9512.net

甜梦文库

甜梦文库

当前位置：首页 >> >> # A utility-theory-consistent system-of-demand-equations approach to household travel choice

A Utility-Theory-Consistent System-of-Demand-Equations Approach to Household Travel Choice

by Kara Maria Kockelman B.S. (University of California, Berkeley) 1991 M.S. (University of California, Berkeley) 1996 M.C.P. (University of California, Berkeley) 1996 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering – Civil and Environmental Engineering in the GRADUATE DIVISION of the UNIVERSITY of CALIFORNIA, BERKELEY

Committee in charge: Professor Mark Hansen, Chair Professor Daniel McFadden Professor Martin Wachs Fall 1998

The dissertation of Kara Maria Kockelman is approved:

_______________________________________________________ Chair Date

_______________________________________________________ Date

_______________________________________________________ Date

University of California, Berkeley Fall 1998

A Utility-Theory-Consistent System-of-Demand-Equations Approach to Household Travel Choice

Copyright 1998 by Kara Maria Kockelman

1 Abstract A Utility-Theory-Consistent System-of-Demand-Equations Approach to Household Travel Choice by Kara Maria Kockelman Doctor of Philosophy in Civil and Environmental Engineering University of California, Berkeley Professor Mark Hansen, Chair

Modeling personal travel behavior is complex, particularly when one tries to adhere closely to actual causal mechanisms while predicting human response to changes in the transport environment. There has long been a need for explicitly modeling the underlying determinant of travel – the demand for participation in out-of-home activities; and progress is being made in this area, primarily through discrete-choice models coupled with continuous-duration choices. However, these models tend to be restricted in size and conditional on a wide variety of other choices that could be modeled more endogenously. This dissertation derives a system of demands for activity participation and other travel-related goods that is rigorously linked to theories of utility maximization. Two difficulties inherent in the modeling of travel – the discrete nature of many travel-related demands and the formal recognition of a time budget, not just a financial one – are dealt

2 with explicitly. The dissertation then empirically evaluates several such demand systems, based on flexible specifications of indirect utility. The results provide estimates of activity generation and distribution and of economic parameters such as demand elasticities. Several hypotheses regarding travel behavior are tested, and estimates are made of welfare effects generated by changes in the travel environment. The models presented here can be extended to encompass more disaggregate consumption bundles and stronger linkages between consumption of out-of-home activities and other goods. The flexibility and strong behavioral basis of the approach make it a promising new direction for travel demand modeling.

iii Dedication This work is dedicated to my family and to my best friend, Steven Glenn Rosen. Their constant support has made an extraordinary difference.

iv

TABLE OF CONTENTS

List of Tables List of Symbols Acknowledgments vii ix xiv

CHAPTER ONE: INTRODUCTION CHAPTER TWO: REVIEW OF RELATED LITERATURE

General Models of Trip Generation Systems of Equations Hybrid/Simulation Models Discrete- and Discrete + Continuous-Choice Models Value-of-Time Models Summary of Related Literature

1 6

6 6 7 9 10 13 15

CHAPTER THREE: RESEARCH METHODOLOGY

Microeconomic Foundations Roy’s Identity in a Two-Budget Framework Theory-Implied Constraints Non-Negativity Concavity of the Expenditure Function Homogeneity Summability Separability Symmetry Validity of Utility Maximization Hypothesis Estimating Benefits and Costs Functional Specification Model Specifications Type 1 Model Specification: Modified Linear Expenditure System Type 2 Model Specification: Modified Translog

18

18 22 27 28 30 32 33 33 36 37 38 40 42 43 44

v

Type 3 Model Specification: Modified Translog with Constants 45 Type 4 Model Specification: Modified Translog with Constants, using Wage and Total Time Data 46 Statistical Specification Integer Demand Observations and the Poisson Assumption Generalizing the Poisson Assumption through Use of a Negative Binomial Implication of the Assumption of Multiplicative Error Component for Indirect Utility Data Set Definition of the Consumption Space Trip Chaining 47 47 49 55 57 58 63

CHAPTER FOUR: EMPIRICAL ESTIMATION AND MODEL VALIDATION

Estimation Techniques Likelihood Maximization Acquiring Starting Parameter Values Variance-Covariance Estimation Results to be Estimated Results of Type One Model: Modified Linear Expenditure System Results of Type Two Model: Modified Translog Economic Implications of the Type Two Model Results Results of Type 3 Model: Modified Translog with Constants I. Discretionary Activity Participation Model Comparisons: Case Example II. Modeling Tours Explicitly

71

71 71 71 72 73 73 76 79 81 81 85 88

Results of Type 4 Model: Modified Translog with Constants, Using Wage and Total Time Data 91 Comparison of All Model’s Elasticity Estimates Discussion of Value of Time Estimates A Functional Conflict between Behavioral Indicators Identification of All Income/Wage Terms in Indirect Utility Other Reasons for Incorrect Marginal Utility of Income Estimates Further Qualifications 94 96 97 98 102 103

CHAPTER FIVE: HYPOTHESIS TESTING AND COST-BENEFIT ESTIMATION

Hypothesis Testing using the Type Two Model Hypothesis 1 Hypothesis 2 Hypothesis 3 Hypothesis 4

110

110 111 114 115 117

vi

Hypothesis 5 Cost-Benefit Analysis using the Type Two Model 118 119

CHAPTER SIX: LIMITATIONS AND EXTENSIONS

Data Deficiencies Modeling Expanded Choice Sets Inclusion of Automobile Ownership in the Model Location Choice Decision Modeling Activity-Participation Times Incorporating Different Preference Structures Recognition of Other Constraints on Behavior

127

127 128 129 130 132 132 133

CHAPTER SEVEN: CONCLUSIONS REFERENCES APPENDIX

A-1: List of Possible Functional Form Specifications A-2: Derivation of Roy’s Identity A-3: Description of Data Set Used A-4: Description of Negative Binomial Distribution A-5: Examples of the Estimation Algorithms, as programmed in SPlus3.3 A-6: Example Algorithm for Travel-Time Cost Calculations, as programmed in Matlab A-7: Estimating the Variability in the Results

137 140 149

150 155 157 160 161 165 166

INDEX

165

vii

List of Tables

Table 3-1: Regressions of Travel Time and Distance on Measures of Urban Form Table 3-2: Description of Variables Used Table 4-1a: Parameter Estimates of Modified Linear Expenditure System, as applied to Discretionary-Activity Participation across Four All-Jobs Iso-Opportunity Contours Table 4-1b: Economic Implications of Modified Linear Expenditure System, as applied to Discretionary-Activity Participation across Four All-Jobs Iso-Opportunity Contours Table 4-2a: Parameter Estimates of Modified Translog Model, as applied to Discretionary-Activity Participation across Four All-Jobs Iso-Opportunity Contours Table 4-2b: Economic Implications of Modified Translog Model, as applied to Discretionary Activity Participation across Four All-Jobs Iso-Opportunity Contours Table 4-3a: Parameter Estimates of Modified Translog Model with Intercept Terms, as applied to Discretionary Activity Participation across Four All-Jobs Iso-Opportunity Contours Table 4-3b: Economic Implications of Modified Translog Model with Intercept Terms, as applied to Discretionary Activity Participation across Four All-Jobs Iso-Opportunity Contours Table 4-3c: Comparison of Log-Likelihood Values across Different Models, based on the Modified Translog Model with Intercepts Table 4-4a: Parameter Estimates of Modified Translog Model with Intercept Terms, as applied to Trip Tours to Four All-Jobs Iso-Opportunity Contours Table 4-4b: Economic Implications of Modified Translog Model with Intercept Terms, as applied to Trip Tours to Four All-Jobs Iso-Opportunity Contours Table 4-5a: Parameter Estimates of Modified Translog Model with Intercept Terms and Wage and Total-Time Information, as applied to Discretionary-Activity Participation across Four All-Jobs Iso-Opportunity Contours Table 4-5b: Economic Implications of Modified Translog Model with Intercept Terms and Wage and Total-Time Information, as applied to Discretionary-Activity Participation across Four All-Jobs Iso-Opportunity Contours Table 4-6: Summary of Elasticity Estimates: Median Values across Households Table 4-7: Summary of Value-of-Time Estimates: Quartiles across Households Table 5-1a: Quartiles of the Estimates of the Derivative of Total Travel Time for Participation in Discretionary Activities with respect to Activity Access/Travel Times Table 5-1b: Quartiles of the Standard Error Estimates of the Derivative of Total Travel Time for Participation in Discretionary Activities with respect to Activity Access/Travel Times

viii Table 5-1c: Quartiles of the T-Statistics of the Derivative of Total Travel Time for Participation in Discretionary Activities with respect to Activity Access/Travel Times Table 5-2a: Quartiles of the Estimates of the Derivative of Total Discretionary-Activity Participation with respect to Income Table 5-2b: Quartiles of the T-Statistics of the Derivative of Total Discretionary-Activity Participation with respect to Income Table 5-3a: Quartiles of the Estimates of the Derivative of Total Activity Participation in Discretionary Activities with respect to Activity Access/Travel Times Table 5-3b: Quartiles of the T-Statistics of the Derivative of Total Activity Participation in Discretionary Activities with respect to Activity Access/Travel Times Table 5-4a: Quartiles of the Estimates of the Derivative of Indirect Utility with respect to Discretionary Time Table 5-4b: Quartiles of the T-Statistics of the Derivative of Indirect Utility with respect to Discretionary Time Table 5-5a: Quartiles of the Estimates of the Derivative of Indirect Utility with respect to Income Table 5-5b: Quartiles of the T-Statistics of the Derivative of Indirect Utility with respect to Income Table 5-6a: Quartiles of Indirect Utility Estimates Table 5-6b: Quartiles of Indirect Utility Changes Following 50% Increase in Travel Times Table 5-6c: Quartiles of Equivalent Variation, in Money and Time Units Table 5-6d: Median Travel Times and Low, Median, and High Budget Levels for Sample Table 5-6e: Comparison of Equivalent Variation Changes for Specific Household Types Table 5-6f: Comparison of Time Valuation for Specific Household Types

ix

List of Symbols

The notation for variables used in this dissertation is provided here, in order of appearance: Chapter 2: v Y VMTt PVMT Chapter 3: v A v T v t v Z v PA v Ptrvl v PZ w

Yun

= Vector of number of different activities participated in per period = Vector of time spent participating in different activities per period = Vector of access/travel times to different activities = Vector of number of non-activity goods consumed = Vector of prices for participation in different activities = Vector of travel costs to different activities = Vector of prices for non-activity goods = Wage, in money per unit time = Unearned income per period = Time spent participating in activity i per period = Time spent participating in the work activity per period = Total time budget (Amount of time available for expenditure in different activities and in travel, per period) = Indirect utility = Income or total monetary expenditures = Vehicle miles traveled in period t = Price per mile of vehicle miles traveled in period t

Ti Tw H

Y

= Income available, per period

x v (?)

Ai

*

= Indirect utility function (i.e., maximized utility) = Optimal consumption of activity i per period = Time budget for set of activities considered, per period = Lagrangian function = Optimized Lagrangian function = Direct utility function = Lagrange multiplier on time constraint, in units of utility per unit time = Lagrange multiplier on money-budget constraint, in units of utility per unit money = Vector of optimal consumption of activities per period = Vector of optimal activity participation times per period = Vector of optimal consumption of other goods per period = Optimal work-activity participation time per period

T L Lopt

u

λ Time

λ Money

v A* v T* v Z*

Tw

*

v v e H (t , Yun , w, u) = Minimum time budget needed to achieve utility u at travel times t ,

unearned income

Yun , and wage w

Td

= Discretionary time available = Vector of number of foods consumed = Vector of number of non-food goods consumed = Vector of prices of foods = Vector of prices of non-food goods = Total monetary expenditures of foods = Minimum monetary expenditure to achieve utility u at prices = Money-compensated Hicksian demand for good i per period

v X Food v X Other

v PFood v POther

YFood

v e$ ( P , u ) v hi ,$ ( P , u)

v P

v v v eT (t , P, Y , u) = Minimum time budget needed to achieve utility u at travel times t ,

v prices P , and income Y Ti ,T (?) v AT (?) EV$ EVT v e$ (t , T , u ′)

= Time-compensated demand for participation time in activity i per period = Time-compensated vector of demands for number of activities per period = Equivalent variation, as measured in money units = Equivalent variation, as measured in time units = Minimum money budget needed to achieve utility and available time T

xi

v u ′ at travel times t

v v eT (t , Y , u ′) = Minimum time budget needed to achieve utility u ′ at travel times t

and available money budget Y

v hi ,$ (t , T , u ′) = Money-compensated Hicksian demand for good i per period v hi ,T (t , Y , u ′) = Time-compensated Hicksian demand for good i per period Xi

*

= Optimal consumption of activity i per period

α , α i , β ij , γ iY , γ iT , γ TY , γ iH , ? o , ? i

= Unknown parameters of the indirect utility function, to be estimated

Xi

= Observed integer level of demand of good type i = Vector of observed integer demand levels = Mean optimal rate of demand for good type i per time period = Vector of unobserved variation, characterizing an observation’s vector of optimal demands, relative to the population mean

v X

λi

v ε

XT

= Observed total of integer demands = Mean total optimal rate of demand across good types, per period = Size parameter characterizing the negative binomial distribution = Probability parameter characterizing the negative binomial distribution

λT

m

p*

v p

Γ (?)

xii

= Vector of probabilities characterizing the multinomial distribution = Gamma function (defined in Appendix, section A-4) = Gamma error term characterizing individual observation’s optimal demands, relative to population mean

ε

P E (?) V (?) COV (?)

= Odds-ratio parameter characterizing the negative binomial distribution = Mean or expectation operator = Variance operator = Covariance operator

X Autos = X A = Observed automobile ownership PA PT

= Odds-ratio parameter characterizing the negative binomial distribution for automobile = Odds-ratio parameter characterizing the negative binomial distribution for total activity participation

pA

= Probability parameter characterizing demand for automobiles, relative to total activity plus automobile demand

XA XT

*

= Optimal automobile ownership level = Optimal rate of total activity participation = Vector of zeros for non-chosen locations and a one for chosen location = Vector of levels of housing attributes consumed = Vector of prices of housing locations = Vector of prices of different housing attributes = Vector of travel times observed for a given neighborhood, to different activities = Vector of travel times perceived/“felt” by an individual/household, to different activities = Overdispersion parameter characterizing a negative binomial distribution (equals 1/m) = Mean parameter characterizing a negative binomial distribution (equals mP)

*

v L v H v PL v PH

v t obs ’d

v t felt

α

?

xiii Chapter 4:

$ θ

wn

= Maximum likelihood estimator of unknown parameters = Estimate of gradient of likelihood function for the nth household, using

$ θ $ θ

W X i ,n v ti vTd

*

= Estimate of gradients of likelihood function across all sampled households, using = Optimal rate of consumption of activity i for household n = Marginal utility of travel time to activity i = Marginal utility of discretionary time = Unknown parameters characterizing expanded indirect utility function = Identifiable functions of unknown parameters and prices which characterize the expanded indirect utility function

δ iP , δ iPY , δ Y , δ YY

δ iP ′ , δ iPY ′ , δ PYT

Chapter 5:

E ε (?)

= Expectation operator, over unobserved/error term

ε

xiv

Acknowledgments

The creation of a dissertation can involve many, many people, and this one certainly did. I am indebted to the National Science Foundation and University of California at Berkeley, for their multi-year fellowships, and to the University of California Transportation Center for its generous dissertation grant. These sponsorships afforded me the security to focus on my intellectual interests throughout graduate-school. I also had great fortune in securing Professors Mark Hansen, Daniel McFadden, and Martin Wachs to advise me in this project. All have provided significant insight and substantially facilitated this work. Michael Mauch, Phil Spector, and Simon Cawley were very generous with their time and programming aid. Chuck Purvis and others in the San Francisco Bay Area’s Metropolitan Transportation Commission’s modeling group provided hundreds of megabytes of critical data. Finally, I am grateful to Professors Elizabeth Deakin, Carlos Daganzo, Tom Golob, David Gillen, Steve Goldman, Ken Train, and Browyn Hall – for their advice and expertise on specific topics.

