewide Traffic Forecasting Model, Statewide Public and PrivateEFacility File (Lansing, Michigan, January, 1974). Also, refer to Volume IX,’Statewide Socio-Economic Data File, and Volume I-D, Proximity Analysis: Social Impacts of Alternate Highwanglans on Public Facilities, in this same series. 139 where "i are weights assigned particular site attributes to provide emphasis in the composite index formation. (When a decision threshold exists in the evaluation procedure, and this threshold has an effect upon the evaluation of the composite index, a more general form may be A I=§Wj[Ij-Dj] The decision elements Dj introduce a degree of control over composite index formation in addition to that provided by the weights w, It should be noted that while addition of decision thresholds to indices can provide greater accuracy in terms of "fit" to measured data, complexity of the index increases rapidly when these terms are added, placing greater burden upon measurement precision of each term of the index to control error in the final composite.12 Composite indices such as site productivity can be pre- determined and placed in tables for access using table look-up procedures. In the example which follows, potential productivity for a site requires determination from field data on type of beach and water quality recorded as ranked observations. A pre-calibrated table relating ranked site indices to productivity measures is formed: 12See Alonso, op. cit. 140 Factor . Category Numper Factor Descr1ption Class A B C D 1 Type of Beach I 90 3O 65 15 II 76 15 4O 5 2 Water Quality I 70 24 15 10 II 52 38 20 6 The site productivity index is computed as follows: Ip = 2:] TABLE (CLASS, CATEGORY) where CLASS and CATEGORY are directives to TABLE. For the example, type of beach is recorded Class 1, Category C and water quality is recorded Class II, Category A: this calculation yields 65 + 52 = 117 as a measure of site produc- tivity. The procedure finds application when site ranking of land areas relates in a non-linear manner to measures of properties such as economic potential. Calibration of the table depends on care- fully chosen test areas where all site properties can be accurately measured.13 13A second table of estimated precision of the productivity values is needed. Final index computation should include a measure of estimated precision of the composite index determined from the component precisions. For example, if the estimated precision of the table entries is :_6.5 for the value 65 and :_5.2 for the value of 52, the precision of the result is computed to be 2 2 6.5 5.2 _ _ vA65) + (—5—2-) X 117 - 0.14 X 117 - 115.5 See, for example, E. B. Wilson, An Introduction to Scientific Research (New York, 1952), pp. 232-258. 141 The method is a preferable one for composite index formation in that linearity of intervals between table values need not be assumed. The use of fitted regression lines to ranked data, for example, may presume a mathematical relationship which does not exist. The table look-up procedure avoids certain types of specification error and does not require elaborate calibration, but does require careful validation and control against bias. On the other hand regression models can and should be used when pr0per1y specified. For example, certain types of non-linearity of the dependent with respect to the explanatory variables can be managed in a regression equation by dividing the sampled data into a series of sub-samples as in Figure 19 and estimating separate model 14 Each sub-sample refers parameters for each subsample in turn. to a linear segment and can be expressed with a linear equation of the form Y1 = a + Z Bixi° ..< Dependent Var1able Independent Variable X Figure l9.--Examp1e of Piece-wise Linear Regression Model of the Type Used to Estimate Site Productivity. 14Refer to J. Kmenta, Elements of Econometrics (New York, 1971), pp. 466-472. It is necessary that certain conditions in the higher derivatives of the population curve be met to avoid dis- continuities or break-points in the model. 