AGGREGATION PROBLEMS IN BENEFITS ESTIMATION: A SIMULATION APPROACH William J. Vaughan • John Mullahy Julie A. Hewitt Michael Hazilla and Clifford S. Russell Cooperative Agreement CR 810M66-01 Project Officer Dr. George Parsons Office of Policy Analysis U.S. Environmental Protection Agency Washington, D.C. 20460 Resources for the Future Washington, D.C. 20036 ------- AGGREGATION PROBLEMS IN BENEFITS ESTIMATION: A SIMULATION APPROACH Executive Summary Background This project was undertaken in response to several concerns about the potential perils of aggregating and disaggregating in the context of pollution control benefit estimation. The oldest of these concerns involved the fairly common practice of using results from national-level studies as the basis for regional benefit estimates. (For example, earlier RFF work—Vaughan and Russell 1982—in which national participation equations for recreational fishing were estimated, was used by another EPA contractor to assess damages to northeastern states from acid deposition.) The mirror image of this practice that is the "blowing up" of regional studies, which are often seen as cheaper or easier pieces of research, to obtain national benefit estimates, was also to be investigated. As the research proceeded, however, it became clear that a prior aggregation practice cried out for examination; that is the use of average aggregate resource-availability measures as explanatory variables in benefit estimation in either national or regional studies. These measures have in the past been used routinely, if without much formal justification,. because a link was necessary between measured participation behavior and the results of pollution control. The resource availability variables served conveniently as such a link, because a reasonable case could be made that availability must matter to recreators and that at least rough account could be taken of pollution control by showing availability increasing as a result. As it turns out, however, it is possible to prove that availability, measured by density of recreation acres per acre of total area, is a conceptually correct proxy for the expected price of obtaining a recreation experience for the recreator around whose home the density is measured. In practice, state or county-level density measures are the only ii ------- ones available, however, and must be used for every resident. The aggregation question, therefore, is how are the results affected by using the average density over some politically or geographically defined area as a price proxy for every resident of that area. Because of the ubiquity of these several practices, it was seen as important by EPA to have some idea of how much violence they each do to results. Unfortunately, for no actual cases could this question be answered. While it is clear that dome of the practices involve the introduction of errors in variables problems into statistical analysis, there is no benchmark against which to compare the results of an aggregation exercise in any specific setting. This lack is not an inevitable one or the result of a problem of principle. It is simply the reflection of the practical problems, generally data problems, that drive researchers toward the aggregation approaches in the first place. But in the absence of real-world benchmarks, this project was designed around a simulation model from a hypothetical world. Approach To create a simulation model appropriate to the examination of the aggregation problems just described required at a minimum that the following features be captured: consumer/recreators, located in space, with known utility functions defined over the included recreation and consumption activities recreation sites, located in space, and initially either available or unavailable because of pollution an overlay of jurisdictional boundaries that would define the areas over which recreation site density would be averaged. On the basis of the "data", generated by the reaction of the (utility-maximizing) consumers to the relative prices of general consumption and recreation at particular sites, the activities going on in the regions could be calculated. The situation in the pre-pollution control situation could be compared to that obtaining after pollution control—mimicked as the making available of previously unavailable iii ------- sites—was implemented. The "true" results then would form a benchmark against which the results of various aggregation practices could be compared. It was also possible to establish intermediate situations to provide additional comparisons, as for example by supplying each consumer with a correct recreation-site price proxy as well as the true travel-cost prices of the nearest recreation site. The practices the simulation model was designed to address were: the use of jurisdictionally-averaged travel price .proxies instead of individually correct travel-cost prices of recreation the use of models estimated for one region to project benefits for a "nation" of differing regions the use of a model estimated for the "nation" to project benefits for specific subnational regions. Model and Calculations A sample of five hundred consumer/recreators, each with a known quadratic utility function, with two kinds of recreation and a composite other consumption activity as arguments, were assigned randomly to locations on a rectangular plane representing a "nation." For one set of runs the distribution of the consumers was uniform within subnational regions that would later be taken to be the smallest "political" jurisdictions. For another set the distributions used were truncated normal, designed to mimic the peaking of population density in urban areas. One of the recreation activities was taken to depend on "water" which was in turn subject to pollution. Recreation sites for each type of recreation were also distributed on the plane on the basis of uniform distributions associated with subnational regions. (These regions were intended to represent different geologic provinces.) For the water-based recreation activity the sites may be thought of as equal-sized lakes. Each lake was assigned to either the iv ------- class "available for recreation" or the class "unavailable by virtue of pollution" for the base case. (Two different extents of pollution were assumed for alternative cases. Three and 30 percent unavailable.) The price of the composite consumption good was taken to be one, in the units of consumer income. Travel cost was taken to be 0.10 per mile and no costs were assigned to site access. The set of consumer maximization problems were solved for true equilibrium choices of consumption and recreation, in days, in the with-pollution situation. The correct travel-cost price for each individual calculated from his or her location relative to unpolluted sites was used for this benchmark. In addition, for the with-pollution situation, several other pieces of data were recorded: the correct recreation site density for each individual based on actual availability around the individuals home local the average density of available sites within each jurisdiction. Aggregation of the average travel cost proxies was further explored by combining the smallest jurisdictional elements into larger states of a variety of average sizes, on the basis of a randomized choice routine. The average sizes used were 5, 10 and 18, where the entire nation consisted of 36 elemental jurisdictions. To produce the benchmark benefits of pollution control for the simulated situation required the following steps: applying the pollution control policy by making available for recreation all the sites placed in the polluted/unavailable class in the base case. re-solving the recreator-consumer optimization problems for the correct new travel cost prices calculating the correct measures of the welfare change resulting from the policy (as equivalent and compensating variation and several approximations thereto). ------- For the jurisdictionally averaged price proxies (recreation site densities), alternative estimates of the welfare changes were established using a version of the participation method. That is, participation equations were estimated from the participation data generated in the with-pollution case, with the key independent variable being the