1

Chapter One: Introduction

This research examines a methodology for modeling household travel demand, as tied to out-of-home activity participation. The investigation adheres to the microeconomic theories of rational behavior and utility maximization1 by the household and incorporates constraints on time, in addition to the common constraint on monetary expenditures. The methodology is tested empirically for several model specifications, using data from the San Francisco Bay Area. The results provide estimates of optimal trip generation and distribution (that is, destination choice) by households together with multiple economic variables, including cross-travel time elasticities, values of time, and welfare changes. Little prior travel-behavior research has taken into account a time constraint or explicitly recognized travel demand as driven by demand for activities at physically separate destinations. Much of the research regarding time constraints has been theoretical, with little empirical support (e.g., Becker 1965, Johnson 1966, DeSerpa 1971). A primary reason for the absence of empirical method is the difficulty satisfying utility maximization theory while permitting estimation. Other methods of analysis have resorted to substantial simplification of behavior based on strong assumptions such as bindingness of a single constraint (either the money or the time constraint is binding, but not both) and/or strongly additive preferences (e.g., Zahavi 1979a, Zahavi et al. 1981, Gronau 1970). Accordingly, these methods have lost many relations of interest. Discrete-choice models can accommodate the simultaneous (rather than sequential) nature of a variety of decision types and can be consistent with utility-maximizing behavior (McFadden 1974, Ben-Akiva and Lerman 1985, Train et al. 1987). However,

2 many choice variables are ordered or continuous (for example, the number of dining trips per month, square footage of home parcel). Ordered logit and probit models have been estimated for a single choice and for error-correlated simultaneous choices (e.g., Yen et al. 1998), but not for a set of simultaneous choices where the parameters are constrained across equations or where the outcomes are cardinal (such as the number of person-trips by a household to different activity types over a day or more). If ordered choices were modeled simultaneously as non-ordered choices, the independence of irrelevant alternatives (IIA) property of the logit model would not be tenable; and the probit suffers from intractability for large numbers of (non-ordered) choices. Thus, this dissertation takes a different approach and seeks to illuminate the interactions and trade-offs among demands for out-of-home activities and, therefore, travel. The methodology is sufficiently flexible that other consumption can be incorporated as well. The approach employs models consistent with utility theory so that the basic model structure and resulting predictions yield behaviors that are economically rational under a wide range of circumstances. Moreover, utility theory provides numerous extensions, supplying, for example, estimates of welfare changes, cross-time demand elasticities, and values of time. In this research, systems of demand functions are derived from flexible functional forms of the indirect utility function through parallels to Roy’s Identity. Continuous (though latent) demand levels underlie the system of interdependent equations, and these equations are simultaneously estimated so that cross-equation parameter constraints and correlated error structures are accommodated. The system is estimated as a set of negative binomial regressions, produced from mixing independent Poissons with

3 stochastic gamma terms and thereby providing for unexplained heterogeneity in behavior. These gamma terms are correlated, recognizing the correlation of unobserved information across multiple responses for a single observational unit. The methods developed here are intended to further the state of the art in traveldemand modeling. The behavioral foundations of the investigated models are stronger than those of many existing models, lending greater credibility to the results and predictions. And the incorporation of relevant market “prices” (in the form of travel times) as well as two distinct budget constraints makes the models applicable to a variety of policy scenarios.2 Moreover, the requisite data are commonly available to metropolitan and local planning organizations, so the methods advanced and applied here can be implemented in the short term. Additionally, the resulting models allow for various tests of hypotheses concerning travel- and activity-related consumption, such as the existence of constant travel-time budgets. Application of microeconomic theory using the model’s estimated (scaled) indirect utility functions also permits evaluation of “welfare” changes due to policy changes (e.g., Hausman et al. 1995, Burt and Brewer 1971, Cicchetti et al. 1976). For example, through inversion of the indirect utility function with respect to either one of the constraint levels, measures of a project’s social “cost” or “benefit” can be estimated in units of time and money by using differences in the constraints’ respective expenditure functions across households. This model’s recognition of simultaneity in decision-making, time constraints on choice, and the discrete nature of travel data, along with its rigorous microeconomic

4 foundation, offer significant advantages in travel modeling. The ensuing chapters detail the model’s specification and illustrate its application.

5 ENDNOTES:

1

The assumption of utility-maximizing behavior, implicit in many models and their constraints (e.g., crossequation constraints on parameters), can often be tested using empirical results. For examples of such tests, see Christensen et al., 1975, and Deaton and Muellbauer, 1980a. For example, if sales and service opportunities were to re-locate, the travel-time environment would change. These changes are incorporated directly in the proposed model, permitting immediate estimation of a household’s response, via substitution and time-constraint effects.

2

6

Chapter Two: Review of Related Literature

General

Over the years, travel behavior has been modeled in a number of ways. Many of the earliest models were developed primarily for prediction; their virtue is that they are easy to apply. Later models are theoretically sounder, based on hypotheses concerning human behavior and focusing on causation. Some of the most plausible travel models acknowledge simultaneity in decision-making by avoiding strictly sequential estimation, hypothesize distinct behavioral mechanisms, and/or suggest new ways of adhering to microeconomic theory. However, shortcomings in existing models persist, and this research seeks to overcome the deficiencies. The purpose of this chapter is to summarize the strengths and weaknesses of existing models.

Models of Trip Generation

In the standard Urban Transportation Planning Model (UTPM), the first step is trip generation – estimation of the number of trips made for different purposes by households. The sequential, rather than simultaneous, estimation of such models and their lack of transportation-supply variables have long been recognized as inherent weaknesses in this mainstay of planning practice (e.g., Dickey 1978, Gur 1971), yet these practices continue in the present day (e.g., MTC 1996, Purvis et al. 1996, ITE Journal 1994). In their comprehensive book Modelling Transport, Ortúzar and Willumsen (1994) point out that while one’s access to opportunities affects trip generation and “offers a way to make trip generation elastic (i.e., responsive) to changes in the transport system”, it “has rarely been used....” (1994, p. 117) For example, in a two-stage “recursive” model of trip and tripchain generation, Goulias and Kitamura’s (1991) explanatory variables are almost

7 exclusively demographic in nature; for non-demographic data, they use a rural-versuslarge city dummy variable and segment their trip-chain model by three city sizes.1 Few recent methodologies consider total trip demand before addressing other aspects of behavior, such as trip chaining, distribution, timing, and duration. The absence of interest may be due to the apparent inelasticity of total demand with respect to access costs. Following an extensive review of past literature on trip frequency as a function of several rather simple measures of location type and accessibility (such as local-area densities and distance to central business districts) and a correlation-based analysis of their own, Hanson and Schwab (1987) conclude that “accessibility level has a greater impact on mode use and travel distance than it does on discretionary trip frequency” – an “unexpected” result given “the strong trip frequency-accessibility relationship posited frequently in the literature” (1987, p. 735). And Ortúzar and Willumsen observe that the incorporation of typical measures of access “has not produced the expected results, at least in the case of aggregate modeling applications, because the estimated parameters of the accessibility variable have either been non-significant or with the wrong sign.” (1994, p. 147) These results may be questioned, however, since the models and measures used to examine this relationship generally are unrefined. In order to estimate the elasticity of travel demand with respect to access, more sophisticated, behaviorally based models should be used.

Systems of Equations

A set of model equations is estimated as a system when a correlated error structure is hypothesized, there exist endogenous explanatory variables, and/or cross-equation parameter constraints exist. Researchers have applied the technique of structural equation

8 modeling to predict multiple travel choices in a manner similar to the modeling methodology developed here, but without cross-equation parameter constraints or a strict behavioral basis. For example, Golob and McNally (1997), Golob and Meurs (1987), Golob and van Wissen (1989 and 1990), and Lu and Pas (1997) regress variables such as vehicle-miles traveled (VMT), time spent per day in different activities, mode share, and auto-ownership on exogenous socioeconomic variables as well as on several endogenous variables. Much of the software used by these researchers allows for latent-variable techniques, such as the Tobit and ordered probit. However, the foundation for such systems in a utility-maximizing framework is missing. In a recent paper, Kitamura writes that existing structural equations models “offer no explicit treatment of the decision mechanisms underlying activity engagements.” (Kitamura 1996) One finds that “prices” are absent from these models, and measures of benefit cannot be constructed from their results. Outside of transportation, there are many simultaneous-equations models of demand for goods and services. Optimal shares of monetary expenditures are typically estimated after applying rigorous microeconomic theory (e.g., symmetry in compensated substitution, homogeneity in prices and income, summability and concavity of expenditures); however, time constraints are not considered. Abundant experience with these models has resulted in an understanding of the limitations of different functional forms and the need for specific cross-equation parameter restrictions for conformance with neoclassical economic theory (such as demands’ zero-degree homogeneity in prices and income). For detailed examples, see Lau 1986, Deaton 1987, Deaton and Muellbauer 1980b, Stone 1954, and/or Pollack and Wales 1978 and 1980.

9

Hybrid/Simulation Models

Recent, so-called “hybrid” models hypothesize traveler decision mechanisms which require less information than utility maximization yet satisfy spatial and temporal constraints. For example, Recker’s (1995) Household Activity Pattern Problem (HAPP) algorithm minimizes a generalized time cost function (which he calls “disutility”) subject to linear coupling, connectivity, temporal, and budget constraints. However, his method takes demand for participation in activities (as well as their duration and location) as given and neglects actual behavior for calibration of the model or its objective function. While the model is detailed and able to accommodate a variety of constraints, it avoids consideration of the basis for travel demand and is effectively a scheduling problem. STARCHILD (Recker, et al. 1986a, 1986b) and SMASH (Ettema et al. 1993, 1995a) are similar to Recker’s HAPP model in that an activity program is provided exogenously, decision rules to choose among alternatives are relatively simplistic, and the models determine scheduling. Another model, AMOS (RDC 1995), can be classified similarly, but it requires more inputs and is tailored for response prediction in a limited policy setting. In comparing these models to econometric models, Bowman and Ben-Akiva (1996) observe that with the “hybrid” models the sample of considered alternatives is often inadequate, the response or decision process is probably too simplistic, and many significant, related decisions must be determined exogenously (e.g., activity type, location, and travel mode).

Discrete- and Discrete + Continuous-Choice Models

Following McFadden’s seminal linkage of the logit model specification to microeconomic theory (1974), many discrete-choice models have been developed for the

10 purpose of travel demand modeling. The strict application of these models requires a complete specification of the feasible choice set, restricting the simultaneous and flexible estimation of total demand. Notwithstanding this limitation, many of these models remain microeconomically rigorous by assuming and applying the principles of utility maximization, though some of the strongest applications are not in the area of transportation. For example, Cameron (1982 & 1985) tests flexible2 indirect-utility specifications in nested logit models for her analysis of home-weatherization choices. While rigorous, the size of her problem is limited; she evaluates two choices – the installation of energy-conserving appliances and, when applicable, the appliance package chosen. In an early study of travel behavior, Adler (1976) relies exclusively on a multinomial logit across “all” possible non-work trip patterns for households, but the independence of irrelevant alternatives (IIA) assumption implicit in the logit formulation is unlikely to hold in his model. Fortunately, the nested logit structure has provided a useful way to avoid imposing the IIA property. Domencich and McFadden (1975) detail a four-level nested logit specification by modeling shop-trip mode split, time-of-day choice (peak vs. off-peak), destination choice, and “frequency.” Still, their model’s permitted shoppingtrip frequency allows just one or no shop trips per household per day, which may be too limiting for many applications. Incorporating the choice of trip purpose, but assuming fixed total demand, Kitamura and Kermanshah (1984) sequentially estimate a nested logit for trip-purpose and tripdestination choices. In the destination-choice model, a negative and statistically significant coefficient on the time-of-day-times-distance variable, after controlling for

11 distance by itself, causes them to conclude that longer trips are less likely toward the end of a day, as time constraints become more binding. Their recognition of a possible timebudget effect is important; however, their assumption of the time-of-day variable’s exogeneity is questionable, and the time constraint is accommodated obliquely. Damm and Lerman (1981) recognize travel as a derived demand and combine discrete-choice models of activity participation with the continuous choice of activity duration. This model offers the advantage of providing information on the time-of-day for an individual’s travel and a system of simultaneous equations for estimation of the five periods’ activity durations. However, while the authors discuss the indirect incorporation of a discretionary-time constraint via an individual’s socio-economic characteristics, this constraint is not made explicit. Moreover, the research considers only the choices of workers on a workday given five distinct periods3 during which to choose participation in a non-work activity, and the authors specify linear utility functions with additive separability across each of the five choices. Kitamura’s work in this area (1984) is similar to that of Damm and Lerman (1981), except in the functional form of the time-allocation equation and in the discussion of model set-up. Kitamura’s models are more fundamentally linked to economic theory and avoid selectivity bias in parameter estimates (by weighting responses in the duration model according to observations’ likelihoods in the discrete-choice model). Nevertheless, due to the substantial complexities of the model, Kitamura relies on very specific functional forms for indirect utility and error structure in order to readily derive activity-participation-time demands. He also considers only two classes of time use: mandatory and discretionary.

12 In the context of auto ownership and use, Mannering and Winston (1985) also combine a discrete with a continuous choice. Making use of Dubin and McFadden’s (1984) appliance-purchase-and-consumption model specification, they specify a linear functional form for demand of a single good, vehicle miles traveled in period “t” (VMTt), and, using Roy’s Identity ( dv dY × VMTt + dv dPVMTt = 0 [where variables here and throughout the paper are as defined in the List of Symbols, immediately following the Table of Contents], Roy 1943), determine the implied functional form for indirect utility (v). They then use this indirect utility in a nested logit model for the number of cars owned – and the type or “class” of vehicle, given the number owned. After estimating the logit – and thus an indirect utility function, the estimated levels of VMT are easily obtained. This modeling method provides another example of a semi-simultaneous mixed discrete-with-continuous model of travel, and it incorporates some basic economic theory for a behavioral basis. Unfortunately, for a case of multiple goods, working “backwards” to derive indirect utility can be very difficult unless one begins with highly constrained demand equations; the connection is more clear if one moves from a functional form for indirect utility to a form for demands. Moreover, Mannering and Winston’s necessarily specific choice of functional form for VMT demand leads to a rather limiting indirect utility function, one that is not, for example, homogeneous of degree zero in income and prices (which is a theoretically required condition discussed in Chapter Three). And, unlike this research, their focus is not on activity participation, the influence of time constraints, or the accommodation of multiple, integer demands. Harvey and Deakin’s STEP analysis package (1996) does not simultaneously combine discrete and continuous choices, but it does apply discrete-choice estimation to a

13 wide array of travel-related decisions by individuals, including location choice and time of travel. Notably, STEP incorporates an entire region’s travel “prices” (i.e., interzonal travel times – peak and off-peak, and intrazonal parking prices) into its models of trip distribution. However, STEP is not fully simultaneous and pays little attention to the implications of microeconomic theory for model form.

Value-of-Time Models

For a long time microeconomics and utility theory focused on the money budget and monetary expenditures. In the 1960’s and 1970’s time valuation, the labor-leisure tradeoff, and activity participation choices began to be studied in a variety of ways, using microeconomic principles. Becker (1965), Johnson (1966), DeSerpa (1971), Oort (1969), and Bruzelius (1979) provide theoretical derivations of time’s valuation across different activities. However, their hypothesized models typically treat travel as a single activity and/or emphasize the time spent participating in (rather than accessing) the other activities. Moreover, their focus is on the theoretical value of time, rather than a working system of demand equations for participation in out-of-home activities. Becker (1965) argues that time use is a highly relevant aspect of household decisionmaking and that “total-income losses” due to non-income-producing uses of time are very significant. Thus, he advocates the incorporation of time in economic models of the household. He suggests that a household’s “full income is substantially above money income” (1965, p. 517) and acknowledges people’s pursuit of “productive consumption,” such as eating and sleeping (activities which Golob and McNally [1997] and others have termed “maintenance”). Becker also comments on the intra-household allocation of consumption and production activities, arguing that members offering relative

14 efficiencies in different areas (e.g., high-wage earners) will contribute relatively more time in those pursuits. More recently, Jara-Díaz (1994) extends time-valuation models into a setting which relies on travel times and can illustrate modal trade-offs. However, his results remain similar to those just mentioned: largely theoretical, based on directutility functions, and rarely tested empirically – except when employing random-utility discrete-choice models. Train and McFadden (1978) look specifically at the labor-leisure trade-off using a discrete mode-choice model. Their work demonstrates how wages might reasonably enter the conditional-utility specification, as well as how workers optimize their time use. But the model only considers the choice of workers and employs a restrictive, two-good, Cobb-Douglas direct-utility specification. Golob, Beckman, and Zahavi (1981) acknowledge the imposition of both time and income constraints in a setting that uses microeconomic theory, but they either consider only one at a time to be binding or assume travel expenditures are negligible relative to time and/or money budgets. Such assumptions may rarely hold: one can reasonably expect that both constraints are binding, as long as people value time and do not experience satiation in consumption. For example, in order to maximize utility, a person can sell his/her time to increase income (while reducing discretionary time available), spend more time in enjoyable activities (e.g., leisure) and/or buy time-saving goods (such as prepared meals). This assumption of the bindingness of constraints is testable in the proposed research4. Additionally, Golob, Beckman, and Zahavi neglect activity participation as the underlying basis for travel demands, rely on additive utility functions,

15 and model total distance traveled, rather than distinguishing trip types or estimating the number of trips made.

Summary of Related Literature

There is a well-documented interest in the modeling of travel-related behaviors. Moreover, substantial progress has been made in the topics of time constraints, simultaneity of travel-related choices, the modeling of both continuous and discrete behaviors, and the implications of microeconomic theory. Still, deficiencies exist. Most prominently, the existing literature does not consider integer consumption of multiple goods based on a continuous and cardinal latent response in a microeconomically rigorous framework; behaviorally-based time-use research remains largely theoretical; models of simultaneous choices which are consistent with utility maximization tend to be of discrete choices; and supply-side variables have been lacking in models of trip demand. In contrast, the present research prominently incorporates supply-side variables (in the form of travel times to iso-opportunity contours) while allowing simultaneous estimation of trip generation and trip distribution, based on continuous, underlying demands derived from rigorously applied microeconomic theory5. This research provides estimates of numerous behavioral descriptors, such as demand elasticities; and it allows for a variety of extensions, such as estimation of access times’ effects on a household’s total travel time and on its welfare. The methods and model specifications used are considerably different from those found in previous work, and they are described in the following chapters.