142 Sample Bias: Monte Carlo Estimation Site quality measurements are often made by classifying and recording visual observations on an ordinal or I'ranked" scale. For example, a site might be judged below average, average, or better than average on a scale whose corresponding values are 1, 2 or 3. Regression analysis is often employed to validate a mathe- matical model which expresses site productivity as a function of observed site quality. While repeated samples allow computation of sample variances that center around the means of rank values in the sample set, bias introduced when an observer regularly and syste- matically mis-classifies observations (e.g., regularly records sites ‘ of value 2 as value 1 or sites of value 3 as value 2) cannot be estimated by standard statistical procedures. It is sometimes useful to know to what extent bias may influence a particular survey's results prior to the actual survey, or alternatively, what level of bias is acceptable to the analysis before the analysis is begun. For this purpose experiments called Monte Carlo experiments are performed which allow testing of various levels of bias against a background of simulated real-world data. In the experiment evaluating site index data we are given three ordinal ("ranked") values on the observation scale. We assume site productivity to be given by the equation Y = B0 + BIX + e 143 where Y is site productivity, X is the observed site classification value, 80 and B1 are regression coefficients and e is a random dis- turbance introduced from natural variations in the trial data. We simulate e by reference to a random number source. (It is the simulation of the random disturbance term e which produces a Monte Carlo problem.) If e has values symmetrically distributed about zero as for example from a normally distributed population with mean equal to zero and variance 02 then the mean of "expected" value of Y will be When the classical assumptions of the linear regression model are met, variance of Y will depend only on the variance of e. Since population variance can be estimated by the sample variance 52 given by the formula —-2 2:2(xi-X) n-l S we see that there exists a tendency for estimated sample variance to decrease with increasing sample size (or number of trials) n. Therefore, sample variance is itself a function of n and the overall problem becomes one of determining, for a given population variance whose estimate is the sample variance 52, how a given bias will influence the results of applying the model to the data. The two independent parameters in the experiment are population variance and amount of bias. We wish to determine N, the critical sample size 144 at which bias becomes significant as a function of population variance and bias. (Bias can be introduced between any of the ordinal observa- tion values 1, 2, or 3.) The output of the experiment will be a set of curves such as those given in Figure 20. Critical 2 Sample Bias + Figure 20.--Critical sample size N as a function of sample bias for given population variances o2 in a simulation of field observations of ordinal ranked values. Critical sample size is that number of independent observations required to reduce the sample variance to the point wherein sample bias becomes an important factor in tests of hypotheses at a pre-determined level of confidence. In the conduct of the experiment, for each given population variance 0 and for a given degree of sample bias, a number of sets of data for each n over a range of values of n are run. This 145 process is repeated for different levels and positions of bias and different population variances. The Monte Carlo procedure allows the investigator insight into the effects of systematic bias which may be present in the data prior to model evaluation of actual field survey data. The procedure is especially useful when nonlinear or transformed equations are to be used in the model structure. Measurement of Relative Economic Productivity Water Related Recreation Index Economic productivity is difficult to estimate. Receipts of firms may provide an initial estimate of productivity of active land-using operations, and can assist the investigator to determine reasonable representation of an industry in the regional economic mix. Potential economic productivity is difficult to determine when a carefully surveyed, previously developed test area is not available for comparison. In such cases judgment alone may have to serve for purposes of estimation.15 This kind of judgment formed the basis of a study conducted by the Soil Conservation Service (SCS) in estimating the potential 16 of Michigan counties for future recreation develOpment. In the 15Accounting procedures used to determine the various con- , tributions of the different productive factors should follow pro- cedures consistent with formation of the overall regional accounting framework. Refer to Harry W. Richardson, Input-Output and Regional Economics (London, 1972), Chapter 5. 