recreation site density for the jurisdiction of residence of the individual. Such equations were estimated for each level of jurisdictional aggregation. The change in recreation days due to the increase in availability caused9 by pollution control was projected in the usual way. This change could itself be compared to the change in days of recreation from the equilibrium solution to the consumer maximization problems. In addition, the welfare measures were approximated using the average value of willingness to pay for recreation days based on the results of the individual optimizations. (This willingness to pay figure may be thought of as representing the results of a separate willingness-to-pay (contingent valuation) survey.) To explore the effect of using national studies on the subnational level, the national participation equations for alternative average jurisdiction sizes were used to project changes in participation in each region taken separately. These results were compared, for both days and willingness to pay values, to the true changes for that region. Exploring the effect of going the other way, blowing up regional study • results to the national level was slightly more complicated than simply reversing the national-to-regional chain of calculation. This was the result of the small number of individuals within the average elemental jurisdiction. To obtain enough observations to do a subnational equation estimation it was necessary effectively to make up a new, larger nation from four of the original nations. Each of these original nations became a region in the new conglomerate. For each such region, the participation equation was used to estimate national participation change due to pollution control. Results The results of using RECSIM to explore the aggregation problems of central concern are interesting but more than a little disquieting. First, vi ------- in using aggregated price proxies it appears that we run very great risks indeed. While time and budget constraints did not permit a fine-grained search for the level of aggregation at which the situation deteriorates markedly, it is clear that by the time aggregation involves five elemental units or more, we are in trouble. At this point, sizes and even signs of participation change projections have become unreliable. While for some individual trials, the aggregated proxies produce equations with excellent predictive ability, these are clearly random events. More likely are events from the same distributions that produce predictions very wide of the mark. This, it should be emphasized, is not a problem with the proxies per se, for when the correct proxy is assigned to each observation, the results are very close to those based on actual travel-cost prices. It is a problem arising from assigning an average (or aggregated) proxy value to each observation in a jurisdiction. As such, it is not surprising. But since data sets on recreation participation have never been rich enough to allow calculation of individually tailored prices and since it is common practice to use average price proxies (availability measures) this particular manifestation of the aggregation problem must be viewed with great concern. The exploration of the practice of applying models at different levels of aggregation than those from which they were estimated leads to similar concerns. This applies both to "scaling up" from a regional case study to a national participation (or benefit) estimate, and to "scaling down" from a nationally estimated model to attack a regional problem. Again, both procedures are commonly suggested. Overall, then, this study suggests that benefit estimation is even harder than is commonly assumed. While defensible methods for doing participation-based benefit studies are available, the data necessary to support those methods usually are not. This implies that if benefit estimation is to become a long term and believable part of the policy choice process, some substantial investment in data generation will be required. Such investments do not come cheaply, but this study suggests just how large their payoffs could be. vii ------- CONTENTS Executive Summary i Chapter 1 - Aggregation Problems in Benefit Estimation: Simulation for Better Understanding 1 RECSIM Model Structure . 2 Problems to be Explored 4 Alternative Methods of Approximating Welfare Changes 4 Aggregated Proxy Price Variables 4 Using Regional Case Study Results to Estimate National Benefits 5 Using Nationally Estimated Relations at a Regional Level 6 Plan of the Report 6 Anticipating the Results 7 References 9 Chapter 2 - Some Theoretical Background 10 General Considerations in Designing Simple Simulation Models of Recreation Choice 10 Recreation Decisions and the (New) Theory of Consumer Behavior 12 The Utility and Production Functions: Wants Versus Characteristics 15 Wants or Characteristics 15 Indirect Utility, Demand Functions for Wants and Goods, and the Structure of the B Matrix of the Lancaster Model . . 23 viii ------- The Utility Function: Selection of a Specific Formulation for Simulation 29 The Indifference Map and Marshallian Demand Curves from the Additive Quadratic 30 Demand Functions from the Additive Quadratic Function 38 The Indirect Utility Function from the Direct Additive Quadratic Utility Function 40 The Treatment of Time and Visits in the Recreation Simulator 41 Concluding Remarks 46 Appendix 2.A - The Lancaster Model: An Overview 48 References 55 Chapter 3 ~ Econometric Considerations 59 Situations Where a Subset of the Regressors are Observed Only as Group Averages 59 Distinction Between Classical Errors-in-Variables Problem and the Disturbances with Nonzero Means Problem. ... 61 Parameter Bias in Mixed Models Using Individual-Specific and Group Average Regressors: The McFadden and Reid Approach *..... 62 Another View of the Parameter Bias Problem in Mixed Models: The Theil Approach 66 Implications and Obstacles 69 Unknown Geographic Regions 72 The Value of Additional Information 77 Methods for Analyzing Demand and Hence Welfare Changes 79 Single Equation Methods 81 The Dual System-Wide Approach 84 Concluding Remarks 36 Appendix 3.A - The Role of Recreation Resource Availability Variables in Participation Analysis 88 IX ------- Relating Density and Distance 89 Some Implications for Aggregation: Measuring the Proxy for X 96 Conclusion 97 Appendix 3-B - The AIDS Model 98 Estimation of AIDS - Some Specific Examples 103 Calculating Welfare Changes: Exact, Almost Exact and Approximate Measures 110 References 119 Chapter 4 - RECSIM Model Design: The Data-Generating Modules 123 Creating Information on Individuals • 125 Elemental People and Geographical Grids 126 Poisson Module: Geographical Grid Placement of the Universe of Recreation Sites 128 Steps to Place Fishing Sites . 128 Policy Module: Selecting a Subset of Pre-Policy Fishable Sites from the Universe of Post-Policy Sites 134 Steps for Policy 134 People Module: Locating Individuals in Space 135 Steps for People 135 Passive Module: Locating Passive Recreation Sites in Space. ... 139 Euclid Module: Connecting Recreation Sites and Individuals. ... 140 Socio Module: Assigning Socioeconomic Attributes to Individuals 142 Aggregate Module: Combining Elementary Political Units to Form Aggregated Political Units 145 Chapter 5 - Model Design: Optimization and Welfare Calculation in RECSIM 152 Preliminary Structure of Optimize 152 Zero Consumption Levels 157 Scaling the RECSIM Utility Maximization Problem 159 Application to RECSIM 160 Basic Data. 161 Welfare Changes 162 ------- Golden Section Search 165 Approximations to Average Values Per Recreation Occasion 168 Concluding Comments 174 Appendix 5.A - Pitfalls in Applied Welfare Analysis with Recreation Participation Models 175 Origins of the Two-Step Method 178 Valuation Issue ; 180 Valuation with Marginal Unit Values . . . 