16 ENDNOTES:

1

One, somewhat typical exception is Safwat and Magnanti’s (1988) Simultaneous Transportation Equilibrium Model (STEM), where total trip generation is estimated as a function of a log-sum accessibility measure (derived via the calibration of their trip-distribution logit model). Cameron (1982) investigates household preferences using the rather flexible translog and Leontief functional forms to describe indirect utility. Note that these functional forms are summarized in the Appendix, section A-1, and are discussed in Chapter Three. The periods for non-work activity participation that Damm and Lerman model are: prior to the home-towork trip, during the home-to-work trip, during work, during the work-to-home trip, and following the work-to-home trip. The bindingness of constraints is tested in Chapter Five by calculating the T-statistics for the derivatives of the estimated indirect utility function with respect to the constraint levels for each household; these derivatives theoretically represent the shadow prices of these constraints, which come out of the utility maximization. In his ground-breaking time-valuation work, DeSerpa suggests that “(d)espite (its) difficulties” a systemof-demands approach to the problem, where travel times effectively represent the minimum amount of time required for participation in/consumption of an activity, “has considerable merit” because “‘noneconomic’ factors, such as comfort and convenience are ... implicitly considered”, aggregation of demands “does not depend on any arbitrary assumptions about the individuals comprising the group”, and, “most importantly, the measure (of time’s value) is compatible with the hypothesis of utility maximisation. No other (time-value) measure can make that claim.” (1971, pg. 842) It appears that DeSerpa would strongly support an approach fundamentally very similar to the one proposed here.

2

3

4

5

17

Chapter Three: Research Methodology

Microeconomic Foundations

In this research household activity and other, related consumption trade-offs are posited to adhere to microeconomic theories of utility maximization. Estimates of travelrelated behaviors, such as trip-making rates and trip distribution, are derived from empirical analyses of statistical models based on this theory. In a rather general formulation of the utility-maximization problem, a household may be assumed to derive its welfare (i.e., utility) from consumption of/participation in a vector of distinct, out-ofv home activities A (which are location specific, include the household’s work activities,

and are indexed by i), the time spent participating in each of these activities Ti (and, in particular for the work activity, Tw ), the total time spent accessing all of these activities

vv v t A (where t is the vector of fixed travel times to access the activities1), and

v consumption of all other goods Z . It is helpful to think of the consumption/decision variables in this problem as rates; for example, one activity might be the number of shopping trips in the local neighborhood per day. Under the general model, households are subject to unearned income ( Yun ) and available-time (H) constraints which are also rates (e.g., dollars per day, hours per day), and these constraints lead to trade-offs between consumption of the different goods. In equation form, the problem can be written as the following:

v v vv v Max Utility ( A, T , t A, Z ) v v A ,T , Z v v v v v v v v v vv s. t . PA A + Ptrvl A + PZ Z ≤ Yun + wTw , ∑ Ti +t A= H , and A, T & Z ≥ 0.

i

(3-1)

18 Note that time spent for activity participation is of two types: travel to non-home sites ( t i ) and during participation itself ( Ti ); both of these enter explicitly in the direct utility function, though only the participation time, Ti , is an endogenous variable. The work activity contributes to the income budget level via the wage earned, w; but participation in most other activities is likely to cost money (with Ptrvl ,i + PAi representing the monetary price per unit of participation in activity i, due to travel costs and direct participation costs). There is an equality in the time constraint since all time not spent in accessing and participating in activities outside of the home counts as time spent in athome activities2. The general model is subject to various modifications. For example, if one wishes to focus on discretionary activity choices and assume work and income exogeneity in such decisions, one would not explicitly model work as an activity and would substitute total income, Y, for unearned income, Yun , and discretionary time, Td (total time minus, for example, work and school time), for total time, H. Also, there are many other constraint possibilities; for example, minimum participation-time constraints may exist for certain activities (such as working, dining out, or seeing a movie in a theater) and only fixed levels of consumption may be permitted (such as working or going to school five times per week).

19 FIGURE 3-1 here *******

20 Figure 3-1 is an illustration of what the utility maximization looks like in a simplified, two-activity case; in this illustration a single, per-unit price and timeexpenditure exist for each of the two activities and income and time budgets are exogenous/given so that t 1 A1 + t 2 A2 = H & P1 A1 + P2 A2 = Y . In a more-realistic, Ngood case, the intersection of the two budget constraints is an N-2 dimensional hyperplane; so the optimal choice “bundle” of activities will not appear as a single point of intersection, as it does in the illustrated case of Figure 3-1. Furthermore, choice of activity participation times (over a given period), rather than just optimal rates, expands the decision space substantially, yielding a hyperplane of dimension 2N-2. In practice, a closed-form/analytic solution to constrained maximization of direct utility functions of more than a couple goods is rare, because solution of the Lagrangian equation’s set of first-order conditions is often intractable. In order to derive a system of (optimal) demand equations, it has been found significantly more convenient to work with the indirect utility function, as defined in Equation 3-2 (with arguments defined as for Equation 3-1 and in the List of Symbols, which follows the Table of Contents).

Indirect Utility ={ Max Utility Budget & Time Constraints} v v v v = v ( PA Ptrvl , PZ ,t , Yun , w, H )

Three:-2)

(Chapter

By beginning from a specification of indirect utility, one can then rely on a relation called Roy’s Identity (Roy 1943) to provide individual demand equations. The derivation of the entire system from a single indirect utility specification imposes many crossequation parameter constraints automatically (because many parameters are likely to show up in two or more of the demand equations). However, there are a variety of other

21 constraints implied by long-held microeconomic theories for the typical, money-based applications of these methods, and these constraints tend to be more subtle; they are discussed shortly, in a section titled Theory-Implied Constraints. Roy’s Identity in a Two-Budget Framework Roy’s Identity is the method for deriving demand functions, whose dependent variables (consumption) can be observed, from indirect utility, which is unobservable and ordinal – rather than cardinal – in nature. Fortunately, Roy’s Identity continues to hold in a two-budget framework, although more restrictively than in the typical, single-budget framework. Given a functional specification for indirect utility, v, as well as exogenously determined available time (T) and income (Y) constraints, the relations one can use to identify optimal demand, Ai , are shown in Equation 3-3. Details of this equation’s derivation are provided in section A-2 of the Appendix. dv dv d Ptrvl ,i + PAi dt Roy ’s Identity: Ai * = ? i = ? , ?i , dv dv dY dT * where Ai = Optimal , long ? run rate of consumption per period , v = Indirect utility , t i = Travel time to Activity i , T = Time available per period , Ptrvl ,i + PAi = Unit Price to participate in Activity i

*

(

)

(Chapter

(due to travel & participation costs),& Y = Income available per period . Three:-3) When income and time budget levels are exogenous and observed, the derivation of optimal demand levels is reasonably straightforward. However, income and discretionary time are likely to be endogenous to the decisions to participate in non-work/discretionary activities; in other words, households probably make choices of how much time to spend

22 working – earning income while giving up discretionary time – when determining the amounts of other activities they might engage in. In such a situation, the identities allowing one to identify demands do not look so similar to the common form of Roy’s Identity, and the estimations of value of time and compensated demand are complicated. Imagine a situation where total time available to a household’s members (e.g., 24 hours each day a member is surveyed), marginal hourly wage of the household, unearned income, travel times, and activity-participation prices are observed. The Lagrangian equation and several of its first-order conditions for utility maximization would look like the following:

v v v v v vv v v v? ? L ( A, T , Z , λ Time , λ Money ) = U ( A, T , t A, Z ) + λ Time ? H ? ∑ Tk ? t A? ... ? ? k v v v v v v + λ Money (Yun + wTw ? PA A ? Ptrvl A ? Pz Z ) v v v v Lopt = v(PA , Ptrvl , PZ , t , Yun , H , w) = v* v v v v v* v v v v ? ? A P , P , P , t , Y , H , w , T A trvl Z un A PA , P trvl , PZ , t , Yun , H , w ,? ? L? v* v v v v ?, ... Z P , P , P , t , Y , H , w ? ? un A trvl Z ? ?

[

[

] [

]

]

dLopt dv dL dL = = + dt i dt i dt i dAi dLopt dv dL dL = = + dH dH dH dAi dLopt dv dL dL = = + dw dw dw dAi

Z =Z

v v * v v* = A ,T = T , A v v*

× × ×

dAi = ? λ Time Ai * + 0, dt i dAi * = λ Time + 0, dH dAi * * = λ Money Tw + 0, and dw × dAi = λ Money + 0. dYun

*

*

Z=Z

v v * v v* = A ,T = T , A v v*

Z =Z

v v * v v* = A ,T = T , A v v*

(Chapter

dLopt dv dL dL = = + dYun dYun dYun dAi

Three:-4)

Z=Z

v v * v v* A = A ,T = T , v v*

23 The endogeneity of discretionary time leads to a form of Roy’s Identity which differs from that shown in Equation 3-3. Following some simple manipulation of the first-order conditions found in Equation 3-4, one has the following form:

Roy ’s Identity with Discretionary ? Time Endogeneity: dv dv Ai * = ? d Ptrvl ,i + PAi dt i =? , ?i , dv dv dH dYun

(

)

where Ai * = Optimal , long ? run rate of consumption per period , v = Indirect utility , t i = Travel time to Activity i , H = Total time available per period , Ptrvl ,i + PAi = Unit Price to participate in Activity i (due to travel & participation costs), & Y un = Unearned Income available per period . Three:-5)

The above identity is not the only one that can be derived from this model

(Chapter

specification. Incorporation of the wage variable, w, allows one to identify optimal work time, Tw , as the ratio of the derivative of indirect utility with respect to wage and with

v respect to total time available. And the vector of other goods ( Z ) remains identifiable (as

*

it is under a situation of exogenous income and discretionary time); demands for these goods equal the negative ratio of the derivative of indirect utility with respect to their prices and with respect to unearned income. Under a situation of endogenously determined budgets, the value-of-time computations change; if unearned income and total time available are observed but discretionary time is endogenous, one can use the following:

24

v dv de H (t , Yun , w, u) λ Time dH Value of Time = = = dYun λ Money dv dYun Three:-6)

(Chapter

However, if unearned income is not observed in the data set (but total time available and wage are, and discretionary time is endogenous), one will need to rely on the following equation:

dv dv λ Time dH dH Value of Time = = ≈ λ Money ? dv ? ? dv ? ? dw ? ? dw ? ? Tw * ? ? Tw ? ? ? ? ? Three:-7)

(Chapter

Note that in this equation one may care to use the observed amount of time worked ( Tw ) to approximate value of time rather than the optimal level of working hours ( Tw ), because unearned-income information may not be available and/or may be measured with significant error. Since unearned-income information is not available in the data set used here for empirical analyses, the approximation in Equation 3-7 is used in those models of Chapter Five that endogenize time expenditures. Assuming that households are able to optimize their time expenditures and activity participation3, how will models which assume exogenous total expenditures/income and discretionary time compare in their value-of-time computations with those which incorporate these variables endogenously? One way to look at the difference is to manipulate the ratio of derivatives in the income-and-discretionary-time endogenous case; for example:

*

25

dH dT ? ? d dY dv dv ? ? dH dYun dTd ? ? dH = × = dTd dv dv dv dYun dYun dY dH (dY dYun )

(Chapter

dv

? ?dT ?? ?1 + w? w dY ? ? un ? ? ? = Value of Time × ? ?1 ? dTw ? ? dH ? ? ? Three:-8)

So, if the second term in the last part of the above equation is greater than one, one will over-estimate the value of time. It seems reasonable that, as total time available (H) increases, a household’s members will work somewhat more, but not all of the newly available time. Thus, the denominator of the second term is likely to be less than one but not necessarily very close to zero (especially if work restrictions – such as a forty-hour week maximum paid week – are imposed). And, as unearned income increases, one may expect work time to decrease, perhaps so much that wage multiplied by work time exactly cancels unearned income, making the top part of the equation close to zero and causing one’s value of time estimate (with the assumption of work-time and income exogeneity) to be much lower than the actual. If work time is exogenously determined for households, then work time is unresponsive to changes in total time available to a household, H, and unearned income, Yun, and one will be estimating the true value of time, without inflation or deflation. Unfortunately, without observing the variable of unearned income across the sample, it is difficult to analyze how work time depends on total time and unearned income. However, one can crudely estimate work time’s response to changes in total time

26 available by modeling observed work time for this sample as a function of wage, travel times, total time available, and a coarse estimate of unearned income; a simple ordinary least squares model across households with one or more workers produces a derivative value of just 0.0808 hours of work time per hour of total time available to the household (with a T-statistic of 29.5). 4 The estimate of unearned income on which this crude model relies is a value equal to the household’s income if the household has no surveyed workers and zero otherwise. Running this same model specification for all sampled households produces a coefficient estimate of just -0.145 hours per $1,000 of unearned income (with a T-statistic of -27.3). These results suggest that the derivatives of work time with respect to both income and wage are small; in fact the ratio of the derivatives of indirect utility with respect to discretionary time and total income available to the household (as in Equation 3-8) are estimated this way to be about nine percent higher than the true value of time, on average5. If this is a good estimate of the bias in this measure, it makes sense to deflate the value-of-time results for models which taken income and discretionary time to be exogenous by five to fifteen percent.

Theory-Implied Constraints

The models estimated here are not as general as the formulation presented in Equation 3-2, due to a lack of data on monetary prices and an inability to microeconomically identify non-work time expenditures in activities; but they are described by a system of equations which determines the optimal number of out-of-home activities accessed per day by household members. In order for a system of demand equations to be consistent with microeconomic theory and common sense, the equations must generally be compatible with several types of constraints; not only do such

27 restrictions impose consistency with theory, they can be helpful in reducing the dimensionality of the problem (i.e., the size of the parameter space)6. Non-negativity of optimal demands is a feasibility limitation, and concavity of total monetary expenditures in prices is a requirement when prices are exogenous and constant; these conditions are generally checked following model estimation. In contrast, zero-degree homogeneity of demands (in prices and expenditures/income) is typically imposed a priori and automatically in the functional specification, and summability of expenditures (to equal total budget) and symmetry (of compensated cross-price effects) are often imposed through parameter constraints. If the conditions of summability and symmetry are not needed for parameter identifiability, their viability can generally be tested using differences in the constrained and unconstrained likelihood values. A final constraint on many estimated models is the implicit assumption of separability of preferences from other, non-considered goods. These various constraint types and their usefulness in the models investigated here are discussed below. Non-Negativity Generally, people cannot consume negative amounts of a good, unless, for example, they own some and sell or give it to others. In the context of this research, one can argue that people sometimes pay others to participate in out-of-home activities for them (such as food shopping). However, the available data do not provide information on such transactions so all observations are non-negative and this condition is imposed on the estimates. The method of ensuring this condition via the estimation process used here is an assignment of a very low likelihood value every time the iterative maximumlikelihood search mechanism tries parameter sets which produce negative demand

28 estimates for every demand type of at least one household. If some, but not all, demand types are estimated to be negative for a given household, the parameter set is permitted but optimal demands which are initially predicted to be negative are set to a positive level very close to zero. The optimal demand rates are not set to exactly zero since it is expected that, for the demand types specified, every household will eventually have to consume at least one such good. For example, a demand set of four iso-opportunity contours for all types of discretionary trips represents a partitioning of destinations for a type of trip virtually all households eventually make. However, if trip purposes were partitioned quite narrowly, segregating purposes like “education,” “work,” and “childcare”, one would need to incorporate zero-level demands since households without students, workers, and/or children would not reasonably be expected to make such trips7. Before concluding this discussion of non-negativity, one should recognize that the rather ad hoc choice of a close-to-zero level of demand to assign to households with a predicted-to-be-negative optimal demand level is not theoretically satisfactory, particularly when the other demand levels are left as initially predicted. In reality, such households find themselves at a corner solution, where Roy’s Identity no longer applies to all demand types at once; instead, theory suggests that an optimization over limited choice sets is undertaken and the maximized utilities of distinct scenarios are compared. This added complexity can be accommodated in the models presented here, though it has not been in the estimated models provided in Chapter Four. In fact, Chapter Four’s predicted demands are well above the close-to-zero value for all demand types in almost all the models estimated.