16U.S.D.A. Soil Conservation Service, Recreation Potentials in Michigan (East Lansing, Michigan, 1972). 146 SCS study, a panel of persons familiar with each county area followed careful guidelines for estimation of county potentials to support future recreation enterprise. A typical arrangement of factors used in the SCS study to derive a composite county index score is shown in Table 15. The list of key elements and their multipliers was provided by the SCS prior to the estimation. Evaluations followed strict guidelines provided by the SCS. TABLE 15.--An Appraisal of Water Sports Areas for a Michigan County. Key Elements Multiplier Rating Score A. Climate l 8 8 B. Scenery l 9 9 F. Water Areas Existing 4 8 32 Impoundment sites ...: O} 05 H. Population Size and Distribution 2 l 2 Age and Occupation l 4 4 I. Proximity to Cities _l. 9 _g_ Totals 11 7O The total possible score in the category was 110. County evaluations were ranked according to score values; scores between 76-110 ranked High, scores between 40-75 ranked Medium, and scores 0-40 ranked Low Potential. The factors employed in the SCS survey to derive county indices for warm and cold water fishing appear in Table 16. 147 TABLE 16.--A Determination of Indices for Warm and Cold Water Fishing for a Michigan County.* .Warm Waters Cold Waters (M) (R) (S) (M) (R) (S) Key Elements A. Climate 1 10 10 1 8 8 8. Water Areas Existing '3 8 24 3 8 24 Impoundment sites 1 6 6 l 6 6 G. Wildlife Populations (fish) 2 9 18 2 7 14 H. Population Size and Distribution 1 6 6 Sportsman Type 1 7 7 1. Proximity to Cities _1_ ._9 _;g _l_ _9_ ._9 Totals 10 73 10 68 M = Multiplier; R = Rating; S = Score. * From SCS Study of Recreation Potentials in Michigan. The composite index by county appears in Table 7, p. 104. Winter Sports Index The climate of the Region generally assures an adequate snow cover for comprehensive winter sports development. Local topography plays an important role in making the region attractive for winter sports development. Portions of the Region have excellent hills and lepes suitable for the development of intensive winter sports activity, while other areas offer possibilities for extensive use of 17 forest lands for winter recreation. To form an initial winter 170*. Chapter II. 148 sports location index, data on local relief were assembled from a study by Pawling, who inventoried average local relief in 2% minute-square cells throughout the State.18 The local rank for each 2% by 2% (6 square mile) area was counted and sums tabulated for each county of the Study Region (Table 17). An aggregate county local relief index was computed by multiplying the number of cells in each rank category of Table 17 by the mean interval of the category as listed in Table 18. These figures were summed and divided by the total number of cells in each county to form the county Local Relief Index (Table 19). The formula for average county relief IR is R. C1 IR=j:L_.—_ X R where Ri is the number of county cells in local relief category 1: Ci is the mean relief difference of category 1, Table 19 summarizes the calculations for each county. Transformed, the county relief indexes result in the spatial dis- tribution shown in Figure 21.19 Transportation accessibility may provide another important factor in determination of winter sports potential. An index of 18John W. Pawling,"Morphometric Analysis of the Southern Penninsula of Michigan“(Ph.D. Dissertation, Michigan State University, 1969), pp. 33-39. 19See footnote, p. 109. 149 .uNe .ee .mcvpzee :_ even see. eeueepNemceu. 0.0 0.m 0.00 0.00 0.0_ N.NP e.0 0.. 0.0 0.. 0.0 0_ eeoexez 0.0 0.0 0.PN 0.0. N.0N 5.0F 0.0P 0.0_ 0.. 0.0_ N.P NP eexsemmez 0.0 0.0 0.0_ 0.0, N.mp 0.0, 0.0N 0.0 0.. 0.P 0.0 0. 0000.00: 0.0 0.00 0.0m 0.0, 0.0 0.0 0. 0.0 0.0 0.0 0._ 0_ 0000.000 0.0 0.0 0.N 0.0, N.mm _.0~ 0.0N N.N 0.. 0.0 0.0 0, memexNex 0.0 _.m 0.N N.mp 0.0. N.NN 0.00 0._ 0._ 0.0 0.. m_ emee>02e 000.0 0.0N 0.0_ F.0N 0.N_ 0.5 0._ 0.0 0.0 0.0 0.. 0.0 N_ eeEEe 0.0 0.__ 0.N_ 0.N 5.0 0.0 0.0 0.0 0.0 0.0 0.. NF xeo>e_eeee 0._ N.. ..0_ 0.0 0.0 0.0 N.0 0., 0. 0._ 0., 0_ uwneem 0.0 0.ep 0.N~ 0.__ 0.N m.__ 0._ 0.N 0.N 0.0 0.. 0 eeeee< 0.0 0.0 0.0 . 0.0 0.0. e.0~ N.0e 0.00 N.0 0. 