182 Valuation with Average Unit Values 185 Concluding Remarks 189 References 191 Chapter 6 - Model Design: The Estimate, Evaluate and Compare Modules 194 The Estimate Module 194 Functional Form 194 Variables 195 The Evaluate Module: Predicting with the Estimated Demand Models 201 Single Demand Equation Model in Prices 201 Single Demand Equation Model in Proxies for Price 204 The Compare Module 205 Criteria for Evaluation of Econometric Model Performance 206 The Argument for Non-Parametric Model Evaluation Procedures 209 Non-Parametric Model Evaluation Criteria 211 Some Formal Non-Parametric Tests of Homogeneity 217 Concluding Remarks 221 Appendix 6.A - The Correct Calculation of Welfare Changes from the Estimated Single Demand Equation"Models 223 References 225 Chapter 7 - Results and Discussion 228 Aggregation of Price Proxies 230 Aggregated and Disaggregated Use of Estimation Results 237 Summary and Conclusions 239 Appendix 7.A 242 Chapter 8 - Summary and Implications for Future Work 404 Summary of Results ' 405 Further Possible Research Using RECSIM 406 Aggregation of Different Response Relations for Same Activity 406 Aggregation of Different Activities Under a Single Name 406 A Possible Focus of Longer Run Work 407 XI ------- CHAPTER 1 AGGREGATION PROBLEMS IN BENEFIT ESTIMATION: SIMULATION FOR BETTER UNDERSTANDING If we wish to estimate the benefits of a public policy that accrue-to society via routes involving reactions by individuals to prices, we are on conceptually familiar economic ground. If we have data on the individuals involved — their characteristics, the prices they face before and after policy implementation, and the consumption choices they made — we can estimate the relevant set of demand equations and extract the individual 4 changes in consumer surplus attributable to the policy. The sum of the changes over all affected individuals will be a proper measure of the benefit of the policy. Even in cases of policies affecting market goods, however, we are never in such an excellent position. Often we lack data on the characteristics of individuals, or on the price changes faced. Or our data on all- other prices and choices may not be comprehensive enough to support a complete demand system. Since these problems interfere with our*ability to correctly estimate policy benefits in market situations, it should hardly be surprising-to find that the difficulties are enormously greater when policies affect unpriced activities or resources, as they do in the case of pollution-control policies.. Then we cannot find market prices to attach to particular choices. Rather, for example, travel costs of available recreation alternatives must be calculated. Neither are data on choices actually made thrown up automatically by market operation but must be collected by special surveys. Further, it may be impossible to be sure that the categories of choices we presume are appropriate, and therefore gather data on, are in fact the categories used by consumers. Additional complications often arise when we try to match the benefit estimates we can produce with the needs of our policy analysis. For example, we may, because of budget constraints, do a regional study of a policy's ------- impact though what we want is a national benefit estimate. Or a national study may have been accomplished when a subsequent question calls for answers specific to a regional problem. Many of these problems that arise as we move away from the idealized benefit estimation situation can be thought of as involving aggregation, whether of data or of the estimates themselves. This broad compass of the word can easily lead to misunderstanding, as one person's aggregation problem need not be another's. (Several of these aggregation issues in the travel demand context are discussed in Koppelman, et. al., 1976.) Beyond* that, it remains an open question how important any one of the aggregation problems may be relative to the others .and relative to other departures from the ideal. These questions cannot in general be answered using actual data, because the benchmark for comparison does not exist. We have no actual data sufficiently detailed and comprehensive to support idealized benefit estimation. This, in a nutshell, is why we have undertaken to design and build a simulation model reproducing the essential characteristics of a pollution control policy context, in which benefits accrue via individuals' reaction to prices sensitive to the policy choice. This model, RECSIM was designed to produce true, if hypothetical, benchmarks against which to compare results reflecting the operation of one or another methodological compromise, the use of surrogates for prices, or the existence in front of the available data of one or another aggregation "veil." In this brief introduction we acquaint the reader with the structure of RECSIM. Then we describe the problems, of both methodology and aggregation, that RECSIM can address and rank these problems .according to our a priori judgement of their importance and the extent to which simulation as opposed to theory is necessary in examining them. The judgement on importance reflects a logical • ordering of the problems. If the first ones, model structure and variable specification, cannot be shown to be handled satisfactorily by the available data and approaches, the latter ones concerning aggregation of results are merely curiosities. RECSIM MODEL STRUCTURE The recreation simulation model is designed to generate data on the choices made by hypothetical individuals faced with alternative consumption 2 ------- P033ibilltie3 that include active recreation with the characteristics of "fishing" and "camping," an urban recreation alternative one might think of as movies, and a generalized alternative for all other ways of spending money. The fishing and camping activities take place only at sites to which travel is necessary (in general) from place of residence. Fishing is sensitive to an hypothesized "pollution" which makes some sites unavailable for the activity in the base case. Camping is not sensitive to this factor, and site availability does not change when pollution-control is hypothesized to occur and increase the availability of the sites for fishing. The urban activity also takes place at specific places, and these are located within areas of higher than average population density. The composite activity is site-less. The distribution of water "sites" and of consumers is the initial problem tackled by the model. These are placed randomly on a plain in accordance with density function parameters that may vary over sections of that plain. The sites and people are then contained within jurisdictions, which are artificial subdivisions of the plain at a finer level than" the divisions used in the assignment problem. One of the model's key capabilities is the production of larger jurisdictions out of the initial units in random ways so that shape and size vary across model runs. Each consumer is endowed with income and time constraints and each faces a set of travel costs depending on residential location relative to sites. The consumer's problem is to maximize utility, under a quadratic utility function and subject to the constraints. Pollution control policy, as suggested above, has the effect of increasing the number of available sites and thus decreasing Cor at least not increasing for any person) the travel cost to the closest fishing site. Exact before and after welfare calculations are possible in the simulation format, and they may be compared with a variety of approximate measures based on methods in general practical use. \.- The welfare measures of interest are dollar equivalents of utility changes — the compensating and equivalent variations; the approximation represented by simple consumer surplus; and a variety of two-step method for aggregate change, based on participation and unit values. 