29 Concavity of the Expenditure Function Price extremes are preferable to balanced prices; this characteristic is manifest in quasiconvexity of the indirect utility function and concavity of its inverse8, the expenditure function. While this characteristic is not immediately intuitive, it is theoretically expected. It is expected because at “average” prices, one can buy no more than one could buy across the combined feasible space of the two price extremes which produced the average, subject to a single budget level; so one cannot be better off at balanced prices that at a combination of the two extremes. Thus, the indirect utility resulting from a weighted average of price vectors can be no higher than that achieved from a weighted average of indirect utilities resulting from the two extreme price sets. Moreover, if one or more prices increase, one is at least as well off if one’s budget increases in an amount equal to the price change (a vector) times the vector of previously optimal quantities; this amount of added income will allow one to consume the old bundle of goods and thereby be just as well-off. But, due to substitution effects, one will likely shift away from consumption of the relatively more expensive goods and be able to be just as well-off, so the amount of expenditures needed to achieve a given level of utility is less and thus concave in prices. These relations translate to the matrix of second derivatives of the money-expenditure function in prices being negative semi-definite. (For further discussion of these conditions, see, e.g., Varian 1992.) How do these conditions apply in the present model, where time characterizes costs? If one were to consider all time use, one would expect humans to require more time in a day in order to be just as well-off if travel times increase. However, the amount of additional time required is not necessarily less than the quantity of activities consumed

30 times their change in travel times. Humans directly experience time use, including travel time, so time expenditures are arguments in the direct utility function. This aspect of time use also arises in the following discussion, on homogeneity, and affects the application of many microeconomic theories in a time-expenditure setting. In reality, more time spent accessing opportunities/activities may require more than a full compensation of total time to keep welfare constant; the direct impact on one’s welfare may be sufficiently negative9. Thus, concavity of time expenditures in travel times is not a required property. And, as one might expect, the sister property of a quasiconvex indirect utility function with respect to travel times does not apply here either. While the time-budget constraint resulting from a weighted averaging of two travel-time vectors leads to a feasible consumption space which is a subset of the union of the two feasible spaces of the original two vectors, one may be better off because the indirect utility function shifts when the time vector changes! There may be a preference for better-balanced travel times because, for example, one can then spend better balanced amounts of time participating in a variety of activities (versus being “stuck” in the few activities which are relatively travel-time inexpensive). Changes in iso-utility contours due to changes in the travel times can bring this about. For these reasons, the conditions of time-budget concavity and indirect utility quasiconvexity are not imposed or expected for the models estimated here.10 Homogeneity Since money is just a unit of exchange and does not itself hold value, rational humans are expected to not alter their choices under pure inflation. The theory is that indirect utility and all demands are homogeneous of degree zero in prices and income; so,

31 if prices and income all change by the same factor, a household’s welfare/utility and consumption choices do not change (see, e.g., Deaton and Muellbauer 1980b, Varian 1992). A typical specification of indirect utility and its resulting system of demand equations show prices everywhere divided by total expenditures, so that homogeneity is implicit in the formulation; section A-1 of the Appendix details several such specifications for a money-expenditure setting, but a general description of such a model in a time-and-money-homogenous setting is the following:

v v ? t P? Indirect Utility = v? , ? ?H Y? Three:-9)

(Chapter

The idea that pure inflation should not change one’s consumption patterns is theoretically acceptable in an environment where people pay for goods with money, but this is probably too strong an assumption for consumption which involves time expenditures, since time is not instantly tradable – people directly experience their spending of time11. For this reason, several modifications were made to the typical model specifications, providing greater functional flexibility by not imposing homogeneity with respect to travel times and the time budget; these functional forms are shown in the section titled Model Specifications. Note that if information on monetary prices were available in the data sets, one could include these and impose homogeneity over prices and monetary expenditures. Summability The very common assumption of non-satiation12, that a little more of a good is a positive thing, no matter how much a person already has, implies summability of monetary expenditures when one is considering consumption across all demand

32 alternatives. Summability is also the condition that the sum of all demands considered in a model times their prices equals total expenditure on the set of goods considered. In a system of activity-demand equations where one is modeling all uses of time (or all uses of, say, discretionary time), one would probably want to impose summability to ensure that results are consistent with reality (e.g., a 24-hour day). However, when one considers only the number of activities accessed, as in this research, rather than also modeling the amount of time spent in each, summability’s imposition – in this case across travel-time expenditures – puts the focus on allocating an exogenous total travel time, rather than allocating total time available. Thus, summability would be unnecessarily limiting and is not imposed here. Separability The neglect of other goods’ price information generally necessitates an assumption of separability and shifts the modeling focus to substitution and trade-offs within a subset of consumption over an exogenously determined subset of the budget. Separability

exists when direct utility is a function of sub-utility functions having distinct good sets as arguments; if utility is an additive function of these subutility functions, strong separability exists.13 As an example, one may collect detailed data on households’ consumption of food items but not have any information on their consumption of clothing, lodging, transport, and utilities. To be able to apply the rigorous microeconomic theories associated with utility maximization and estimate a system of demand equations across this limited data set, one would need to argue for separability of preferences and rely on food expenditures

33 as the exogenous budget constraint, rather than total budget. The utility function and demand functions would then be written as the following: v v v u( X ) = f (u1 ( X Food ), u2 ( X Other )), v X i , Food = X i , Food ( PFood , YFood ). Three:-10) Separability is a strong assumption; it implies that consumers can order their preferences in each, distinct subset of choices independent of the amounts of other goods consumed. Strong or additive separability is even more restrictive; it rules out the possibilities of inferior subsets of goods and complementarity across subsets while imposing approximate proportionality between own-price and income elasticities. A more detailed discussion of separability can be found in Deaton and Muellbauer (1980a). A model which assumes separable preferences can be significantly more limiting than a model considering the role of the entire budget available to a consumer. However, if one assumes that prices of all non-considered goods are the same for all households, preference separability is unnecessary. In the case at hand, this condition requires that only the travel-time environments differ across the sample population. The constancy of other goods’ prices across the sampled observations means that their effects are not identifiable empirically; so, even if these prices were known, their invariance would effectively conceal their distinct parameters within the set of identifiable effects. One of the limitations this assumption places on model estimates is that the effects of changes in relative prices of the non-considered goods will not be predictable with the results established here. (Chapter

34 How valid is the assumption of price invariance across non-considered consumption in the models estimated here? The price of a McDonald’s hamburger may be the same regardless of where purchased in a region, but the prices of other goods, such as restaurant dining and food shopping may vary according to land rents, freight delivery costs, and local shoppers’ preferences. However, if, for example, demand types are defined sufficiently broadly in a spatial sense (e.g., destination zones are large), average price variability may be rather negligible, with enough opportunities present to match the prices found elsewhere. If prices of goods not considered in the demand system do vary significantly, one may assume that separability holds and replace the variable of total expenditures with that of the subset’s expenditures. Or, if prices move proportionally together, according to one’s location (e.g., central-city versus suburban dwellers), one may consider deflating or inflating income measures according to a price index, across sampled consumers. These approaches are not taken here, however, because it is virtually impossible to argue separability of goods consumption and activity participation (since many activities are complements of consumption – for example, recreational activities and entertainment expenses) and because price and monetary-expenditure information is lacking in available data sets. Symmetry Symmetry is a condition that arises in the typical system-of-demands frameworks, i.e., in those where only a monetary constraint governs. It refers to the condition of symmetry of compensated cross-price effects (Slutsky 1915). Income-compensated or Hicksian demands can be derived simply by taking the first derivatives of the typical,

35 money-expenditure function; the derivatives of these demands with respect to other goods’ prices are the cross-price effects, and these are symmetric thanks to Young’s Theorem (which says that the order of differentiation is not important). Thus, the second derivatives of the expenditure function with respect to prices Pi and Pj are symmetric, as illustrated in the following equations: v v Expenditure = e$ ( P, u) = Money needed at prices P to achieve utility u; v v de$ ( P, u) Money ? Compensated Hicksian Demand = hi ,$ ( P, u) = ; dPi v v v dhi ,$ ( P, u) d 2 e$ ( P, u) dh j ,$ ( P, u) Compensated Cross ? Price Effect ij = = = . dPj dPi dPj dPi Three:-11) The matrix of compensated-demand derivatives is called the Slutsky matrix, and theory implies that it is negative semi-definite, since total money expenditures are concave in prices. However, in the model structure investigated here, the first derivative of time expenditures with respect to an activity’s travel time is not the compensated demand for that activity. In the common application, expenditures equal prices times amount of goods consumed; but in the decisions considered here, time expenditures are the sum of access costs/travel times multiplied by the number of out-of-home activities consumed plus the amount of time spent in each activity (in- and out-of-home). Thus, the derivative of time expenditures with respect to any travel time is no longer equal to the time-compensated demands for activities, so the matrix of second derivatives of the timecompensated expenditure function is no longer the same as the time-based Slutsky matrix and symmetry is not a condition imposed on the demand system estimated here. The following equations illustrate this property:14

(Chapter

v v v v Typically, e$ ( P , u) = P ′h$ ( P , u); and , v since e$ ( P, u) is homogeneous of degree one in prices and v hi ,$ ( P, u) is homogeneous of degree zero in prices , v v v de$ ( P, u) de$ ( P, u) Pi = e$ ( P, u) = ∑ hi ,$ Pi , and ∑ = ∑ hi ,$ , ∑ dPi dPi i i i i v v de ( P, u) so $ = hi ,$ ( P, u). dPi v v vv v v v v However , eT (t , P, Y , u) = t AT (t , P , Y , u) + ∑ Ti ,T (t , P, Y , u),

i

36

(Chapter

and is not homogeneous of degree one in access times and involves unidentifiable participation ? time demands. Three:-12)

Validity of Utility Maximization Hypothesis It is important to recognize that many empirical analyses of demand systems, analyzing different consumption sets’ shares of monetary expenditures for aggregate, serial data sets and using a variety of common forms (such as the translog and generalized Leontief), have failed to support results satisfying basic economic theories (e.g., Guilkey et al. 1983 and Caves and Christensen 1980). For example, imposition of symmetry constraints may reduce the likelihood of the observed sample substantially or the concavity of expenditures in prices may not be satisfied at many observations. (Lau 1986, Deaton and Muellbauer 1980a & 1980b, Pollack and Wales 1978 & 1980) Lack of support for well-accepted economic theory by a model suggests that the model specification is substantially incorrect and/or the households/individuals observed are not economically “rational” (according to a utility-maximization hypothesis of behavior). Any modeler should be conscious of these possible inconsistencies and check for them where practicable. However, as discussed throughout much of this section on theoryimplied constraints, very few of the theories which are expected to apply in a money-

37 expenditure setting are likely to hold here. Without symmetry and summability, Roy’s Identity is the origin of virtually all restrictions imposed in the models estimated in Chapter Four; these restrictions are implicit by virtue of the common parameters found throughout the estimated demand equations and are due to the system’s derivation from a single indirect utility function. Non-negativity of demands and positivity of the marginal utility of time are the only other conditions imposed here; however, marginal utility estimates are considered for their conformance with theory, and the concavity of estimated expenditure functions and convexity of estimated indirect utility functions are examined briefly. Estimating Benefits and Costs “Equivalent” and “compensating variation” are measures of welfare changes following price changes, each using a difference in expenditure functions but at different reference levels of indirect utility. The author knows of no empirical examples where equivalent and/or compensating variation has been quantified with anything other than a money metric. Actual welfare change is not measurable in known units, since it is generally agreed to be the change in utility associated with price/cost changes.15 The distributional effects associated with policy changes are very important. Total benefits exceeding costs/disbenefits only signifies a potential for Pareto superiority, i.e., the possibility of a Pareto-preferred redistribution of the benefits so that no one is worse off following a positive-net-benefits change. (Varian 1992) As economist Steven M. Goldman describes it, “Cost-benefit analysis as a welfare measure which is done independently of distributional effects is fundamentally flawed.” (Goldman 1998)

38 Measurement of welfare changes using a money metric favors projects benefiting those who have the most monetary resources available, rather than those who might experience the most welfare benefit, because those with the most money can place the highest monetary value on a change in conditions16 (see, e.g., Heap et al. 1992, Price 1993). This is of particular concern in the evaluation of projects producing significant time-expenditure differences, such as transportation infrastructure alterations (e.g., Daganzo 1997). A fortunate result of recognizing a time constraint in utility maximization is that the indirect utility function can be inverted with respect to this budget variable and welfare impacts can be assessed with a time metric. Equation 3-13 provides the definitions of equivalent variation which are used here, in terms of money ( EV$ ) and time ( EVT ). As illustrated, equivalent variation can be written as the difference in expenditure function values at reference price levels, as well as, under constant budget levels, the integrals of the compensated/Hicksian demand equations. 17

t v v v v v Welfare Change$ ≈ EV$ = e$ (t o , T , u ′) ? e$ (t o , T , u o ) = ∫ h$ (t , T , u ′ ) ? dt t′

v vo

t v v v v v & Welfare ChangeT ≈ EVT = eT (t o , Y , u ′ ) ? eT (t o , Y , u o ) = ∫ hT (t , Y , u ′) ? dt t′

v

vo

(Chapter

Three:-13) Note that the negative of equivalent variation can be understood to mean the maximum amount of money or time a household would be willing to give up to avoid the change in prices/travel times, if budgets levels are unchanged. Chapter Five’s section on cost-benefit analysis uses both the income and time definitions of equivalent variation to estimate the welfare impacts of an increase in travel times.

39

Functional Specification

Assuming that the demand equations arise from derivatives of the indirect utility function, one may wish to select functional forms for the indirect utility function, v, which are flexible to a second (or greater) order18. This flexibility permits estimation of cross-price and income elasticities, in contrast to non-interactive functional forms, which produce only non-zero direct elasticities. The transcendental logarithmic’s functional form (i.e., the translog) is commonly used in practice (e.g., Cameron 1982, Christensen et al. 1973 & 1975, Pollack and Wales 1980) and is quite flexible19, but it has some drawbacks. Under a situation of no cross-parameter constraints, the number of translog parameters increases with more than two times the square of (rather than linearly with) the number of goods, which may result in statistical insignificance for many parameters and low confidence in estimation – depending on sample size. For empirical estimability over limited sample sizes, one may need to make some a priori assumptions as to relationships and assume a relatively parsimonious form for estimation. Other functional forms for indirect utility are also possible and have been used in money-expenditure systems of demand. A variety of forms are shown and discussed briefly in the Appendix (A-1), but the simplest to estimate impose untenable assumptions implicitly. For example, in a theoretically consistent linear-in-unknowns demand system of monetary expenditures on three or more goods, all income elasticities must equal one. (Lau 1986) And, in the traditional consumption framework where only a monetary budget governs, the Cobb-Douglas and utility-consistent Rotterdam (Barten 1964, Theil 1965) functional forms impose additivity and homotheticity assumptions on preferences – along with a constant, unitary elasticity of substitution20 across all pairs of goods! (Greene

40 1993, Christensen et al. 1975) In reality, the substitutability of consumption goods may vary widely, given different relative combinations (for example, at extremes of goods ratios, less substitutability is expected than at better-balanced levels). The generalized Leontief functional form may be more flexible than the translog when unequal or low elasticities of substitutions exist across the choices (Guilkey et al. 1983, Caves and Christensen 1980), but can be rather intractable in its most general form (Diewert 1971 & 1974) and is not expected to perform as well when high and unequal substitution elasticities exist (Caves and Christensen 1980). There is no particular reason to expect similarity of substitution across different activity types, but there is an expectation of high substitutability across certain choice definitions. For example, in the empirical investigations pursued here, activities are distinguished by the iso-opportunity zones in which they take place, rather than the activity type or purpose; therefore, one may expect very high substitution effects across zones and opt for a translog specification. Substitution is expected to be less when one considers very distinct activity types, such as personal business versus social, so the Generalized Leontief may be most useful in these cases; however, some trip types, such as non-food shopping and recreation, may to a certain degree still act as substitutes. Among the models estimated here, in Chapter Four, one of the specifications resembles Stone’s Linear Expenditure System (1954), while the others are based on modifications of the translog specification.

Model Specifications

Four distinct model types are tested empirically in Chapter Four, and their functional forms are specified here, with typical economic notation for demand ( X i ) replacing the

*

41 notation for optimal rate of activity participation, Ai . The first of these four models is an attempt at a relatively simple specification using an indirect utility specification similar to that which generates Stone’s Linear Expenditure System (1954). The other three are modifications of the translog model (Christensen et al. 1973 & 1975), and they are presented here in order of increasing generality. All models are used to estimate longrun, optimal out-of-home activity participation rates (per day) for households, and all but the third are used only once, to model participation in discretionary activities. The third of these specifications is also used to model entire home-based tours of activities, rather than just individual stops, and these tours can include non-discretionary trip-making. All specifications shown, except the fourth, rely on discretionary time (total time minus work and school time) and income as exogenously provided arguments. The empirical results from those analyses are provided in chapters Four and Five. Type 1 Model Specification: Modified Linear Expenditure System In an effort to begin with as simple a functional specification as possible, Stone’s Linear Expenditure System or “LES” (1954) was examined for use. However, without the ability to impose homogeneity in the time dimension and due to the presence of two budget variables, the resulting demand system is not nearly as simple as Stone’s. The indirect-utility specification and resulting demand equations used for this modified-LES specification are as follows:

*

Indirect Utility = v ={ Max Utility Budget & Time Constraints} =

42

∏t α

i i

i

1 ? Td ? ∑ β ij t i t j ? ∑ β iY t i Y 2 i, j i

, so...

? ? 1 ?Td + ∑ β jk t j t k + ∑ β jY t j Y ? ? ∑ β ij t j ? β iY Y , 2 j ,k j ? ? j where t i = Travel Time to Activity i , Y = Income, & Td = Discretionary Time Available, Xi* =

αi ti

(Chapter

and β ij = β ji ?ij ( for identifiability of parameters ).