0._ 0 0.00 eaameea 0.0 0.0 0.00 N.NP 0.0_ 0.0_ 0.0_ 0.0 0.0 0.0 0.0 N 000.00 0.0 N.N 0.0_ 0.00 N.0N 0.FN 0.0P ..m 0.0 0.0 0.0 0 eeeem0 0., 0.0 F.0F N.0F 0.0_ 0.0P 0.0. 0.0 e._ 0.0 0.0 m NueeeeEeeoz 0.0 0.0 0.N 0.__ N.0N 0.00 0.0_ N. N. 0.0 0.0 0 eeeezeee 0.0 0.. 0.0 0.~F 0.0m 0.0m 0.00 0.0_ 0.0 0.0 0.__ 0 000000000 0.0 0.0 N.N 0.0 0.0. N.mp 0.0m 0.0_ 0.0_ 0.0 0.. N weeep< 0.0 0.0 N.NP N.0_ 0.PN N.NN 0.NN 0.0 0.~ N.P 0.0 _ aeoup< __ 0_ 0 0 N _ 0 0 0 m N _ _e>emeeH Neeeee «.000000 00 eeneee0ee _e>eeeee .e._00 some 0. mppee epez mm »0 mm .0 2.00:2--.NN 00000 150 TABLE 18.--Local Relief Intervals. Interval Number Interval (ft) Average Interval OkDCDVOSCfl-hwmd —J O 25 59 75 100 150 200 250 300 400 500 - 24 - 49 -' 74 - 99 149 - 199 - 249 - 299 - 399 - 499 - 599 12 37.5 67.5 87.5 125 175 225 275 350 450 550 151 .uNe .ee .mewpzee =0 «pee see. eeueeNNemeeu k. ee.mNN mm.eem 00.00 egemxez mm.mN NN.NNN eN.N0 eexeemmwz N¢.Nm om.e- oe.Nm eeumwcez o¢.~mp NN.¢mm em.ee 30:0Neee Nm.mm mm.me~ oe.mm exmex—ex vm.mm ee.mom em.Nm emge>eck eeecw mo.meN «N.Nmm oe.mN peEEu Ne.mmp em.m_m ee.Ne xwe>e_gegu mN.NNN mo.mm~ om.Nm mNNeem Nm.mm_ N~.Nem om.em ewcpe< N~.Nm Nm.mmN em.eNN eNmN omega em.N0_ mm.mmm om.Nm emempo me.~m «a.mNN eN.me eeeemo mm.ee 0m.¢mm eN.eeN Neeeespeez Nm.ew me.Nom em.em eeewzegu «N.NN Nm.meN ee.eeN cemaeeese «a.Nm mm.¢m_ ee.weN eceep< 00.00 Nm.o~N em.NNN eeeeN< Ne>gepeN an eeuzmwez mueeeu :0 NNV Nee \ NNV Nee mNNeu sauna weppea mN—eu Eeuee eceeem Neceeu .605 00:00 :83 :83 ...0 .50. “a .3 we :38 E E :0 ..xwucN wwNNwm Name; aw::OU11.mN m;mencc «mm IIIIIII mom N ............................... 00000000000 ..................... 000000000000000000Noounnn c IIIIIIIIIIIIII m N ............................... 00000000000 .................. 000000000000020000000000 I IIIIIIIIIIIIIIII N . ................................ 00000000000 .............. 0000000000000000000 000 II III-“IIIIIIIIIIII . 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N ........N......... ........ 00000000000000000000000000 moommmwwm 0000 “III m N .. .................... .......... 00000000000000000000000 0woooww¢m mnwo II N 1.1.1.11...11.1.... ............ eeeeeeeeeeceeeeee neeeeeere eu - .0 N H 0.00.00.00.00...oooooooo.ooo OOIOOOOOOOOIOOOII UOOOOUUUUUOOOU Orwwwww®¢® I - N n ............................ .................. 00000000000 cmommmom¢ I n N ............................. .................... 000000000 00000000 I I N H 0.0.0.0.......IOOOOOOOIOOOII. ......‘O..........O. COUOPOOU 0¢W®Q®®® . . H N ......OOOOOOOO....OOOOIOOOOOOO. .....O............. UUUOOCU 001$”me - H H 0.0.0.0000....ooooooooooooooo.oo OIOOOOOOOOOOOOOOO OOOOCO Qt®®®®0 I. H . 0.0.0.0000...OIOOOIOOIOOOIOOOIOOOO. ....OOOOO.OOOOO OOOQOO wawmwm - O H ......OOOOOOIIOOOOOOOIOOOIOOOOIOO.I.I. ......OOO... OOUUO ¢WQ®¢QQ - H H 00.0.0000...IOOOOIIOQOIOOIOOOIOIIO.I..... ...OOOOO OOCUU ¢W®®WW® . 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N N eeemw I W m..--.--.-~---'.---IM----..0--H.--...-.-“9-.--.-0--0I.-IO.I.-N---’O----\w-.--.I--Cm|-0-.I---’----O-.-Im-IUOOOIUONO---¢|I-U”0-0-6-9--. 153 transportation accessibility was developed by aggregating all trans- portation land in each county, and forming a ratio of this area to total county area. Multipliers ranging from 1.0 to 1.5 in value were applied to each county raw transportation score to reflect the influence of proximity to major regional highways as shown in Figure 22. Figure 23 shows the distribution of proximity multipliers for each county. The formula is used to derive the transportation index. AT expresses county transportation land area, A total county land area, w the county proximity weighting factor. The results of the computations appear in Table 20, and the regional distribution of this index appears in Figure 24. For winter sports recreation activity, access is important, as many county roads, passable in the summer season months, are non-accessible in the winter season. Counties having close proximity to the major north-south I-75 freeway exhibit a high index value, while counties in the eastern portion of the Region show diminished transportation accessibility (Figure 24). Regional areas located in the western counties show relatively high accessi- bility due to the several major routes into the Region from the south, as well as influences from the I-75 corridor. The indices for local t0pographical relief IR and trans- portation access IT combine to form a composite winter sports 154 <0.