3 ------- The model is structured to 'allow comparison of welfare measures in a variety of contexts, including: model specification or general variable speciftcation, average level of jurisdictional aggregation, and level of study contrasted with level at which an answer is being sought. The comparisons are made using formal tests to reduce the impressionistic component. PROBLEMS TO BE EXPLORED We' intend to use RECSIM to investigate the seriousness for benefit estimation of four common second-best practices. It appears that theory•can tell us what is correct in each case but cannot tell us in advance the direction or size of the error introduced by approximate methods. RECSIM will provide some of that missing information. Alternative Methods of Approximating Welfare Changes The most important because most fundamental question we shall put to RECSIM is, what penalty do we pay for the use of such ad hoc methods of benefit estimation as the participation equation approach? Because we shall have both true benefits (compensating and equivalent variation measures) and a "correct" approximation via a Marshallian demand function, we can be quite precise about the impact of going the participation route (often the only practical route). Further, we can see how this impact varies with the simulation model's initial conditions, for example: the severity of the pre-policy pollution problem, the structure of the household production "technology" matrix, and the extent of variation in the surrogate distance prices (availability densities). This last opportunity is closely linked t'o our second major question for RECSIM. Aggregated Proxy Price Variables In recreation benefit studies actual individual-specific micro data on recreation choices along with incomes, ages, sex, and other relevant variables is usually available. But other data, especially data on the prices of recreation alternatives Is usually not available. This is true, for example, of the hunting and fishing data set that we used in an earlier study (Vaughan and Russell 1982), the 1975 MSHFWR survey. In this survey, ------- Individuals and their choices are finely characterized but virtually no information is available on the relative price sets each individual faced in making the reported choices. Surrogates for prices, such as density variables, must be found if econometric estimation is to be performed. (See below, appendix A to chapter 3). Only by (1) locating each person on a very fine grid; and (2), characterizing each person's recreation choice set (at vast expense in time and effort) can we improve on the use of a more aggregated proxy for the price set — something like the density of relevant recreation opportunities in the general neighborhood (state or county). But a priori, we would expect the utility of the method to depend on whether the aggregation level of the available surrogate price data matched the actual choice "horizon" of the recreationista. If, for example, recreationists tend to make choices among sites within 30 or 50 miles of home while the available density data is for averages by'state, we should expect the match to be poor. RECSIM will have the capability of providing data at any level of spatial aggregation from the smallest grid square within which average density is constant up to two or three major- subnational regions. The method for aggregating regions will be random, so that the benefit estimate comparisons can be carried out for a number of different arrangements all having the same average aggregation 1evel. RECSIM- was designed primarily with this type of data aggregation problem in mind. This reflects its intellectual roots in our recreation benefits work and the fact that, for recreation, data on individual choices but not on the price vector behind those choices, are commonly available. Using Regional Case Study Results to Estimate National Benefits A form of aggregation of the benefit estimates themselves involves going from-regional caae studies to national totals. Interest in this possibility might be- said to arise partly because of historic accident and partly because of. research budget constraints. In particular, regional case studies have long been popular with funders and researchers because they were seen as cheaper and more manageable. But once a regional study is done, how can it be used to get a national benefits? ------- For acme, this aggregation problem raises the moat interesting conceptual economic problems although it is not the most challenging econometrically. This is because we are dealing here with "edge effects'* across regional boundaries that are complicated by simultaneous but presumably differential change on both sides of each boundary. In other words, we are flirting with questions of the importance of general equilibrium models. RECSIM could be used to begin exploration of this problem by performing simulated regional studies for differently defined regions and estimating national total benefits based on regional per capita benefits (or regional per capita participation effects) and national populations. The effect of using more or less care to attach per capita effects to income/age/sex strata could be examined at the same time (eg: the use of simple per capita averages could be contrasted with use of a projection equation having the key socio-economic variables as independent variables and per capita benefits as dependent variables.) Using Nationally Estimated Relations at a Regional Level At the other extreme from the previous problem, existing national studies, like our freshwater fishing participation model, seem to promise money saving routes to benefit estimates for less than national situations. For example, in the fisheries case, PIEC used our national model to estimate benefits from acid rain control affecting the northern and eastern tier of states. Investigation of the penalties to be expected from this procedure could also be undertaken using RECSIM. The "best" national participation equation (for any particular level'of choice-set aggregation) could be obtained and then applied to "regional" benefit questions. PLAN OF THE REPORT After this brief introductory chapter, we lay the foundations for the simulation model in chapters on theory (chapter 2) and econometrics (chapter 3). In the former, we discuss consumer theory for situations in which zero consumption is a possibility to be allowed for; describe the utility function and household production approach to be used; and problems in the treatment ------- of time in such optimization problems. The latter chapter deals with the major problem for practical analysis raised by having seme variables available only in aggregated form. That is, the second of the aggregation problems noted above is dealt with on a theoretical level in chapter 3. We then go on to discuss alternative approaches to- the estimation of demand functions and their corresponding econometric implications. Two technical appendices to chapter 3 deal in more depth with (A) the role of recreational availability variables in participation analysis; and (8) the multi-equation demand system referred to as "Almost Ideal Demand System" or AIDS. There follow three chapters describing the design and construction of RECSIM in more detail. Chapter 4 provides an overview and then goes into data generation. In chapter 5 we discuss the optimization routine and related correct welfare calculations. Chapter 6. is concerned with the estimation of demand functions, the evaluation of approximate welfare measures, and the techniques for comparison of answers arrived at by different routes. A discussion of results from RECSIM runs is provided in chapter 7. Finally, in chapter 8 we summarize, our findings and discuss additional research possibilities for the model. ANTICIPATING THE RESULTS The results of using RECSIM to explore the aggregation problems of central concern are interesting but more than a little disquieting. First, in using aggregated price proxies it appears that we run very great risks indeed. While time and budget constraints did not permit a fine-grained search for the level of aggregation at which the situation deteriorates markedly, it is clear that by the time aggregation involves five elemental units or more, we are in trouble. At this point, sizes and even signs of participation change projections have become unreliable. While for some individual trials, the aggregated proxies produce equations with excellent predictive ability, these are clearly random events. More likely are events from the same distributions that produce predictions very wide of the mark. This, it should be emphasized, is not a problem with the proxies per se, for when the correct proxy is assigned to each observation, the results are very close to those based on actual travel-cost prices. It is a problem arising from assigning an average (or aggregated) proxy value to each 7 ------- observation in a jurisdiction. As such, it is not surprising. But since data sets on recreation participation have never been rich enough to allow calculation of individually tailored prices and since it is common practice to use average price