Three:-14) Note that Stone’s original specification produces a system of demand equations whose parameter space increases only linearly with the number of goods consumed, “I”. While Stone’s system requires the estimation of 2I-1 parameters, the modified system used here has a parameter set which grows quadratically with the number of goods considered, requiring the estimation of 2I+I(I+1)/2 parameters21! The assumption of homogeneity saves a modeler many degrees of freedom for estimation purposes; however, there exist many major weaknesses with the LES, as discussed in Section A-1 of the Appendix. Under the modified LES specification used here, the value of time is independent of the budget levels, depending only on access times, and the time-budget elasticities of demand are independent of all variables but the demand’s own access time; such functional inflexibilities pose a serious problem. For example, this model’s estimation results, which are provided in Chapter Four, produce negative values of time for all households in the 10,834-observation sample! A more flexible model is almost certainly necessary.

43 Type 2 Model Specification: Modified Translog Having considered the strengths and weaknesses of various functional forms, many of which were discussed in a previous section, titled “Functional Specification”, a modified version of the Christensen et al.’s translog form (1975) was chosen to represent the indirect utility function for the remaining set of models estimated here. The translog was chosen for its second-order functional flexibility as well as for its ability to flexibly model substitutes well.22 The most restrictive form of this general specification that is analyzed here is termed the “Type 2 Model Specification”, and it is as follows:

v Indirect Utility = v ≈ Translog (t , Td , Y ),

i ij

v = α o + ∑ α i ln(t i ) + ∑ (1 2)β ij ln(t i ) ln(t j ) +

(Chapter

∑γ

i

i

ln(Td ) ln(t i ) + ∑ γ iY ln(Y ) ln(t i ) + γ TY ln(Td ) ln(Y )

i

Three:-15) The optimal demand levels which result from application of Roy’s Identity (with respect to time) to the above formulation are the following:

So, X i * =

? ? ?1 ? ? ? α i + ∑ β ij ln(t j ) + γ iY ln(Y ) + γ iT ln(Td )? ?? ? ti ? ? ? j ? 1 ?∑γ Td ? j

jT

ln(t j ) + γ TY

? ln(Y )? ?

,

(Chapter

where t i = Travel Time to Activity i , Y = Income, & Td = Discretionary Time Available, and β ij = β ji ?ij & γ TY = 1 ( for identifiability of parameters ).

Three:-16) Notice that the number of parameters in this modified translog system increases quadratically with the number of good types considered. The system of equations

44 requires the estimation of 3I+I(I+1)/2 parameters, which is 2I more than in the LESbased, Type 1 model. Type 3 Model Specification: Modified Translog with Constants The Type 2 model specification can not be nested with a no-information model specification (i.e., a model without any explanatory information) since all of its unknown parameters interact with explanatory variables. Therefore, a more flexible model of this form was investigated, adding I+1 parameters to the modified-translog specification to effectively function as constant terms; this change produces the following: Indirect Utility = v = α o ? ∑ ? i t i + ? o Td + ∑ α i ln(t i ) + ∑ (1 2)β ij ln(t i ) ln(t j ) +

i i ij

(Chapter

∑γ

i

i

ln(Td ) ln(t i ) + ∑ γ iY ln(Y ) ln(t i ) + γ TY ln(Td ) ln(Y )

i

Three:-17) The optimal demand levels which result from application of Roy’s Identity (with respect to time) to the above formulation are the following:

? ? ?1 ? ? ? α + β ij ln(t j ) + γ iY ln(Y ) + γ iT ln(Td )? ?i ? ? ? ti ? ? i ∑ ? j , So, X i * = ? ? ? ? ? o + ? 1T ? ? ∑ γ jT ln(t j ) + γ TY ln(Y )? ? d ?? j ? where t i = Travel Time to Activity i , Y = Income, & Td = Discretionary Time Available, and β ij = β ji ?ij & γ TY = 1 ( for identifiability of parameters ).

Three:-18)

(Chapter

The expectation is that this more flexible specification will provide more reasonable estimates of behavior, such as demand elasticities and values of time; it also allows the

45 nesting of the Type 2 specification within the Type 3 and so provides a means of gauging the need for Type 3’s added flexibility. Type 4 Model Specification: Modified Translog with Constants, using Wage and Total Time Data As discussed in the section on application of Roy’s Identity, one’s discretionary-time and income budgets may be endogenous to the choice of discretionary-activity participation. Thus, a model that allows for these choices in a simultaneous manner may prove useful. Taking the most flexible of the model specifications suggested, i.e. that of the modified translog with constants, a specification based on wage rates and total time availability to a household’s members is the following: Indirect Utility = v = α o ? ∑ ? i t i + ? H H + ∑ α i ln t i + ∑ (1 2)β ij ln(t i ) ln(t j ) +

i i ij

(Chapter

∑γ

i

iH

ln( H ) ln(t i ) + ∑ γ iw ln( w + 1) ln(t i ) + γ

i

wH

ln( w + 1) ln( H )

Three:-19) The optimal demand levels which result from application of Roy’s Identity (with respect to time) to the above formulation are the following:

? ? ?1 ? ? ? α i + ∑ β ij ln(t j ) + γ iw ln( w + 1) + γ iH ln( w + 1)? ?i ? ? ? ti ? ? ? j , So, X i * = ? 1? ? H + ? ∑ γ jH ln(t j ) + γ wH ln(Y )? H? j ? where t i = Travel Time to Activity i , w = Wage Rate, & H = Total Time Available, and β ij = β ji ?ij & α 1H = 1 ( for identifiability of parameters ).

Three:-20)

(Chapter

46 Since income is not exogenous in this model, values of time are estimated using Equation 3-8’s approximation, which requires an estimate of the unobserved variable optimal work time, Tw . As described in Table 3-2, a household’s work time is assumed to be eight hours per day for full-time workers plus four hours per day for part-time workers. These results of these computations are shown in the following chapter, Chapter Four.

*

Statistical Specification

Integer Demand Observations and the Poisson Assumption Observed demands can be visibly discrete in limited-period data sets. However, one may expect that continuous and smoothly differentiable preference and demand functions underlie observed behavior, since households are typically free to optimize their choices over relatively long periods of time. This is the assumption made here, so a link to a model of cardinally ordered discrete demand levels is needed for empirical estimation. This link may be best provided via the Poisson distribution, which is defined over the set of non-negative integers. Given an assumption of Poisson-distributed demands, the various activity types i (e.g., near vs. far, or dining vs. social activities) can be characterized as in Equation 3-21. This set of Poisson random variables is simultaneous in nature, since the derivation of all mean demands from a single indirect utility specification introduces common parameters across the demand specifications.

X i ~ Poisson( λ i ), ?i , v v v v * where λ i = X i = f i ( PA , Ptrvl , A , PZ , t , Y , T ). Three:-21)

(Chapter

47 The Poisson distribution arises naturally from counts of independent events that occur at a specified rate, so it would be a plausible distributional assumption if household members make trips at randomly and independently selected times throughout their window of discretionary time. In reality, household members are often constrained to temporally and spatially coordinate their trip-making due to limitations on automobile, driver, and transit availability, closures of activity sites (e.g., stores late at night), and the desire to engage in activities together. Moreover, activity participation and travel take time, undermining the assumption that such events occur independently in time.23 Without independent and identically distributed exponential inter-event times, the Poisson may still characterize the counts of activity participation across households; this may be particularly true over longer periods of time, as the short-term/daily realities of trip chaining and activity coordination take on less importance relative to long-run behavior. Unfortunately, the household travel surveys with sufficient sample size and detail for use in this study tend to be of short duration (e.g., one to two days, typically); so the Poisson remains a significant assumption. However, as described next, the Poisson is mixed here with a gamma distribution, in order to capture unobserved heterogeneity across different households having the same set of observed characteristics. Generalizing the Poisson Assumption through Use of a Negative Binomial One limitation of the typical Poisson regression model is that its variance is constrained to equal its mean. Cameron and Trivedi (1998) describe the failure of the Poisson assumption of equidispersion as qualitatively similar to a failure of homoscedasticity in a linear regression model, but with possibly much larger effects on standard errors. However, allowing variation in the Poisson’s parameter λ by mixing the

48 Poisson with another distribution can help one avoid a restrictive equi-dispersion assumption and accommodate the effect of unobserved factors on each household’s mean trip-making rates. Overdispersion is common in behavioral data (Cameron and Trivedi, 1998), and it was found to be present in the trip-making data sets, after controlling for a variety of market characteristics and demographic explanatory variables and then applying statistical tests described in Cameron and Trivedi (1998). Even though a large set of explanatory variables is used, the dispersion coefficients (the α parameters) are highly statistically significant in all models, indicating a decisive rejection of an equidispersion hypothesis. Factors other than travel times and income and time budget levels play an important role in household activity participation rates. Whether a household is active or inactive, profligate or frugal, may mean significant differences in optimal rates of trip-making. Thanks to these unobserved characteristics, one also would expect there to be some variation in trip-making rates across households with the same observed characteristics. It is therefore useful to add a “second layer” of stochasticity by mixing a Poisson with a second distribution. Additionally, one may reasonably hypothesize correlation across the unobserved components of the various demands by a single household, since one can expect the deviation in a household’s demand for one type of trip to be associated with deviations in its other trip demands. For example, if a household is taking part in a certain out-of-home activity more than one expected (given its time and income constraints and the set of travel times it faces), it may also tend to participate in other outof-home activities with a higher-than-expected frequency. Information on one set of

49 demands observed for a specific household, relative to expectations, is likely to help one better predict other consumption by the same household. The use of a compounded and correlated error structure within a system of Poisson equations is unusual. Few modeling efforts have used a multivariate Poisson form to model demands, particularly in a rigorous micro-economic framework with more than two choice types. Hausman, Hall, and Griliches (1984) use the seminal set-up of Bates and Neyman (1952) in order to model the number of patents received by a panel of firms over time as “fixed-effect Poissons”, which integrate to negative binomials; but there are no “prices” or explicit links to profit maximization in their model. Hausman, Leonard, and McFadden (1995) estimate the choice of recreational sites using a multinomial distribution conditioned on total number of trips, where the total is a fixed-effects Poisson and travel costs are included in the set of explanatory variables. Hausman, Leonard, and McFadden’s model provides measures of welfare/benefits via a logit model’s log-sum maximum-expected-utility. However, their model does not consider other types of trips or related consumption, and the two decision stages (i.e., total number of trips and allocation of these trips across sites) are estimated sequentially, rather than simultaneously. If significant flexibility of the error terms’ covariance structure (e.g., a multivariate normal distribution across the “ ε i ’s”) were permitted, the maximum-likelihood equation’s values would almost certainly have to be computed using numerical integration or distribution simulation over the multiple of probabilities. Such an approach

v is illustrated by Equation 3-22, with g( ε ) representing an assumed joint-density, such as

50 the multivariate normal, with fixed mean and a variance-covariance matrix to be determined.

v v v v Prob X = k λ 1 (ε ),..., λ I (ε ) =

(

)

+∞1

ε 1 =?∞ ε I

∫ ...

+∞

? λ i ki ? v v g (ε )δε exp( ? λ ) i ? ∫ ∏ k i !? ? ? i =?∞

(Chapter

Three:-22) Many have used simulation for estimation of complex specifications; for example, Yen et al. (1998) have used it for a set of correlated-in-unobserveds ordered probits, and Train (1996), McFadden and Train (1996), and Mehndiratta (1996) have used simulation successfully for a random-parameters logit model. However, estimation times tend to be long – and a second simulation will invariably lead to a set of different estimates. Instead, if one can specify the second layer of stochasticity (i.e., the layer within the Poisson’s own lambda parameter) so that the random component can be tractably integrated out, the estimation is much simplified. For this reason, an integrable error structure was sought in this research, leading to the mixing of a Poisson with a gamma to produce a negative binomial distribution; the use of the same gamma error term across all of a household’s demands allows for a cancellation of these terms in the probabilities of a multinomial (which is conditioned on a negative binomial for total demand), as illustrated in the following equations:

51

If X i ~ Poisson( λi = X i ε ), and

*

∑X

i =1

I

i

= X T ~ Poisson ∧ Gamma = Negative Binomial (m, p * ) , where λT

I m(1 ? p * ) * = Xi , ∑ * p i =1

I v v ? *? then Prob(X 1 , X 2 ,..., X I | p1 , p2 ,..., pI ) = Multinomial ( X | p, X T ) Neg .Bin? X T | ∑ X i ? i =1 ? ? ? ? ? ? I XT ! X i ?? Γ ( X T + m) ? * XT * m ? pi ? = I ∏ ? X !Γ( m) ? ?(1 ? p ) ( p ) ? ? I =1 ? ? T ? ? ∏ X i! ? ? i=1

where pi =

∑ (X ε ) ∑ X

I * I j =1 j j =1

Xi ε

*

=

Xi

*

.

* j

(Chapter

Three:-23) As implied in the above equations, two parameters characterize a negative binomial. The parameters m and p * are used in Equation 3-23, and these can be thought of as a size and probability parameter. Appendix section A-4 describes the negative binomial distribution in more detail. The negative binomial assumption has been used in empirical work for several decades. For example, Chatfield et al. (1966) used a single negative binomial regression equation to model household purchases, but Rao et al. (1973) were the first to use a system of equations and thus a specification similar to (yet much simpler than) the set-up followed here. Rao et al. (1973) modeled the number of boys and the number of girls born to a pair of parents as symmetric binomials (i.e., with probability of either equal to 0.5) conditioned on a negative binomial for the total. As mentioned, a negative binomial distribution (NB) can be thought of as a Poisson whose parameter varies as a gamma (i.e., Poisson∧Gamma). And the entire system of demand equations can still be considered a system of Poissons, but with variation permitted in the rates ( λ i ’s). Knowing total count, a system of independent Poissons

52 becomes a multinomial (MN) distribution; knowing total count under a system of correlated Poissons conditioned on a negative binomial (i.e., Poissons∧NB( λ T )) also implies a MN. However, for one to be able to identify the probabilities of the choices (pi’s) with a closed-form solution – and avoid simulation or numerical integration, one must make some assumptions and thereby constrain the system’s “double stochasticity” to a certain form. Here, the assumption that the system conditioned on total count is a MN with pi’s equaling λ i λ T implies that the variation in each λ i is equal to the factor of variation in the λ T times pi. Thus, for such a set-up, each multiplicative gamma error component is the same value as the gamma random variable that affects total trips. Since a gamma variable times a constant is also a gamma variable, all marginal distributions of trips ( X i ’s) are negative binomial(m,piP), with their mean rate having a gamma distribution; in statistical notation: X i ~Poisson( λ i )∧Gamma( m, m X i ).24 The

*

density function for a gamma distribution is shown in Equation 3-24, helping illustrate why the rates for individual demands are also gamma distributed. The stochastic assumptions of observed demands having Poisson distributions whose rates interact multiplicatively with the same unobserved gamma variable, for a given household, imply that individual rates can be thought of as gamma variables with the same size parameter as total demand (m), but with modified scale parameters ( m X i , rather than m X T ).

* *

λT ~ Gamma(m, m X T ) → pdf λ (λT ) =

*

T

me ? λT m

XT*

(λT m X T ) m ?1 , * X T Γ ( m)

*

where m > 0,λT ≥ 0, and Γ(m) = ∫ e ? x x m ?1dx [= (m ? 1)! if m is integer ].

0

∞

(Chapter

So pi λT ~ Gamma(m, m pi X T ) = Gamma(m, m X i ). Three:-24)

* *

53 Typically, a multinomial’s component levels are negatively correlated, because of a fixed sum. However, when the sum or total is allowed to vary as permitted here, the unconditional correlation becomes positive, as shown in the following set of equations. Overdispersion, as previously discussed, is also a property of this distribution and illustrated in the following equations.

1 ? p* v If ( X 1 , X 2 ,..., X I ) ~ Multinomial ( p, X T ) ∧ Negative Binomial (m, P = ), XT p* then E ( X T ) = mP = ∑ X i ,

* i

and V ( X T ) = mP(1 + P) = E ( X T )(1 + P) > E ( X T ) → Overdispersion. E ( X i ) = mPpi = E ( X T ) pi , and V ( X i ) = pi PE ( X T ) + pi E ( X T ) > E ( X i ) → Overdispersion. COV ( X i , X j ) = E X T (COV ( X i , X j | X T )) + COVX T ( E ( X i , X j | X T )) = ? pi p j E ( X T ) + E X T ( pi p j X T ) ? pi p j E ( X T ) 2 = pi p j PE ( X T ) > 0, as expected.

Three:-25) In sum then, the system of demand equations can be termed a multivariate negative binomial, since each of the demands is marginally represented by a negative binomial. Moreover, the special, same-gamma-term assumption allows the system to collapse to a multinomial for the different splits of activity types, conditioned on a negative binomial for total activity participation. As a point of comparison, the third model specification in Chapter Four is run with the same-gamma-term assumption and without it (i.e., as a system of independent Poissons); the correlations of residuals resulting from this later specification are investigated.