¢NN 00.0000N om.N 00.000NN mo.N 00.000000 vgoNxmz 00.nm 00.0NO0N om. 00.0000 00.N 00.00000m wmxsmmmNz 00.n0 me.NNONN 0N.N 00.0NO0N mo.N 00.000000 mmpmN:mz 00.00 mN.00NON 00.N 00.0000 00. 00.Nmmmmm smcmNomA mN.mm 00.000NN om.N 00.0mN0 00.N 00.0NN000 mxmmemg 0N.NmN 00.N000N 0N. 00.000NN 00. 00.0000Nm mmgm>mgN 0:0:0 00.0mN 00.N000N om.N 00.0000N 00. 00.000000 00550 00.0NN N0.Nm~0N om.N 00.0000 N0. 00.000000 xNo>0Ngmz0 .00.N0 No.00NNN 0N.N 00.0000 00. 00.0000NN mNN:mm 00.NNN 00.0000N om. 00.00NON em. 00.0000mm ENLN:< mN.0N Nm.NNNO 0~.N 00.0000 00.. 00.0mmmm: mNmN cmwga 0:.NNN 00.0000N om.N 00.00Nm N0. 00.0m0000 ommmuo N~.N0 N0.0000 0N.N 00.0000 mo.N 00.000000 mcoumo 00.00 00.0NON . om.N 00.0NNO mo.N 00.000N00 Nu:LoEN:oz mN.0~N 00.00m0N om.N 00.0N00 mo.N 00.000000 ngowzmc0 Nm.¢NN 0N.N00¢N om.N 00.0vaN m¢.N 00.0NNON0 :mmzonm:0 Nm.N0 00.0000N 00.N 00.NONNN N0.N 00.000NNO 0:00N< NN.00 0~.mm0N 0N.N 00.0N00 0N.N 00.00N000 acouN< Nucsou NwNNQNNsz mmc< 0:00 mmmgm>< xwu:N xmu:N Nuwewxogm :ompmugoam:mgN mo mmg< o» wmg< 0:00 00:200 vaNNaeLoz 30¢ Nozzmwz 20:300 NNV we aNNmm zu::o0 Nmuop N8 N3 :0 E E N: .xmv:N :owumugoamcogh mo :oN00030500--.0~ 0005 3089/0- .emumxm :owumu0oam:m0k 003000: pm:om0wmun.00 000000 mwhmidz hmomrhmoz x_o>w..mOmmIU/ / Ca 302.502: 156 1.3 1 \EMMET' 1.5 M. 1. 2 -.1 1.c111:_amo_msous LSLE LgflA_L LEVOMH __1__. l 1.3 i I H. 1.0 1.0 ANTRLM 1 0155560 MQ_NTMORENCY _4_L_f_E_NA_ LEELANAU ‘ 7 | I 1.1 1 1.1 1 1.3 I 1.5 1.2 1.1 BENZ_IE lemrywsase K_ALKA5KA_ anyrono 95001: L ALCONA_ 1.3 1.3 MANISTEE —_- wsxrono MISS_AUKEE Figure 23.--Distribution of Transportation Multipliers by County. 157 ..-...0'.'~-..0...-.d----‘----H--.‘.'---¢----.'---o0.--.----K-"..-I--®I--UO---. .‘0 hfiflfiohfiflqH0hflfl4HNMOflnH. 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H _00-_0000- m .0-0_000- .00-.000 . .00-.00 H ......u. u I w a o 0----H----.-‘0-0'---...--Q----0----N--.-..---o----.----m--..0 .---3----0'.--M -..I000.-NI.. -0..-. .0-...--C. neuoo Owwtwwommwwmwwma oocono UOQOOOChOOOOUO. 161 Digital processing and digital combination of data from several data sources including remote sensing will permit modeling efforts to achieve greater detail and specification while providing the modeler with important options of flexibility in selection of aggregation, detail level and the cost effectiveness of data acquisition. Formation of the Data Base Flexible and cost efficient data base systems require care- ful design. There exists a trade-off between costs of data base system Operation and system flexibility. Flexibility must be attained at the expense of system through—put efficiency. Yet proper design will allow either option to be selected by the system user while providing the lowest operating cost attainable with current technology. By taking maximum advantage of the design of modern computer hardware the costs of digital image processing of remotely sensed 20 data will decrease substantially. Bennet has elaborated five basic expense categories associated with the formation of a data base: Development Costs Acquisition and Operation Costs Processing Costs Interpretation Costs Application and Presentation Costs 20Marilyn Bennett, "Evaluating Data Collection Costs; Emphasis on Remote Sensing," M.S. Thesis, Department of Resource Development, Michigan State University (East Lansing, Michigan, 1974 . 162 These costs are inherent to all data base systems, and require detailed analysis in the process of cost-effective system design. Within the framework imposed by these five cost categories the regional modeler and data base system designer must select the least-cost combination sufficient to meet the required application. Procedures to permit variable sampling intensities of data bases provided by satellite remote sensing, digitized photo interpretation, or ground-based surveys need incorporation into the data base manage- ment system to achieve, for a given application, greatest cost efficiency. Each data file should have an associated precision file for maximum effectiveness in precision determination. As stated before in this study precision determination allows the modeler important control over the optimization of the level of detail of data and hence, to the data base system designer, optimization of the costs associated with data acquisition and utilization. The complete data base system should incorporate methods of precision determination at all points in the data manage- ment process, especially when two or more data bases are combined together to form a composite.21 Knowledge of the precision of combined data bases allows meaningful decisions to be made when model results are applied. Summary The chapter has provided a discussion of previously developed, general methods for land-use inventory, and has outlined 2‘Spe Ackoff, op. cit., pp. 218-250. 