proxies (availability measures) this particular manifestation of the aggregation problem must be viewed with great concern. The exploration of the practice of applying models at different levels of aggregation than those from which they were estimated leads to similar concerns. This applies both to "scaling up" from.a regional case study to a national participation (or benefit) estimate, and to "scaling down" from a nationally estimated model to an attack in a regional problem. Again, both procedures are commonly suggested. (To name only one example, we were asked for help, in an attempt by PIEC to apply the national models in Vaughan and Russell, 1982 to the analysis of the recreational fishing benefits in the northeast to be expected from acid rain control.) Overall, then, this study suggests that benefit estimation is even harder than is commonly assumed. While defensible methods for doing participation-based benefit studies are available, the data necessary to support those methods usually are not. This implies that if benefit estimation is to become a long term and believable part of the policy choice process, some substantial investment in data generation will be required. Such investments do not come cheaply, but this study suggests Just how large the-ir payoffs could be. ------- REFERENCES 1. Haitovsky, Yoel; 1973. Regression Estimation from Grouped Observations, New York: Hafner Pre33. 2. Koppelman, Frank S., Moshe Ben-Akiva and Thawat Watanatada. 1976. j Development of an Aggregate Model of Urbanized Area Travel Behavior, Phase 1 Final. Report to U.S. Department of Transportation, Cambridge: MIT Center for Transportation Studies. 3. Stewart, Mark B. 1983. "On Least Squares Estimation When the Dependent Variable ia Grouped," Review of Economic Studies, vol. 50, pp. 737-753. 4. Vaughan, William J. and Clifford S. Russell. 1982. Freshwater Recreational Fishing, Washington, D.C., Resources for the Future. ------- CHAPTER 2 SOME THEORETICAL BACKGROUND Before a useful simulation model of recreation participation can be constructed it is necessary to clarify some underlying theoretical issues. In particular, we shall wish to explore the related questions of how to represent the utility function presumed to lie behind observed decisions and how to allow for the possibility of zero consumption of some subset of available goods and services. Along the way the issue of how to include time in the problem may usefully be addressed. These background matters are taken up here. GENEBAL CONSIDERATIONS IN DESIGNING SIMPLE SIMULATION MODELS OF RECREATION CHOICE In order to exploit the calculus, conventional utility theory makes the implicit assumption that the consumer's optimal consumption bundle will represent an interior solution in the space of available alternatives. That is, the maTrtimmt of the consumer's utility function occurs at an interior point, of the budget plane where all goods are consumed in positive amounts, not at a corner where one or more commodities are not consumed at all (Russell and Wilkinson, 1979, p. 36). Quandt (1970) observed that this implicit assumption is unrealistic in travel-oriented applications-, since consumers do not "undertake a little bit of travel by every mode on every link in a network" (p. 5). The same observation could be made about leisure activities since it is a rare individual indeed who dabbles in each and every possible leisure pursuit across the spectrum of possibilities from sky diving to bird watching. Rather, individuals pick and choose, and engage in some recreation activities at the expense of others; an observation which is repeatedly confirmed by surveys of recreation participation. 10 ------- Thus, the implicit interior solution assumption of conventional utility theory must be relaxed, or .the theory itself reformulated, in order to incorporate the phenomena of non-participation (i.e., zero consumption) in a simulation context. The first alternative is to remain within the confines of traditional utility theory, relaxing the interior solution assumption. The corner solution rationale for zero consumption in leisure pursuits is made by Ziemer et. al., (1982) based on the Kuhn-Tucker conditions (see Silberberg, 1978, Ch. 12). Essentially this means ruling out the class of utility functions where the marginal rates of substitution between pairs of goods are everywhere defined and equatable to the respective goods price ratios. For example, members of the Bergson. family of utility functions which are all transformations of the additive (in logarithms) homothetic utility function U -• Hx are ruled out, since their indifference curves never cut the goods axes,'and corner solutions cannot occur. Another route to explain the same phenomena, rather than restricting attention to utility function formulations which allow for corner solutions and excluding those which don't, is to reformulate neoclassical utility theory along household production lines. Lancaster's consumption theory, which was initially brought to bear on travel demand problems in the 1970 Quandt volume, is such a route. The general form of the Lancaster model sketched in appendix 2. A guarantees zero consumption of some goods, independent of the class of utility function specified. But conventional utility theory can be regarded as a special case of the general Lancaster model. In this instance, corner solutions can be- produced by an appropriate formulation of the utility function. Therefore, the flexibility of the Lancaster model to represent either, the neoclassical case with corner solutions or the "pure" Lancaster case makes it an obvious choice for simulation, since one need not "believe" in either model. But, as explained below, the general form of the Lancaster model introduces complications of its own for the econometric analysis of the outcomes it generates. The econometric analysis becomes more tractable when the Lancaster model is structured to represent the neoclassical utility ------- maximization problem as a special case, with corner solutions allowed by selection of the appropriate type of utility function. RECREATION DECISIONS AND THE (NEW) THEORY OF CONSUMER BEHAVIOR In addition to the conventional theory of consumer behavior presented in most micro texts, where utility is defined in goods space, there are at least two general lines of theoretical attack on the consumer's choice problem in the context of the optimal selection of market goods and leisure activities, including recreation. The first, and perhaps most popular theoretical construct in the recreation literature, is the Becker (1965) household production model (Deyak and Smith, 1978, Desvouges et. al., 1982, for example). The second, perhaps more appealing but less utilized approach is the Lancaster household production model (Rugg, 1973, Mak and Honour, 1980, and Greig, 1983* for example). Both of these theoretical models (reviewed in a general context in Deaton and Muellbauer, 1980 and in the recreation context by Cicchetti and Smith, 1976) are particular variants of the general approach to consumer behavior called household production theory. In this "new" approach to consumer theory, the household does not obtain utility directly from goods purchased in the market. Rather it employs these goods, along with its own time, to produce output of utility yielding, non-market goods over which the utility function is defined. The new theory of the consumer may be decomposed into three basic components: A utility function, a production function, and resource and time constraints. The utility function has as its argument a vector of entities which may be variously defined to be those processes, events or objects from which the individual or household directly derives utility. The production function is the technical relation which depicts the manner in which market good inputs and time are combined to produce the vector of utility generating entities. The resource constraint may be a simple function of market goods, prices and household income or may incorporate additional constraints on household time. Becker's version of the new theory of the consumer assumes neoclassical production functions and smooth convex utility surfaces. Such assumptions concerning the shape and differentiability of the functions permit the use