2 2

(Chapter

54 Implication of the Assumption of Multiplicative Error Component for Indirect Utility The assumption that households having the same observed characteristics can have different long-run, optimal rates of activity participation according to a gamma distribution implies something about the variation across these households’ indirect utility values. Since the average optimal rates, X i ’s, are derived via Roy’s Identity, the gamma error component must come out of one or both of the derivatives which are used: the derivative of indirect utility with respect to travel time or that with respect to available time. One stochastically convenient theory is that the households, although well aware of their marginal utility of available time, observe their travel-time environment with some error such that the travel time they perceive is really distributed like the inverse of a gamma random variable around the “true” or observed travel time. Another possibility is that the travel time data observed and used to estimate the models provide the mean travel times within different neighborhoods, but the actual, household-specific travel times within that neighborhood are inversely gamma distributed around that neighborhood’s mean. These stochastically equivalent assumptions translate to the following:

*

Let t i ,obs ’d = ε t i , felt , where ε ~ Gamma (m, m). v v Then v = v (t felt , T , Y ) = v (t obs ’d , ε , T , Y ), and ? ? ? dv dv ? ? ? ? dt ? ? dt ? ? ? i ,obs ’d i , felt or actual * Xi = ? =? ? dv ? ? ? ? dT ? Three:-26) × ? ? dv ? ? ? ? ε dt i , felt or actual ? ? ? dt i ,obs ’d ? =? ? dv ? ? dv ? ? ? ? ? ? dT ? ? dT ? dt i ,obs ’d

(Chapter

The above is likely the simplest method of integrating back from the error assumption on the observed demand system to the unobserved indirect utility function;

55 but other processes also could lead to the multiplicative gamma specification in the demand system. However, no matter what method of accommodating the gamma error specification at the level of the indirect utility function, welfare analysis is likely to be complicated and one should be wary of using the indirect utility functions and their inverted expenditure functions as originally specified, without explicitly acknowledging the stochastic components. The need for averaging a measure like equivalent variation over its unobserved, random components is generally important when the error does not enter additively with a mean of zero, since expectation is a linear operation. McFadden (1996) provides a nice discussion of this situation in discrete-choice models. Note that a simulation-of-likelihoods estimation method would allow one to estimate a more distributionally complex model, without imposing the same gamma error term on all the Poisson rates at the level of observed demand. For example, one could begin by specifying the unobserved heterogeneity to occur in the indirect utility function and then see what that implies for the demand functions. An error term which arises additively and which is independent of travel times and the total-time-budget variable would not show up in the demand equations for number of trips (though it would typically be relevant in average welfare estimates). A more reasonable assumption would be of random parameters, i.e. of unobserved differences in household’s preference structures; such an assumption would be very similar in nature to Train’s (1996) and Mehndiratta’s (1996) random-parameters logit models.

Data Set

The 1990 Bay Area Travel Surveys (BATS) was used for the empirical portion of this research. They detail trip-making of over 10,000 households in the San Francisco

56 Bay Area for periods of one, three, or five workdays. While the BATS are not surveys of activities, per se, BATS households’ activity participation can be inferred from the trip purposes and the start and end times of consecutive trips. Since survey lengths vary across the BATS households, a time component is included explicitly in the likelihood. For example, if the Poisson rates, X i ’s, are for a one-day period, one must multiply them everywhere in the Poisson specification with the variable days, where “days” is the number of days for which the household was observed participating in activities. In the multivariate negative-binomial likelihood equation used here (shown in Equation 3-23) the multinomial portion of the likelihood remains the same, but the negative binomial’s probability for total observed trips ( X T ) changes. The expectation of a multi-day survey’s total number of trips, X T , is days times the single-day level, but the variance increases more than linearly (unlike a Poisson). The parameter p * for a multi-day survey must be replaced by m m + (days × X T ) everywhere, so the variance remains equal to ? + α? 2 , but the mean, ? , has become X * × days . Note that the process remains a negative binomial with the same gamma term, as long as one assumes that the heterogeneity for an household is constant for each of the days the household is surveyed.25 The BATS data set is described in more detail in the Appendix section A-3, and a definition of all variables used is provided in Table 3-2. Definition of the Consumption Space Interestingly – but not too surprisingly, investigations for this research indicate that access times for activities distinguished simply by type or purpose (e.g., dining versus recreational) are endogenous, given a household’s location. In other words, even given

*

(

*

)

57 their relatively fixed residential locations, households can, to a significant extent, choose how long they spend accessing different types of activities. Initially, per-trip travel times and distances for the San Francisco Bay Area were regressed on a wide variety of urban form variables (e.g., accessibility to all jobs by automobile, accessibility to sales and service jobs by walking, entropy across the proportions of half-mile-radius-neighboring land uses, mix of neighborhood land uses, and developed-area densities, as defined in Kockelman 1996 & 1997) in order to instrument for the travel times and costs associated with different locations, after controlling for trip purpose/activity type. The predictive power of these models was minimal; for example, ordinary least squares regressions of per-trip travel times and distance on the large set of detailed urban-form variables produces R-squareds of just 0.002 and 0.016, respectively. The R-squared results of OLS regressions controlling for mode and trip type are shown in Table 3-1, where it appears clear that such models are effectively useless for prediction. This set of access measures does not predict statistically significant reductions in per-trip travel times or distances, even after controlling for mode and/or trip purpose!26 The first of these two general results is in agreement with the combination of Zahavi et al.’s constant travel-time-expenditures hypothesis (Zahavi 1979a & 1979b, Zahavi and Talvitie 1980, Zahavi and Ryan 1980) and the travel-time-inelastic nature of trip demands described by Ortúzar and Willumsen (1994) and Hanson and Schwab (1987) (as mentioned in Chapter Two’s literature review). As a result of all these indications, the possibility of instrumenting for the travel costs needed for the system-of-demands approach by using characterizations of a household’s environment appears very remote.

58 The evidence suggests that people travel further than they need to; this may very well be because they wish to expand their choice set of activity sites and thereby increase the expected “quality” of the activity they do engage in, at their chosen sites. For example, while one probably will travel only to the closest of a very specific activity type (such as eating out at a McDonald’s), one will not often travel to the closest dining establishment.27 As long as the marginal value of travel time plus the monetary cost of travel remains below the marginal value of increased opportunities brought about by traveling further, people can be expected to lengthen their journeys. Table 3-1: Regressions of Travel Time and Distance on Measures of Urban Form

59

TIME Regression’s Dependent Variable: PV-Trip Travel Time for Personal-Business Trips PV-Trip Travel Time for Social Visit Trips PV-Trip Travel Time for Dining/Eat Trips PV-Trip Travel Time for Recreation Trips PV-Trip Travel Time for Grocery/Food Shop Trips PV-Trip Travel Time for Non-Food Shopping Trips Non-PV-Trip Travel Time for Personal-Business Trips Non-PV-Trip Travel Time for Social Visit Trips Non-PV-Trip Travel Time for Dining/Eat Trips Non-PV-Trip Travel Time for Recreation Trips Non-PV-Trip Travel Time for Grocery/Food Shop Trips Non-PV-Trip Travel Time for Non-Food Shopping Trips DISTANCE Regression’s Dependent Variable: PV-Trip Travel Distance for Personal-Business Trips PV-Trip Travel Distance for Social Visit Trips PV-Trip Travel Distance for Dining/Eat Trips PV-Trip Travel Distance for Recreation Trips PV-Trip Travel Distance for Grocery/Food Shop Trips PV-Trip Travel Distance for Non-Food Shopping Trips Non-PV-Trip Travel Distance for Personal-Business Trips Non-PV-Trip Travel Distance for Social Visit Trips Non-PV-Trip Travel Distance for Dining/Eat Trips Non-PV-Trip Travel Distance for Recreation Trips Non-PV-Trip Travel Distance for Grocery/Food Shop Trips Non-PV-Trip Travel Distance for Non-Food Shopping Trips R-Squared 0.006 0.004 0.006 0.009 0.002 0.004 0.020 0.073 0.008 0.009 0.061 0.033 R-Squared 0.011 0.011 0.010 0.016 0.014 0.020 0.011 0.026 0.009 0.006 0.027 0.028

Unfortunately, in virtually all existing travel data sets there is no information regarding the quality of activities pursued. For example, except for general activitypurpose categories, there are no survey questions regarding the grade or class of establishments visited or the unit prices of activity consumption. To deal with this lack of detail in the data, one may choose to segment activities by some measure of quality, relative to an observation-specific origin (e.g., the household’s home location). One such measure is the number of choices a trip-maker has, which increases with time and/or distance traveled28. Thus, the number of jobs has been used here to distinguish activity

60 quality for trip types. Discretionary trips to locations within bands of 60,000, 300,000, 900,000 and two million jobs serve as the four types of trips in the models investigated. The variables derived from the travel surveys and from travel-time and employment data are described in Table 3-2. The focus is on the household as a unit, rather than intrahousehold trade-offs and decisions. So the total time available and income budget29 apply to the entire household, and the sum of activity engagements over the households’ members is the observed demand. Travel times for the four good groups distinguished in the data set represent average travel times to access the four different iso-opportunity contours from a household’s home location. In addition to the number of jobs, the amount of land area in different uses can be used to measure opportunity levels, particularly for activities like outdoor recreation. One may also wish to include trip-making from non-home trip-making bases, such as work. However, the size of the demand set may increase multiplicatively; for example, tripmaking to four iso-opportunity contours from the home and work bases of a one-worker household across all trip types would mean eight different demand types (and eight different travel times upon which to apply Roy’s Identity).

61

Table 3-2: Description of Variables Used

Dependent Variables: Number of Person Trips - Number of trips by surveyed household members (i.e., those members aged five and over) in the region on the survey days(s); does not include trips to home location to: Discretionary Activities, including: Medical/Dental Activity Social Visit Dining - Eat meal Recreation Grocery Shopping Non-Food Shopping Non-Discretionary Activities, including: Work and Work-Related Activities Personal Business Activity Education Other - Child care, serve passenger, change travel mode, other reason Explanatory Variables: Income “Y” - Pre-tax household income in 1989 Marginal Wage “w” - Estimate of average wage per hour for household ($/hour) = Income/50/(40×#full-time workers + 20×#part-time workers in household) Discretionary Time “Td” - Estimate of non-work-related and non-school time in a day available to a household’s members age five and older (hours/day) = 24×Household Size - Time in work-related & school activities Total Available Time “H” - Household Size (#members age five & older) × 24 hours (hours/day) Travel Times to Iso-Opportunity Contours - Average total travel time by single-occupant vehicle during free-flow conditions to access successively further sets of opportunities, relative to household’s home traffic analysis zone (TAZ); computed sequentially to nearest TAZs in turn (and exclusive of travel times to TAZs lying in other iso-opportunity contours). Contour Levels constructed at: 60,000, 300,000, 900,000 and two million total jobs, cummulatively.

Another way of creating more detailed consumption sets involves segmenting isoopportunity contours by modes of travel and by trip type. For example, the different modes available would generate different travel times, recreational trips’ travel times would come from contours based on entertainment and other recreational employment, and shopping trip travel times would come from those based on sales jobs. Clearly, there will be very high substitutability among these classes, which can be accommodated using

62 a flexible system of demand equations, as described in the section on functional specification in this chapter. Trip Chaining The chaining of trips into “tours” is a common phenomenon which complicates the analysis of activity-participation demands by altering access times. Within the Bay Area Travel Surveys (BATS), 36.6% of home-based trip tours involve more than one nonhome stop. 51.6% of the BATS person sample are full-time workers (and six percent are part-time workers); so a large percentage (12.3%) of the BATS trips are between work and some non-home purpose, and 5.74% of sequential trip pairs represent a tour from work and back (i.e., they have “work” as the first trip’s origin and as the second trip’s destination). However, more than half (56.4%) of the chained trips are unrelated to work. The marginal cost of adding a stop to one’s tour can be relatively small, if that stop is anywhere near the general path between primary activity locations. The nature of home-based tours found in the data set was investigated and it was found that most tours contain a single major leg from home, even though the average number of stops per tour is close to three, at 2.71. The mean and median total travel time per tour30 are 21.4 and 11.9 minutes, respectively, across all tours made (which number almost 40,000); and the travel time from home to the furthest destination accessed in each tour (with “furthest” measured by travel time) are 9.6 and 5.2 minutes. Thus, a single leg of the tour accounts for about 45-percent of the tour’s travel time, which can be taken to mean that about 90percent of the tour time is spent accessing a single destination. These results suggest that a single destination accounts for much of the tour’s travel time, while additional stops are relatively marginal in travel time cost.

63 Weekday non-work trip-making by workers tends to not be very complicated. For example, in their 1981 data set of Nagoya, Japan, workers, Kwakami and Isobe (1990) found that of the 15% of workers making non-work trips before or after work, only 2.3% made a trip on their way to work, 6.7% made one stop on their way home, and 4.2% made a single trip after arriving home. Kwakami and Isobe’s simulation results, which took work time to be exogenous, predict that as the time spent working during the day falls, workers travel further per non-work trip in addition to making more non-work trips; this is likely due to the loosening of the discretionary-time budget constraint and the ability to consume a higher quality of activity by traveling further. In a related study of tripchaining by workers, Kitamura et al. (1990) found that mid-chain stop locations between work and home “tend to cluster along the line segment than connects the home and work bases as commuting distance increases” (1990, p. 153); they also found that intensity “peaks” of stop location form toward the home and work ends of the segment. These same sorts of tour characteristics were found in the analyzed data sets, leading to a specification which accommodates chaining behavior; further description of this model’s definition of demands, along with empirical results, are presented in the following chapter.

64 ENDNOTES:

1

The chaining of trips as well as the linking of activities at a single opportunity site (e.g., shopping and entertainment at a shopping center) complicate the analysis since access times can be reduced and, to a significant extent, endogenized. To accommodate this effect, one can introduce variables for the possibilities of linking trips and/or model endogenously the number of chained trips to better account for the impact such travel behaviors have on a household’s choice set and utility. Chapter Four presents and estimates a model, using the Type 3 model specification described later in the current chapter. In-home activities are included in the vector of activities, travel costs.

2

A , but they have zero travel time and zero

3

One should be aware that these formulations assume a two-constraint case. If other constraints apply and lead to corner solutions for variables such as work time, the specified model will be insufficient and equations such as 3-7 and 3-8 will not apply. Moreover, if a household’s perceived wage or marginal return to an extra hour of work is unobservable, one may need to construct a model which accommodates this fact. For this regression model, observed work time is the amount of hours spent at work and in work-related activities during the survey day(s) for each household with workers, across its members. Table 3-2 defines the wage variable, w, used in these regressions (which is estimated using income and the number of full- and part-time workers) as well as the total time available to the household (per day), H. The median wage estimate for the household sample is $15.58/hour in 1990 pre-tax dollars, and this was substituted for the wage variable, w, in Equation 3-8. Substituting -0.000145 hours/dollar and +0.0808 hours/hour for the derivatives of work time with respect to unearned income and total time, respectively, yields a bias estimate of +9.04 percent. There is significant debate as to the validity of constraints implied by the theory of demand, such as homogeneity and symmetry (of the substitution matrix). For example, empirical tests of aggregate, serial demand systems by Deaton and Muellbauer (1980a) and Christensen et al. (1975) reject these restrictions. Deaton and Muellbauer suggest that their model’s “rejection of homogeneity may be due to insufficient attention to the dynamic aspects of consumer behavior.” (1980a, p. 312) They suggest adding time-trend variables, lagged values, and stocks as explanatory variables. And Polak and Wales (1978 & 1980) cite the importance of analyzing stocks, rather than flows, of durable goods, which requires a rigorous dynamic treatment of behavior. Thus, one should use care in the analysis of some of the goods of interest here, for example the number of automobiles or size of home; and any results using basic methods of analysis for such goods should be considered with some caution. However, the data likely to be available for the research at hand will not be serial, so there can be little consideration of these effects. Additionally, the current research will not experience the problems of high collinearity in prices and aggregation biases, which aggregate serial data are prone to (e.g., Deaton and Muellbauer 1980a, Barten 1977). Note that the log-likelihood equation must be re-written to accommodate certain zero-level demands, since one or more multinomial probabilities will equal zero and the logarithm of zero is undefined. If a demand level is optimally zero, the multinomial’s choice set collapses, eliminating the zero-level possibilities. An Rn→R1 function f(x) is convex/concave if f(αx1+(1-α)x2) is less/greater than or equal to αf(x1)+(1α)f(x2), for α∈[0,1]. Therefore, concavity of the money-expenditure function implies that the amount of expenditures needed to achieve a given utility level is no lower at a set of average prices than at two initial sets of unbalanced prices. Very similarly, quasiconvexity of the indirect utility function implies that the upper contour set of a such a function is a convex set; so, if two sets of prices, given income, lead to the same level of indirect utility, any average of those prices can be no better for the consumer. In a money-expenditures setting, this condition can be written as the following: the set of prices P such that k ≥ v(P,Y) is a convex set. Since added consumption of some activities (without binding budget constraints) may require so much added travel time and produce a net negative impact on utility, the commonly assumed property of “nonsatiation” or “monotonicity” is not likely to always be viable here. After a certain point, strictly

4

5

6

7

8

9

65

more of an activity is not necessarily a good thing. This is important to note because even if preferences are complete, reflexive, transitive and continuous (as described in Varian 1992), there may not exist a continuous utility function which represents those preferences.