163 the essential features of each of the methods. The chapter details the use of indices for site description and outlines mathematical methods for index evaluation. The chapter discusses methods for estimating site economic productivity using indices and details a method employed by the Soil Conservation Service to estimate county rankings for summer season recreation development. The chapter details a method for determining a winter sports location index by measurement of local topographic relief and tranSportation accessibility and develops a spatially ordered location of winter sports activity using gradient profile maps. The chapter discusses the design of a data base for regional land-use studies and outlines factors which must be considered in the design of an adequate, cost effective data base system. The chapter also outlines how the combination of remotely sensed, satel- lite obtained imagery with data obtained from other sources depends upon sample intervals and variable sampling rates which in turn are dependent upon adequate measurement of data precision at all levels of the data management process. The chapter concludes with a discussion of the importance of precision analysis to the optimal operation of a regional land- use data system. CHAPTER VI SUMMARY, CONCLUSIONS AND RECOMMENDATIONS Summary The study has produced a design for an allocative economic and land-use model for sub-state regions, capable of allocating land uses or of responding to regional changes in economic demand with predictions of altered land-use configuration. The 18-county sub- state region selected as the focus of the study is located at the tip of the lower peninsula of Michigan and is typical of natural areas in the Midwest whose scenic attributes and other natural features provide recreation attraction. A large number of visitor trips to these areas originate from the Midwest's major urban population centers. The study began with a discussion in Chapter I of the need for more comprehensive land-use planning in these rural, recreation oriented regions to avoid conflicts of future land use and possible resulting degradation of important forest and water resources. Chapter II provided a description of the physical and economic history of the Study Region, outlining its early development as a lumber producing area and its emergence as a center of recrea- tion activity in the mid-nineteenth century. An outline of current economic trends and important issues concerning growth of seasonal 164 165 or second-home dwellings in the Region helped describe the potential impacts of land-use conversion upon regional forest resources. Chapter III develOped an input-output model based on secondary data derived from the national accounts table to more fully assess recent growth changes in the regional economy. The input-output model was constructed and evaluated for economic trends over several key years. Growth sectors were noted to be manufacturing, construction, and lumber and paper products, with regional agricultural output virtually constant over the period. A projection of economic output to 1980 included a comparison of leconomic output to regional tourism expenditures, noting the significant and continuing impact of tourism upon the regional economy. Chapter IV outlined a model which combines both the macro- economic structure of the regional input-output model and the micro- economic structure of land-using firms to produce a model capable of land-use prediction in response to exogenous changes in regional demands. The chapter discussed the importance of precision analysis to regional modeling, and included a model application disclosing areas of likely future development of summer-season, water-related recreation in the Region. The chapter provided an initial statisti- cal treatment of the correlation of recreation potential to the density of second or seasonal dwellings in the Region. Chapter V treated several aspects of survey sampling on a spatial data base. The chapter provided review of several previously designed comprehensive