of ------- classical Lagrangian optimizing techniques as opposed to the programming approach of Lancaster. It is interesting to note that the connection between either of these theoretical models and applied econometric work in recreation analysis is often somewhat loose. But, it is also true that the different theoretical models appear to yield roughly equivalent equations to be estimated from survey data explaining recreational trips. Particularly, the inclusion of income, site characteristics, and trip expenses is commonplace (See McConnell and Strand, 1981, Rugg, 1973, and Ziemer et. al., 1982 for superficially comparable "trips" equations derived from different theoretical models). Thus each theoretical model leads, generally speaking, to a roughly similar estimating equation. The principal difference between the two new approaches appears in practice to be that estimating equations are often derived from the Becker model under the highly restrictive assumptions of non-Jointness in household production (Pollak and Wachter, 1975) and a Cobb-Douglas type utility function. Taken together, these two assumptions imply that the only "site prices," or proxies thereto, appearing in the reduced form participation equation to be estimated are the prices of sites supplying the particular service flow of interest (fishing for example) and not the site prices for sites supplying complementary or substitute services. Particularly, the utility function defined on service flows must be of the sort that produces demand functions which are independent of the (shadow) prices of other types of service flows. Similarly, the marginal cost function for a particular service flow derived from the total cost function must be independent of the level of output of any other service flow (Deyak and Smith, 1978). Another feature distinguishing the Becker approach from the Lancaster approach is that the most general form of the Lancaster model posits that each input to the consumption technology produces a set of Joint service flow outputs over which the utility function is defined. For the purpose of constructing a simulation model of recreation choice the Lancaster version is preferable as a manageable way to represent hypothetical consumers and the universe of spatially distinct choices. It can be set up as a programming problem to generate realistic outcomes in the sense that some consumers will choose not to recreate at all and others will ------- choose some subset of available recreation activities- such as fishing and swimming or just fishing alone. . For this simple model the optimal pattern and level of consumer choice of goods (sites indexed by activity category) is a function of income, the nature and particular parameters of the utility function (where in general the latter may be functions of socioeconomic variables like educational attainment, sex, age* race and the like), all goods prices, and goods characteristics. Thus a "loose" econometric specification of equations to be estimated explaining choice of visits or participation intensity in broad activity classes, which are viewed as goods, can be obtained directly, as in Rugg (1973). There is a more elegant route to the same end. However complicated the structure of the model to be estimated, choice theory in general states that the probability of selecting a particular alternative is proportional to the representative utility of that alternative. For a particularly complex model involving a sequential recursive structure for trip destination choice, trip duration choice and trip frequency choice, see Hensher and Johnson (1981, pp. 312-316). The direct utility function in the Lancaster model can be written in terms of goods to produce the individual choice problem: Max U - u (Bx) S.t. y >. px x > 0 The solution to this problem provides the equilibrium values of the x. elements in.the x vector (i - 1,...,n), which are functions of all prices, income and the parameters of the B matrix. If we substitute this system of demand equations g1(p, B, y) for the x.'s in the direct utility function an alternative representation of the preference ordering in terms of the indirect utility function with prices, income, and the B matrix as arguments is obtained. This formulation suggests the arguments in an estimated probability-of-participation choice function: U* - u* (p, B, y). We turn next to some complications and particulars: the meaning of characteristics and the related coefficients in the consumption technology ------- and utility function; the form and properties of the utility function assumed for simulation; and finally, the role of time in our version of the recreational choice model. THE UTILITY AND PRODUCTION FUNCTIONS: WANTS VERSUS CHARACTERISTICS In the Lancaster model, market goods and time are the inputs in the production of joint outputs of characteristics. The characteristics possessed by a good are assumed to be the same for all consumers (Lancaster, 1966a). Market goods themselves yield no direct utility. Instead utility is a function of the characteristics, which are assumed to be measurable on a cardinal scale (Lipsey and Rosenbluth, 1971), and are "in principle intrinsic and objective properties of consumption activities" (Lancaster, 1966b, p. 15). Wants or Characteristics Lancaster's "characteristics" are identified in the literature with Becker's "commodities" (Pollak and Wachter, 1975). Both are outputs of household production. Generally, in reference to the Lancaster-type model the term characteristics is uniformly employed for the outputs (Gorman, 1930, Lancaster, 1966a and b) while when reference is made to Becker's household production model the terms "basic commodities" (Becker, 1965, 1971) "basic goods" (Muellbauer, 1974), "underlying" or "non-market" goods .(Deaton and Muellbauer, 1980) or "service flows'* (Deyak and Smith, 1978) are used interchangeably for the outputs. (An exception is Muth, 1966, who labels inputs into household production "commodities" and outputs "goods"). The essential feature of such an entity is seemingly that,-while it is produced from market goods whose qualities can change, its quality is constant (Muellbauer, 197U). The several authors writing in this area seem to have some difficulty reaching a consensus regarding the practical definition of "commodities" or "characteristics". Indeed, such a practical definition is in part a philosophical question (Edwards, 1955, Ch. 3), and in a way it is irrelevant, since such entities are often immeasurable in the absence of clever definitional legerdemain. A sample of attempts to capture this illusive concept suggest the difficulty: 15 ------- The main difficulty in the traditional theory is the assumption that goods purchased in the market place, food, clothing, theater tickets, heating fuel, medical care, and so forth are the objects of choice that directly enter the preference system. Obviously, this assumption is not literally true; for example, food does not directly give utility, but only contributes to the "production" of meals that do give utility. Preparation time, shopping time, stoves, refrigerators, knowledge of cooking, and many other inputs are also used in producing a meal, and food no more directly produces utility than do these'other inputs. (Becker, 1971, p. 44). A meal (treated as a good) possesses nutritional characteristics, but it also possesses aesthetic characteristics, and different meals will possess these characteristics in different relative proportions. Furthermore a dinner party, a combination of two goods, a meal and a social setting, may posess nutritional aesthetic and perhaps intellectual characteristics different from the combination obtainable from a meal and a social gathering consumed separately. (Lancaster 1966a, p. 133). If we eat an. apple, we are enjoying a bundle of characteristics-flavor, texture, juiciness. Another apple may have the same flavor but associated with a different texture, or be more or less Juicy. (Lancaster, 1966b, p. 15). Suppose that the eggs have certain measurable characteristics A, B, ... and that each egg of type i yields a. units of A, b, units of B, and so on. Suppose moreover that these characteristics are additive and do not interact, so that his total consumption of A is a - la.x. and of 3 is b - £b.x. and