10

Concavity implies that the symmetric matrix of second derivatives has only non-positive diagonal terms (i.e., d 2 eT dt i 2 ≤ 0 ? ). The matrices of second derivatives in prices of the expenditure function which d result from estimation of the Type Two model were computed for all 10,834 households, separately. Rather interestingly, for all 10,834 households, the first three of the four diagonal terms were positive and the fourth was negative; this result may suggest a condition more closely resembling convexity in discretionary-time expenditures with respect to travel times to the first through third contours! Convexity of the indirect utility function is a little easier to examine, since indirect utility is an immediate product of the models’ parameter estimation. In the case of the modified translog model specifications with and without constant terms, the diagonal terms of the matrix of second derivatives (with respect to travel times) is βii/ti2. These should be non-negative if the function is convex, but one finds that the βii (which determine the sign of this derivative) are estimated with negative signs for two to three of the demand types in the four models of this type (as shown in Tables 4-2a, 4-3a, 4-4a, and 4-5a). Thus, convexity of the indirect utility function is not apparent in travel times.

11

With monetary expenditures, people simply “hand over” their money; it is an immediate transaction, not requiring effort at the moment of use and affecting the spender only in how much money he/she has left over. One word of caution as to expectations of non-satiation here: Activity participation can be tiring and eventually undesirable for an individual, so non-satiation may not exist in terms of out-of-home time expenditures alone. Having more goods is easy when compared with experiencing activities, since the former requires storage space (or friends who are willing to cart away your belongings). Thus, the viability on non-satiation in activity participation may not make sense, particularly at the level of the individual. Still, one must experience his/her entire day (in contrast to not having to spend one’s entire income); restorative activities such as sleeping help make up for energy, and summability across all time expenditures is clearly a valid condition. Note that strong separability allows a monotonic transformation of the direct utility function to produce an equivalent direct utility function which is explicitly additive in the sub-utility functions. Even if one were to make the highly heroic and unreasonable assumptions that discretionary time expenditure is homogeneous of degree one in access times, time-compensated activity demands are homogeneous of degree zero in access times, and the sum of the derivatives of the various timecompensated time-in-activity demands [ Ti ,T (t , P, Y , u) ’s] with respect to a single access time equal zero, one can still not argue that the time-compensated activity demands are the first derivatives of the timecompensated expenditure function. The following equations make this apparent:

Even if v v ? dT j ,T ? ? dh j ,T ? deT ( t , P, Y , u) = ∑ hi ,T + ∑ ? ∑ ti ? + ∑ ? ∑ ?, ∑ dt dt dt i ? ? ? i i j i i ? j i i = ∑ hi ,T + 0 + 0 and

i

12

13

14

v v v v deT (t , P, Y , u) t i = eT ( t , P, Y , u) = ∑ hi ,T t i + ∑ T j ,T , ∑ dt i i i i the solution to these two equations is NOT deT = hi ,T . dt i

If the time-compensated activity demands were in fact the first derivatives of the time-compensated expenditure function, one then could impose Slutsky symmetry on the estimable/identifiable demands, derived from one’s indirect utility function. Symmetry of the Slutsky matrix in the common problem formulation (i.e., one with purely a monetary expenditure constraint) is generally very useful because, together with a condition for negative semi-definiteness (i.e., concavity of expenditures in prices), it guarantees integrability (see, e.g., Jorgenson and Lau, 1979); this means that these two conditions guarantee the existence of an indirect utility function that could generate the demand system estimated.

66

However, as discussed in a prior section, one cannot assume concavity of the time-expenditure function, thanks to travel time’s direct effect on one’s welfare. And, without the expenditure derivatives producing compensated demands here, one cannot logically impose symmetry on the Slutsky relation, as illustrated by the following relation.

v v v v * v v Given: hi ,T (t , P , Y , u) = X i (t , P , Y , e T (t , P , Y , u)). Without dhi ,T dt j = de T = hi ,T , dt i dX i dX i de T d 2 eT + ≠ dt j dY dt j dt i dt j

* *

? ? dhi ,T dX i * dX i * * ? ?. ? and dt ≠ dt + dY X j either ? ? ? j j

15

Changes in what is known as “consumer surplus” are a special case of equivalent and compensating variation; the value of change in consumer surplus lies between these two estimates and is defined as the integral of demand (vectors) over a change in prices. (Varian 1992) The argument for a money measure of benefits/disbenefits is that it best accommodates society’s values of benefits/disbenefits to everyone over a variety of impacts experienced. For example, one can argue that the time of high-income persons should be more valued by society than that of other persons, since their elevated incomes are typically due to their higher-valued labor-market activities; in other words, society optimally trades their time at a higher rate. Equivalent variation was originally defined by Hicks (1956) as the difference in expenditures at a reference utility level, rather than at a reference price level. However, when budget levels are held constant, the expenditures under the before and after scenarios are the same, so the definitions provided in Equation 3-13 are then equivalent to those given by Hicks. The definitions used here can be found in Varian (1992); they are the negative of Deaton and Muellbauer’s (1980) definitions, when available income levels are unchanged. Note also that Equation 3-13 assumes income (Y) and available time (T) are exogenous. With income and work time endogenous instead, one would write the lower set of equalities with unearned income, Yun, in place of total income, Y, and both sets of equalities with an added argument of wage, w.

16

17

18

Flexibility to a certain order means that any set of values for that order of derivatives can be achieved (with a single, variable set of parameters). Christensen et al. (1975) note that the translog provides a second-order approximation to any (typical) direct or indirect utility function; thus, the resulting demand functions provide a first-order approximation to any system. The same is true of the Almost Ideal Demand System (Deaton and Muellbauer, 1980a) and the generalized Leontief (Lau 1986). An elasticity of substitution is the dimensionless version of the derivative of the ratio of two goods with respect to their marginal rate of substitution (MRS). MRS is effectively a utility-constant measure of substitution between two goods. The following equations illustrate this definition:

du MRSij = ? du dX j dX i ?X ? d? i X ? ? j? d MRSij = Rate of substituting X i for X j to keep utility constant.

19

20

Elasticity of substitutionij =

(

)

×

MRSij ? Xi ? ? X ? ? j?

.

67

21

The parameterization of the distributional assumptions, which are described in the following section – Statistical Specification, adds additional parameters requiring estimation to each of the model types discussed in this section. Second-order flexibility is not fully realized with the current specification, because it does not include a log(Y)2 term (which would not be identifiable from the demand system estimated). However, the ability of this functional specification to capture substitution relationships is likely to remain superior to that of the other most popular form for such models, that of the Leontief. As discussed earlier, in the section on Functional Specification, substitution is important in the empirical analysis pursued here because the demand sets modeled in Chapter Four differ by quality of destination, rather than by activity type or purpose. So significant substitutability is anticipated. Some researchers are working with activity-duration models, acknowledging that activities endure separately rather than overlap (e.g., Ettema et al. 1995b, Bhat 1996), but micro-economic or other rigorous behavioral linkages are missing from these models. For example, using Weibull-based hazard functions, Kim (1994) models activity duration separately from trip generation but simultaneous with trip travel time. Recall that a gamma random variable can be thought of as the sum of m independent exponential random variables, with each exponential sharing the rate λ. Thus, a constant p (which is less than one) times a gamma can be thought of as the sum of m independent exponentials, each with a longer rate of λ/p; so the inverse of the rate (which is the average time between events) is shorter and the sum of the exponential times between events is shorter. One can think of the sum of “t” days worth of a household’s travel data as being the sum of “t” Poisson random variables, each with the same mean over the population having this household’s characteristics – and each interacted with the same gamma term, which represents the heterogeneity within a population of similar observed characteristics. As is well known, the sum of “t” independent Poissons is Poisson; the “t” days of Poissons considered here for a single household (indexed by n) are independent when one conditions on knowing the gamma error term and the mean rate (over the population with this household’s characteristics). Thus:

22

23

24

25

∑ (X

t s =1

i ,n, s

λ i , n = X i *ε n ~ Poisson λ i , n (t ) = tX i *ε n , and

* ? ? tX ~ Neg. Binomial? m’= m, P’= i m ? . ? ?

)

(

)

26

More accessible environments do appear to lower automobile ownership, reduce total travel distances, and shift mode of travel to slower modes (such as bus and walking), as described by Kockelman (1996 & 1997). The aggregation of trips into the broad categories asked for in surveys (e.g., recreational vs. shopping trips) obscures the subtle but important distinctions across activities and renders travel times endogenous. In theory, if one had a large enough sample of observations, one could model a system of activity demands where essentially every destination-and-mode (and time-of-day!) combination was a possible “good” to be consumed by everyone residing in the region. Travel costs would be fairly obvious (given inter-zonal travel times and distances), and with regional data one could ensure that an individual’s responses to a limited survey would not bias his/her vectors of travel costs while implicitly controlling for quality- and price-of-activity differences. It merits mention that much of Zahavi’s work (e.g., Zahavi 1979a) measures utility by total distance traveled, essentially asserting that it is access to opportunities that determines one’s welfare – an idea similar to those discussed here. However, distance may be a seriously flawed utility measure; for example, who can say with certainty that several short-distance journeys are preferred to a few longer

27

28

68

journeys, just because the first choice involves less distance? The approach advocated in this dissertation allows the data to interpret preferences far more flexibly than a distance metric.

29

The survey used, like most surveys of significant size, does not provide income-per-worker or hourly wage information, so analysis at the individual level would not have been feasible. The travel times referred to in this section are not as reported by survey respondents (who tend to report times in increments of five minutes); instead, they come from interzonal free-flow automobile-travel times provided by the region’s metropolitan planning organization, which is the Metropolitan Transportation Commission in the case of the Bay Area Travel Surveys.

30

69

Chapter Four: Empirical Estimation and Model Validation

Estimation Techniques

Likelihood Maximization The likelihood maximization relies on S-Plus statistical computing software (produced by MathSoft Co.), using a model/trust-region approach described by Gay (1983). Due to the constraint of strict positivity on the Poisson rates ( X i ) and the complexity of the likelihood equations’ first and second derivatives 1, the algorithm employs its own, numerical approximations to the derivatives utilized in a quadratic approximation to the likelihood function over an iterative series of neighborhoods it “trusts”2. Acquiring Starting Parameter Values When a model’s regression equations are not linear in unknown parameters, as is the case for the functional forms considered here, the choice of a method to achieve starting parameter estimates can be quite difficult. This is particularly true when negative estimates of the marginal utility of time and negative demand estimates are effectively disallowed (as discussed in the section on non-negativity in Chapter Three). In the present specification, one could run an iterative maximum-likelihood search procedure on each demand equation individually, using negative binomial stochastic assumptions; however, such individual regressions require their own sets of feasible start values for many of the unknown parameters, along with a likelihood search, without guarantee of unique convergence.

*

70 To begin the maximum-likelihood search routines in the models estimated here, several different sets of parameter estimates were attempted for each system of equations as a whole, with theory often guiding the choice of sign (e.g., the derivative of indirect utility with respect to each travel time should be negative, so signs on the starting values of prominent coefficients of travel times were chosen appropriately). Section A-5 of the Appendix provides actual code and starting values used in typical program files. Fortunately, not too many parameter sets were required to find a feasible set from which to begin the iterative search in each model. Additionally, several very distinct but feasible starting parameter sets were used on each model, and the likelihood values of their solutions were compared in an attempt to avoid convergence to a local, rather than global, maximum. Variance-Covariance Estimation The model complexities complicate the estimation of errors in estimation, so the specification of the log-likelihoods’ gradients and Hessians is not elementary. To facilitate computations, the Berndt et al. (1974) method (BHHH) of Hessian estimation is used, requiring only gradient information. The variance-covariance matrix for the parameter set is thus estimated applying the following equation, but employing the parameter estimates generated from the sampled observations (Greene 1993, p. 326):

AsymptoticVariance ? Covariance Matrix =

?1 ? ?n = N ? ?1 ’? $ ? AVC θ = E ? ? ∑ wn wn ? ? = E [W ′W ] , ? ? ? ? ? n =1

()

(

)

(Chapter

$ is the MLE , w = Gradient of Likelihood of n’th observation, where θ n and W = Matrix whose rows are the wn ’s. Four:-1)

71

Results to be Estimated

To limit the problem size while analyzing a variety of different functional specifications and illustrating use of the method, only four demand types were distinguished here; and they are used with all four model types (Type 1 through Type 4, as specified in Chapter Three). Specifically, the four demand types used in four sets of results are the number of discretionary trips (i.e., non-work, non-education, and nonserve-passenger trips) made to each of four iso-opportunity contours using the Bay Area Travel Surveys. The contours are defined, as discussed in Chapter Three, by (free-flow, automobile) travel times from a household’s specific “neighborhood”/traffic analysis zone (TAZ) out to contours of 60,000, 300,000, 900,000, and two million jobs in the region. The fifth set of results presented here also relies on just four distinct demand types across the four all-jobs iso-opportunity contours described, and it uses the Type 3 model specification; however, it explicitly accommodates trip chaining by focusing on activity tours, rather than individual stops/single activities. These five sets of results are described here now.

Results of Type One Model: Modified Linear Expenditure System

The estimates for a Type 1 model of discretionary-activity participation (represented by Equation 3-14) are shown in Table 4-1a; these represent the likelihood-maximizing set over the full sample (N=10,834) after starting from a variety of parameter values. The median levels of first-order estimates of demand elasticities and the value of time for this model are shown in Table 4-1b. Even though the great majority of the parameter estimates appear to be highly statistically significant in this model, the value-of-time and aggregate mean estimates

72 differ significantly from expectations. For example, the quartiles of the value-of-time estimates are all negative across the sample; these estimates are presented in Table 4-7, along with value-of-time estimates from other models, in the section titled “Discussion of Value of Time Estimates”, toward the end of this chapter. There are other clues that the model is far off. For example, the average X 1 across households is 2.67 trips/day, which is more than twice the observed mean of 1.09 trips/day, and the ratio of the sum of X 2 to X 3 is 2.25 while the model predicts 1.55. Thus, this model does not appear to be sufficiently flexible for our purposes, and its results should not be taken too seriously.

*

73 Table 4-1a Parameter Estimates of Modified Linear Expenditure System, as applied to Discretionary-Activity Participation across Four All-Jobs Iso-Opportunity Contours, using the BATS data set Parameter Final Standard TP-Values Estimates Errors Statistics 0.902 1.94E-01 46 0.000 0.0929 3.26E-02 28 0.000 0.160 4.68E-02 34 0.000 0.0947 2.16E-02 44 0.000 0.0580 3.15E-02 18 0.000 -4.39E-05 2.24E-06 -20 0.000 -5.06E-05 1.74E-06 -29 0.000 -3.21E-05 3.06E-07 -105 0.000 -2.99E-05 6.23E-07 -48 0.000 -2.18E-01 3.79E-02 -5.8 0.000 -6.72E-02 1.52E-02 -4.4 0.000 -8.91E-02 7.19E-02 -1.2 0.215 2.69E-02 5.50E-02 0.5 0.625 5.14E-02 1.87E-02 2.7 0.006 -1.15E-01 1.66E-02 -7.0 0.000 5.49E-02 7.41E-03 7.4 0.000 1.95E-01 2.54E-03 77 0.000 -5.93E-02 2.10E-03 -28 0.000 9.96E-03 3.66E-04 27 0.000

α α1 α2 α3 α4 β1Y β2Y β3Y β4Y β11 β12 β13 β14 β22 β23 β24 β33 β34 β44

L = -46,696 N = 10,834

74 Table 4-1b Economic Implications of Modified Linear Expenditure System, as applied to Discretionary-Activity Participation across Four All-Jobs Iso-Opportunity Contours, using the BATS data set

VALUE ESTIMATED: Discretionary-Time Elasticity of Demand: Immediate Zone Demand Near Zone Moderate Zone Far Zone Income Elasticity of Demand: Immediate Zone Demand Near Zone Moderate Zone Far Zone Cross-Time Demand Elasticities: w/r/t Time of: Immediate Zone: Near Zone: Moderate Zone: Far Zone: Median of Sample 0.389 0.672 0.572 0.304 0.044 0.163 0.411 0.696

Immediate -0.054 0.016 0.284 -0.216

Near 0.081 -0.725 0.814 -0.658

Moderate 0.234 0.575 -2.711 -0.209

Far -0.191 -0.617 1.021 -0.541

Notes: Demands are in trips per day, Discretionary Time is hours, Travel Times are minutes, & Income is before-tax dollars.

Results of Type Two Model: Modified Translog

The translog functional form of Equations 3-15 and 3-16 was used, with the expectation that its larger parameter set would provide more flexible estimation and better results than that of the modified linear expenditure system. For purposes of parameter identification, this model’s γ

TY

parameter was fixed to equal positive one and the β ij ’s

are constrained to equal β ji ’s. The parameter estimates are shown in Table 4-2a. The log-likelihood value for this estimation is -46,431.6, but it cannot be compared with a no-information situation (where each X i is modeled as a constant, independent of time and income information) or even a full-information situation (where each

*

75 household’s X i ,n is modeled as its own constant), because there are no free constants in this model – every unknown parameter is interacted with explanatory variables which vary across households. Observe that the variance in the data is substantially reduced by using explanatory information. For example, the estimate of the overdispersion parameter, α, falls from 1.6383 (for total trips) to 1.001 here, signaling a tighter distribution thanks to explanatory information and the model structure itself. The average X i estimates of this model fall much closer to the sample means than those of the modified linear expenditure system, suggesting much better accuracy in aggregate prediction. The average X i ’s are estimated to be 1.14, 0.66, 0.29, and 0.19, respectively, while the observed per-day average demands are 1.09, 0.62, 0.27, and 0.19. Theoretical considerations aside, this model appears to predict aggregate behavior well.