methods for determining site potential for 166 economic development, followed by a discussion of relevant site measures to be applied in a survey of the regional land base. The chapter addressed questions of control over bias in survey sampling, and included development of preliminary indices to assess the spatial distribution of winter season recreation potential in the Region using local topographic relief and local transportation accessibility as primary determinants. The chapter discussed methods of combining data from remote sensing with data obtained from alternate sources to produce an efficient, responsive land-use inventory system. The chapter identified cost effectiveness of the land-use survey, control over precision of the data, and optimization of the data management system as important elements of the land-use modeling effort. Conclusions Seasonal Housing There exists in the Region a continuing high rate of land conversion for housing, a substantial portion of which is probably seasonally occupied, second home development. However, the region does not suffer from a lack of land for new housing, but does suffer a scarcity of desirable shoreline land. Many new development areas, therefore, are likely to be in lands previously devoted to agri- cultural use or to forests used for wildlife habitat or timber production. 0f the remaining undeveloped regional shoreline a large amount exists in a protected status in state and federal forest 167 preserves. As demand for recreation land continues, more pressure may be created to convert public lands into public recreation areas. This conversion would impact directly upon available commercial timber land and would diminish the availability of certain types of wildlife habitat. The Region currently shows signs of increasing economic infrastructure as industrial growth and exports continue. Manu- facturing has increased dramatically since 1963, resulting in growing export capacity and creating new employment for local labor forces and immigration of skilled and semi-skilled labor. Demand for permanent housing will increase especially in locations near popula- tion centers and centers of industrial development. Among the major land using sectors of the Region's economy, growth sectors appear to be those of Lumber Products, Paper Products, and Construc- tion.- Agriculture continues to produce a static gross output from a diminishing land base, possibly indicating economies resulting from improvements in efficiencies of farming Operations. Agriculture is a major land using sector, but is not a major component of regional gross output. Tourism continues to be a major factor in the economy of the Region. The comparative predominance of tourism expenditures in the Region is reflected in the 1980 projected figure of 600 million dollars or roughly twice the 1980 forecast value of net exports of manufactured goods. Tourism expenditures represent a value between 20 and 25 percent of total projected 1980 regional gross output. 168 The continuing visitor activity in the Region has its counterpart in demand for seasonal, vacation type dwellings. With the continued high incidence of conversion of land for housing, a substantial portion of which is seasonal housing, greater impacts upon forest resources and conversion of large areas of forest land can be expected in the future. Summer Season Recreation Development The preliminary analysis of summer season water-related recreation development provided in Chapter IV shows the effectiveness of treating primary recreation attractiveness features as spatially distributed variables. Similar treatment of attractiveness vari- ables provided the spatially distributed winter season potential index of Chapter V. The results of both summer and winter season attractiveness analyses correlate well with observed location preferences of recreation enterprises. Expansion of these recreation enterprises will occur in areas of the Region possessing less attractiveness features as available land areas in prime locations become scarce. The results of the study's development of both summer season and winter season recreation potential show that the western portions of the Region tend to have greater attractiveness features relative to the eastern portionsmo mgouumm «H Lo. mpepspthapm pppcwo--.~-m m4mMLH as... m_ mo._om.wH ooooom.m~mmmp page. 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