so on. We can envisage these as quantities of Vitamin A, Vitamin B. . ; (Gorman, 1980, p. 843). The term "characteristics1' was chosen for its normative neutrality; in my earliest draft of this idea I called them "satisfactions," but that" has too many connotations. (Lancaster, 1966b, p. 14). The consumer desires to have satisfying experiences and can achieve satisfaction by using commodities [in this context, purchased goods]. The great number of wants that he wishes to satisfy arise from physiological and psychological needs... Springing from the idea of differences between commodities and the idea of differences between wants, comes the idea that commodities have different want-satisfying powers. Some commodities satisfy certain wants but will not do at all for the satisfaction of other which, in turn, may easily be satisfied by the consumption of some alternative commodities. Essentially this idea of want-satisfying powers is the notion that commodities have different qualities. These qualities are partly due to the technical nature of the commodities themselves and partly due. to the nature of the wants they serve and the 16 ------- .consumer'3 valuation of their effectiveness in serving these wants. It is impossible to separate these two determinants of qualities since, although the technical characteristics of commodities are objective, each consumer's evaluation of them in relation to his own wants is subjective. Nevertheless these subjective qualities are the links between commodities and wants. To begin with, these qualities will be regarded as fixed and they completely specify the consumer's technology. (Ironmonger, 1972, p. 15). Calories and alcholic content, for food and beverages, and distance travelled and travel time for transportation services, are examples of easily measurable characteristics, while qualities like tastiness in food would have to be translated into a number of measurable, characteristics such as salt content, liquid content, etc. Italics added (Lipsey and Rosenbluth, 1971, p. 133). A reading of the above quotations makes it obvious that no precise definition of what it is that enters the utility function of the consumer is available. Nor can some of these entities be easily observed * some are objectively measurable characteristics such as vitamin and protein content, but others are vaguely defined, immeasurable (or difficult to measure) entities such as good health, flavor, pleasant climate, and beautiful scenery. Lipsey and Rosenbluth (1971) refer specifically to this measurement problem,, while Hensher and Johnson (1981) present a particularly confusing set of distinctions among objective characteristics of commodities - which they call features- * and the attributes of these commodities which are arguments- in the utility function - which they call consumption services. For Hensher and Johnson, consumption services are functions of objective characteristics and there need not be a one-to-one mapping between them. But, there is another way to view all of this, which is presented in a particularly lucid way in Ironmonger (1972) and ties into the Henscher and Johnson view. Want Power- In Ironmonger's lexicon, purchased goods are called commodities. In the conventional theory of the consumer, happiness (utility) is a direct function of the. quantities of commodities consumed. But for Ironmonger, purchased commodities do not directly produce happiness (utility). Rather, happiness (utility) is determined by the degree of satisfaction of separable "wants" 17 ------- which stand between utility, and commodities^ Separate wants (warmth, shelter, entertainment, companionship, variety, distinction, knowledge, etc.) exist independent of goods and services (commodities), but these commodities possess differential want-satisfying powers. Thus the objective characteristics of commodities do not directly enter the utility function, which instead is defined in terms of wants, which any reasonable person would admit are immeasurable. Often all we can measure are the quantities of market goods purchased which, in the recreation.case, are visits to sites, each with its own "generalized cost". This view recasts the Lancaster consumption technology and utility functions in terms of wants which are no less vague than characteristics. The game then involves how the redefined B matrix is structured; that is, which goods supply which want or combination of wants. • Specifically, the b •* J activity coefficients in the consumption technology matrix are no longer regarded as units of measurable objective characteristic i obtained from one unit (a site visit) of purchased good j. Rather, the b.. are redefined as coefficients representing satisfaction of want i obtained from consuming one unit of good J. Ironmonger views the want satisfaction coefficient values as immeasurable or subjective, being partly determined by the objective characteristics of the commodity and partly by the consumers subjective evaluation of those characteristics. This suggests a complex model with random coefficients in the wants consumption technology matrix, an interesting but unnecessary complication for our purposes. If we assume that all consumers translate objective characteristics into wants in the same way (i.e., are members of what Edwards (1955) calls the same taste community) then the want satisfaction coefficients are related to measurable objective characteristics, albeit in an unknown way. Then, if sites are classified into groups which share roughly the same objective characteristics, we can reasonably assume they share the same want satisfaction coefficients. For example, uncrowded coldwater game fishing sites can all be considered to satisfy wants in a similar way, which is different from, in general, the way crowded warmwater rough fishing sites satisfy the same wants. Additionally, one category of activity may satisfy ------- an overlapping intersection of wants with another activity, while both may at the same time supply wants particular to each. Or, finally, subsets of the want space defined over activity categories may be mutually exclusive. Alternative B matrix structures are discussed below, based on these possibilities. The Nature of the Wants Matrix Representing the Consumption Technology— Lancaster originally assumed the B matrix for any individual contained no linear dependencies, so its rank is the number of columns or the number of rows, whichever is less, implying that no characteristic (or want) be redundant. Juxtaposed to this theoretical consideration is the practical econometric dicta that spatial alternatives (destinations satisfying leisure related wants) and their associated choices (fishing, boating, etc.) must be defined to encompass geographic zones within which the elemental alternatives (sites) are homogenous or at least have equal across-zone variances in their utility measure (Ben-Akiva and Watanatada, 1981). As Hensher and Johnson (1981, p. 73) remark: In practice, it becomes necessary to limit the number of alternatives in a choice set, regardless of the knowledge of the full set of relevant alternatives. In other words, it is necessary to group alternatives by design or randomly select a subset of relevant alternatives. Furthermore, data availability is such that often it is not possible to empirically specify a choice set containing elemental alternatives but only a lesser number of alternatives with some implicit relationship between a "group" alternative and a subset of elemental alternatives. In practice this means that the elemental alternatives (sites) must be grouped in some way into sets of assumed identical alternatives. In our simulation model this is done in two ways by construction. Both methods assume we know a priori the groupings of sites which keep within-group variation: in. the want coefficients to be nearly zero but permit between-group variation. Essentially we assume that all fishing sites are alike, all camping sites are alike, all tennis courts are alike but that the groups differ in their want-satisfying abilities. Note that sometimes the same site can supply wants in boating, swimming and fishing at effectively the same travel-cost based price for all three activities to a certain individual. ------- This means that some multipurpose sites appear as repeat entries in the B matrix. Second, several partitions of the wants matrix are possible, apart from the most general case of linearly independent columns, each column pertaining to a site (e.g., each site a unique entity). This possibility is ruled out by the practical necessity of creating sets of assumed identical alternatives. The first partition of the B matrix is the neoclassical case, where all sites in a broadly distinct activity category similarly satisfy a unique set of wants which does not overlap with the wants provided by another group of sites in any other activity category. In this special case utility can just as easily be defined over goods as it can over wants. Particularly, in the recreation case, the consumer will select the closest site from each activity category (all other sites in the activity category being inefficient) and optimize over sites, since goods (sites) map uniquely into wants, and the utility function can be defined over either quite simply. Given the general wants consumption technology z » Bx the neoclassical variant of the general modal can be written in terms of the 3 matrix partitioned as block diagonal; I" z,! [ Z2] where « xt ... x sites unique to satisfaction of want column vector Zi of row dimension p. X- - x ,... x sites unique to satisfaction of want column m+i . m+n vector Z2 of row dimension q. and, for Btl and B22: b1m' 2m ------- '22 b(p-H)(m+1) " b(p+l)(m+2) (p+2)(m+1) '(p-MKm+nj '(p+2)(m+n) * b (p+q)(m+2) (p+q)(m+n) Within any B. . sub-matrix the elements in any row are equal, so the sites (x's) associated with that submatrix are perfect substitutes in provision of the associated wants. For example, suppose p-q-2, m-2'and n-3. Then we have, dropping the column subscript to denote want coefficient equality: bi bt 0 0 0 b2 b2 0 0 0 0 0 b, bj b, 0 0 b,, b,, b,. Performing the matrix multiplication gives: zz - bjX; * baxa z, - b,x, * b,x% * b,x, Take want zl and let a be any number between zero and one. Then, obviously a given level of want zlt say zlt can be achieved either using at fixed level x\ with x2 equal to zero; or with x2 set at the same fixed level xz with xt equal to zero, or any weighted combination: 21 ------- X2) Thus xt and x2 are perfect substitutes in production of wants and are the same good since they, and they alone, supply wants zt and z2 in fixed proportion (b^bj). The same perfect substitution relationship in production exists among goods x,, x* and x, which alone supply wants z, and z* in fixed proportions (b,/b%). All wants but one in each Zl and Z2 sub vector partition of the z column vector are therefore redundant, since each row in the corresponding submatrices of the partitioned B matrix can- be obtained as a linear combination of the elements in, say, the first row of each submatrix BH and B22. The consumer's problem simplifies to choosing one least cost good (site) from each sub-vector Xt and X2 of the X vector partition, since all goods within a partition satisfy wants in the same way but at different costs. Then, since all wants but one are redundant within the corresponding partition of the 3 matrix the reduced B matrix is square with zeros on the off-diagonal and row (and column), dimension equal to the number of X matrix partitions, because the number of sub-vectors in the partitioned X vector define the number of distinct goods (unique activity categories). From the example, then, any x in Xj is the same as any other, since they are all representative of the same homogenous good, as is true of any x in.X2. The problem is then easily written either in terms of wants or goods, since the former are directly proportional to the latter. Normalizing wants on, say zt and z, to eliminate redundancy we can write two equivalent problems based on the example: I. Max U (zlt z,) S.t. II. z, - o b, » «* y - Ri*i * p»xa Max U (blxl., b,x2) S.t. y - ptx, * p2x2 22 ------- The second problem is clearly the neoclassical case, but for the trivial (and unnecessary) conversion of goods quantities into want quantities via ^ and b,. . When the production technology matrix cannot be partitioned as block diagonal, we have the more general Lancaster model. In our simulation model of the more general case we still maintain the assumption that all sites within an activity category have the same want coefficients as- a matter of computational convenience. This assumption simplifies the process of selecting sites on the efficiency frontier and reduces the size of the • programming model to be solved, since.for each individual only the sites closest to his location (i.e., lowest "priced") in each activity category need be considered in setting up the problem. This within-category homogeneity of sites assumption is adopted only to isolate the aggregation issue front other complications which, along with increased realism, would introduce additional computational costs. In passing, however, it should be noted that the conceptual link between wants and site quality attributes is consistent with studies of angler attitudes (Moeller and Engelken, 1972, Sports Fishing Institute, 1974, Spaulding:,-1971, Hendee, 1974, Hampton.and Lackey, 1975) which have confirmed the notion that there are important dimensions of the sport fishing experience other than catching and eating fish. Similar observations could be made about other general categories of recreation activity such as boating, swimming, and camping. Our simulation model is rather skeletal in regard to these complications, one of which, congestion, has received some attention in Oeyak and Smith (1978). The next section discusses a problem of more immediate importance - the nature of the goods (site) demand functions and the indirect utility function produced by either the general or specifically neoclassical alternative formulations of the Lancaster model, which has important implications for econometric estimation. Indirect Utility, Demand Functions for Wants and Goods,, and the Structure of the B Matrix of the Lancaster Model .The form of the econometric model to be estimated from the simulation data .depends on the structure of the Lancaster model itself, particularly the 23 ------- B matrix.• Econometric details are discussed at length in chapters 3 and 6. For now it is enough to know that it is necessary to arrive at expressions for the indirect utility function, wants demand functions, and goods (trip) demand functions from the Lancaster model in order to properly specify the econometric model(s) to be estimated. A manageably small general Lancaster model with two wants and three goods is adequate for illustrative purposes. The problem is the-familiar maximization of utility defined over wants subject to the consumption technology and a budget constraint.. The x's represent trips to sites in various activity categories and the p's the costs of travel: Max U - F (zlvzs.) . . S.t. zl - bii*i + blzxa + bj,x, za - baix! * baaxa * ba,x, y - ptxt * pzxa * p,x, bij 1 °» pj > °» XJ 1 ° Suppose the optimal solution to this problem involves a choice of a (non vertex) combination of xt and x2, and a zero quantity of x,. The shadow prices of the wants zl and. za can be related to the column vector of goods prices (trips to sites in various activity categories) actually consumed, p, through the matrix B made up of the same goods (i.e., with the columns pertaining to goods not consumed deleted by (Klevmarken, 1977): p - B'ir where ir is a r x 1 column vector of shadow prices. From the example: fPO - 1. The rtn shadow price is equal to 3U/3z converted into a monetary equivalent by multiplication by the marginal cost utility, 1/X, (Deaton and Muellbauer, 1980, p. 250). 24 ------- or •Pi " bl1 ffl * ^21*2 Pi " &12 *2 * &22*2 Also, if r S n, the vector IT is equal to: ir - .The inverse of Bf can be found by first calculating its determinant |B'|: ^ |Bf| » btl baa - bla bat Then, construct the cofactor matrix of all the elements of B and take the transpose to get the adjoint of B: and adj B (Bf) -b 1 2 adj B' - b -b 22 21 so or, the shadow prices of the z's are, at the optimum: 1 it* - 1 If the individual's direct utility function is Cobb Douglas, we can write it as: Ui » zraizaaa , a» + a, - 1 The associated Marshallian demand functions for wants are: ------- z^ifi , ira, y) - (o z2(irt , ira, y) - ( |