* *

*

76 Table 4-2a Parameter Estimates of Modified Translog Model, as applied to DiscretionaryActivity Participation across Four All-Jobs Iso-Opportunity Contours, using the BATS data set Parameter Final Standard TP-Values Estimates Error Statistics 1.00 0.00 123 0.000 1.53 1.29 1.2 0.238 -8.35 1.96 -4.3 0.000 -9.69 1.74 -5.6 0.000 -3.52 1.55 -2.3 0.023 -7.96 0.91 -8.7 0.000 -0.643 0.24 -2.7 0.007 -1.05 0.24 -4.4 0.000 -0.202 0.19 -1.1 0.286 -1.03 0.51 -2.0 0.041 -3.90 0.60 -6.5 0.000 0.858 0.29 3.0 0.003 8.47 1.02 8.3 0.000 -2.27 0.40 -5.7 0.000 1.95 0.48 4.1 0.000 1.60 0.19 8.6 0.000 1.15 0.16 7.2 0.000 -0.576 0.10 -6.0 0.000 -0.836 0.12 -6.8 0.000 -0.178 0.10 -1.9 0.064 0.913 0.23 4.0 0.000 1.44 0.26 5.6 0.000 1.50 0.28 5.4 0.000 1.00 n/a n/a n/a

α α1 α2 α3 α4 β11 β12 β13 β14 β22 β23 β24 β33 β34 β44 γ1Y γ2Y γ3Y γ4Y γ1T γ2T γ3T γ4T γTY (fixed)

L = -46,432 N = 10,834

77 Economic Implications of the Type Two Model Results This model’s estimates’ of elasticities are shown in Table 4-2b. Overall, this model’s results appear reasonable, including the value-of-time estimates across the household sample (whose quartiles are provided in Table 4-7). Discretionary-time elasticities are positive, as one would expect (i.e., more discretionary time available to the household leads to more discretionary activity participation). Income elasticities, on the other hand, are positive for far and moderate zone activities but negative for closer activities; it appears that money is spent on access to consumption of activities further away, rather than near one’s home. It is interesting that near trips are not found to be “inferior” with respect to time, but they are with respect to income (albeit to a minor extent). Note that these results are not definitive because part of this income effect is due to the purchase of automobiles, which effectively reduce per-trip marginal costs and travel times, and part is arguably due to the higher-income households having more specialized workers who must travel further on workdays and so undertake more activities at sites remote from home, but near their work locations. The presentation of the fifth set of model results more explicitly considers this question of trip chaining. Finally, observe that own-travel-time elasticity estimates are generally negative as one would expect of economically “normal” goods, but not for the nearest zone’s activity participation rates. And most cross-time elasticities are positive, suggesting the expected substitution effects (rather than complementarity), since the demands are only defined across “quality” here (i.e., level of opportunity choice), not activity type (e.g., social and personal business activities are less likely to be substitutable).

78 Table 4-2b Economic Implications of Modified Translog Model, as applied to Discretionary Activity Participation across Four All-Jobs Iso-Opportunity Contours, using the BATS data set

VALUE ESTIMATED: Discretionary-Time Elasticity of Demand: Immediate Zone Demand Near Zone Moderate Zone Far Zone Income Elasticity of Demand: Immediate Zone Demand Near Zone Moderate Zone Far Zone Cross-Time Demand Elasticities: w/r/t Time of: Immediate Zone: Near Zone: Moderate Zone: Far Zone: Median of Sample 1.028 0.869 0.678 0.706 -0.294 -0.208 0.086 0.122

Immediate 0.257 0.100 0.242 0.047

Near 0.061 -0.891 0.832 -0.207

Moderate 0.103 0.496 -2.952 0.383

Far -0.033 -0.188 0.442 -1.446

Notes: Demands are in trips per day, Discretionary Time is hours, Travel Times are minutes, & Income is before-tax dollars.

79

Results of Type 3 Model: Modified Translog with Constants

The Type 3 model specification, a modified translog which includes constant terms, (as shown in Equation 3-18) has been applied here to two different demand sets. The first covers the discretionary-activity participation demands used in the previous two models; the second looks at home-based tours of all trip types. Both rely on the four isoopportunity contours used previously, which count all job types as opportunities. I. Discretionary Activity Participation The estimated parameters for a Type 3 model across four divisions of discretionaryactivity participation are shown in Table 4-3a. The estimate of the overdispersion parameter α has dropped to 0.938, suggesting that estimates are falling closer to observations than in the two previous models; and the demand estimates accurately estimate aggregate behavior. The average X i ’s are estimated to be 1.08, 0.62, 0.28, and 0.19, respectively, while the observed per-day average demands are 1.09, 0.62, 0.27, and 0.19. An advantage of the translog specification with constant terms is that one can nest a no-information case within the specification. Table 4-3c provides a summary of the likelihood values resulting from a variety of specifications linked to this particular specification. The log-likelihood value for the no-information case4 is -47,688; in contrast, the log-likelihood of the full model is -46,218. The p-value for the hypothesis that the no-information model is the proper model, given the assumption that this third model specification encompasses the true model as a nested specialization, is 0.000; so one must reject this hypothesis (given the assumption).

*

80 Table 4-3a Parameter Estimates of Modified Translog Model with Intercept Terms, as applied to Discretionary Activity Participation across Four All-Jobs IsoOpportunity Contours, using the BATS data set Parameter Final Standard TP-Values Estimates Errors Statistics 0.938 0.020 47 0.000 0.267 0.049 5.5 0.000 0.930 0.234 4.0 0.000 0.189 0.062 3.1 0.002 -0.075 0.026 -2.9 0.004 -0.033 0.030 -1.1 0.266 -0.831 2.306 -0.4 0.718 -1.066 3.325 -0.3 0.749 -1.112 2.676 -0.4 0.678 -0.944 5.668 -0.2 0.868 -4.191 1.159 -3.6 0.000 0.051 0.329 0.2 0.876 -1.453 0.414 -3.5 0.000 0.714 0.268 2.7 0.008 -0.123 1.287 -0.0 0.924 -4.464 1.007 -4.4 0.000 2.237 0.537 4.2 0.000 7.845 1.513 5.2 0.000 -2.842 0.709 -4.0 0.000 -0.200 1.658 -0.1 0.904 1.491 0.292 5.1 0.000 0.078 0.141 0.6 0.579 -1.126 0.216 -5.2 0.000 -1.187 0.230 -5.2 0.000 -0.813 0.146 -5.6 0.000 0.262 0.310 0.8 0.398 1.443 0.435 3.3 0.000 1.888 0.521 3.6 0.000 1.000 n/a n/a n/a

α ?0 ?1 ?2 ?3 ?4 α1 α2 α3 α4 β11 β12 β13 β14 β22 β23 β24 β33 β34 β44 γ1Y γ2Y γ3Y γ4Y γ1T γ2Y γ3Y γ4Y γTY (fixed)

L = -46,219 N = 10,834

Table 4-3b

81 Economic Implications of Modified Translog Model with Intercept Terms, as applied to Discretionary Activity Participation across Four All-Jobs IsoOpportunity Contours, using the BATS data set

VALUE ESTIMATED: Discretionary-Time Elasticity of Demand: Immediate Zone Demand Near Zone Moderate Zone Far Zone Income Elasticity of Demand: Immediate Zone Demand Near Zone Moderate Zone Far Zone Cross-Time Demand Elasticities: w/r/t Time of: Immediate Zone: Near Zone: Moderate Zone: Far Zone: Median of Sample 0.774 0.652 0.440 0.415 -0.206 -0.039 0.144 0.132

Immediate 0.574 0.020 0.252 -0.073

Near -0.014 -0.630 0.688 -0.316

Moderate 0.124 0.440 -2.608 0.347

Far -0.144 -0.303 0.384 -1.231

Notes: Demands are in trips per day, Discretionary Time is hours, Travel Times are minutes, & Income is before-tax dollars.

Economic Implications of the Type Three, Discretionary Activities Model Results Estimates of various economic implications of this model are shown in Table 4-3b. While elasticity signs and magnitudes appear to be in general agreement with those estimated for the previous, no-constant-terms model for these data, the value of time estimates differ dramatically. Even though this model is more flexible (and offers a significantly higher log-likelihood value, of -46,218 versus -46,432, for a difference of just five parameters), its value-of-time estimates are highly negative and thus contrary to expectations – in clear contrast to the value-of-time results for the previous model. This

82 model’s value-of-time results are provided and discussed along with the value-of-time results for other models, toward the end of this chapter. Model Comparisons: Case Example Ten variations of this Type 3 system of discretionary-activity demands were run for purpose of likelihood comparisons. They differ primarily in their assumptions about the stochastic nature of the unobserved heterogeneity; but, the “no-information” and “fullinformation” variants make assumptions about the access to explanatory information. A broad comparison of likelihoods like this can be done for any of the modified-translogwith-intercept models since their specification includes intercepts, allowing them to be rigorous nested within a full-information case and over a no-information case. Note, however, not all of the ten variants are specialization’s or generalizations of all others. All ten cases are described briefly and compared by means of their log-likelihood values in Table 4-3c. Several interesting results emerge from these log-likelihood values. One is that the imposition of the same-gamma heterogeneity assumption significantly constrains the model as estimated. Without removing any estimated parameters yet allowing optimal activity-participation rates to vary independently of one another (given their population means, for a given set of household characteristics), the log-likelihood rises dramatically, from -46,218 to -42,592, given a difference of just three identifying restrictions5. Following this change in stochastic specification, the overdispersion factor, α or “a” in the table, rises substantially as well, from 0.938 across the set of trips to 1.5 for “immediate” or very local trips, 2.2 for “near” trips, 2.8 for “moderate” trips, and 7.4 for “far” trips. These results suggest that the imposition of the same-gamma assumption for

83 characterizing the unobserved heterogeneity in optimal participation rates is too strong – at least for the data set used, where short-duration observations are likely to be more heavily influenced by chaining and intra-household trip coordination. More flexible models, which still provide for some correlation in unobserved information, should prove useful, though these likelihoods will probably require simulation. Table 4-3c Comparison of Log-Likelihood Values across Different Models based on the Modified Translog Model with Intercepts (analyzing Discretionary-Activity Participation across Four Iso-Opportunity Contours, using the BATS data set)

Model Description: i. Poisson (no unobserved heterogeneity,α=0): ii. No Information (no explanatory variables): iii. MODEL AS ESTIMATED (same-gamma heterogeneity): iv. Semi-Independent Negative Binomials (with same overdispersion "α"): v. Totally Independent Negative Binomials (different "α"’s): vi. Full Information (all optimal rates = observed rates, minimized variance): vii. Individually Estimated Negative Binomials for each Demand

(without cross-equation parameter constraints):

α

0 1.012 0.938 2.13 1.5 to 7.4 0

Log-Likelihood -52,052 -47,688 -46,218 -42,981 -42,592 -32,749

Immediate Trips: Near Trips: Moderate Trips: Far Trips:

1.51 2.20 3.04 6.18 Sum =

-16,640 -12,538 -7,635 -5,864 -42,677

Another way to look at these results is to compute the fraction of total likelihood difference, between the full- and no-information cases (cases vi and ii), that is “explained” by the specified model (case iii). This ratio is often referred to as a pseudoR2, and it is 9.84% for this model6. This is actually higher than one might expect, given the disaggregate and short-term nature of the data. As mentioned in Chapter Two’s literature review, little if any research has found significant elasticities of trip demand with respect to travel times (Ortúzar and Willumsen 1994, Hanson and Schwab 1987). The percentage of explained variation in models of single-day trip-making, other than the

84 model proposed in this research, tend to be on the order of five percent (e.g., Hanson and Schwab 1987). However, one should not put too much stock in this measure of explained variation; simply a longer survey period with fewer zero-level observations of demand would increase the percentage (or R-squared), without any change in parameter estimates, because the zeros do not diminish the full-information log-likelihood at all7. Another point of interest is that the removal of the cross-equation parameter constraints does not do much for the log-likelihood. In case vii, where each demand is estimated completely independently of the others (but with the same general functional form given in Equation 3-18), 18 more parameters are being estimated than in the set of demand equations derived from case v’s single indirect utility specification8 (of Equation 3-17); yet this only translates to a likelihood increase of -42,592-(-42,677), or 85. This difference still provides for a highly statistically significant likelihood ratio test of the difference9, but the magnitude of the difference appears small when compared with the differences other changes in the model create. For as many observations as there are in the data set (10,834 households times four dependent-value observations per household), it is not surprising that one would get a statistically significant result for most tests; what is surprising is the relatively small size of this difference for this particular test. It suggests that the derivation of a set of demands from a single indirect utility specification is not so presumptuous or limiting! However, the value-of-time results remain unbelievable, so the model structure is imperfect. II. Modeling Tours Explicitly Due to the prevalence of trip-chaining or “tour-making” in many observations of activity-participation, the incremental travel time faced by a household to pursue an

85 added activity can be substantially less than the round-trip travel time from home. Since most tours appear to involve a primary stop or leg with a significant travel time from the home location, the data set of demands constructed for analysis here is based on the number of tours made, with the furthest destination visited during the tour determining the tour type (according to which of the four distinct iso-opportunity contours the tour belongs). Since the number of tours that are exclusively non-work related is rather small (about 15,000 tours in the BATS data set) and many of the tours containing a work purpose also contain discretionary-purpose stops, tours of all types were assembled here for analysis, providing roughly 40,000 tours across the BATS households surveyed. Estimation results are given in Tables 4-4a and 4-4b.

86 Table 4-4a Parameter Estimates of Modified Translog Model with Intercept Terms, as applied to Trip Tours to Four All-Jobs Iso-Opportunity Contours, using the BATS data set Parameter Final Standard TP-Values Estimates Errors Statistics 0.076 0.004 20.7 0.000 -0.079 0.035 -2.3 0.022 1.339 0.412 3.3 0.001 0.339 0.120 2.8 0.005 0.187 0.073 2.6 0.010 0.127 0.097 1.3 0.189 -0.044 2.847 -0.0 0.988 -0.931 4.114 -0.2 0.821 -1.101 5.650 -0.2 0.845 -0.729 14.639 -0.0 0.960 -2.702 1.226 -2.2 0.028 1.231 0.510 2.4 0.016 -0.835 0.522 -1.6 0.110 0.812 0.437 1.9 0.063 -0.000 1.594 -0.0 1.000 -4.450 1.457 -3.1 0.002 3.012 0.920 3.3 0.001 7.866 2.696 2.9 0.004 -1.974 1.006 -2.0 0.050 0.445 4.695 0.0 0.924 -0.032 0.119 -0.3 0.789 -0.625 0.186 -3.4 0.000 -1.506 0.378 -4.0 0.000 -2.833 0.722 -3.9 0.000 1.225 0.575 2.1 0.033 1.225 0.575 2.1 0.033 2.343 0.862 2.7 0.007 4.959 1.607 3.1 0.002 1.000 n/a n/a n/a

α ?0 ?1 ?2 ?3 ?4 α1 α2 α3 α4 β11 β12 β13 β14 β22 β23 β24 β33 β34 β44 γ1Y γ2Y γ3Y γ4Y γ1T γ2Y γ3Y γ4Y γTY (fixed)

L = -48,469 N = 10,834

87 Table 4-4b Economic Implications of Modified Translog Model with Intercept Terms, as applied to Trip Tours to Four All-Jobs Iso-Opportunity Contours, using the BATS data set

VALUE ESTIMATED: Discretionary-Time Elasticity of Demand: Immediate Zone Demand Near Zone Moderate Zone Far Zone Income Elasticity of Demand: Immediate Zone Demand Near Zone Moderate Zone Far Zone Cross-Time Demand Elasticities: w/r/t Time of: Immediate Zone: Near Zone: Moderate Zone: Far Zone: Median of Sample 0.970 0.956 0.848 0.704 -0.022 0.039 0.123 0.191

Immediate 0.356 -0.157 0.052 -0.092

Near -0.139 -0.400 0.407 -0.259

Moderate 0.015 0.397 -1.344 0.093

Far -0.196 -0.433 0.070 -0.711

Notes: Demands are in trips per day, Discretionary Time is hours, Travel Times are minutes, & Income is before-tax dollars.

Observe that the overdispersion factor α is very close to zero here, suggesting less of a negative binomial and more of a Poisson distribution; this reduced value also suggests more stability in estimation thanks to less unobserved variation (assuming that a Poisson holds). So tour-making may be less variable than individual stop-making, which makes some sense given the fixed cost of getting ready to leave one’s home and take care of business and activities outside one’s home; the marginal cost of adding stops is relatively small once is already “out and about”. Moreover, the same-gamma-error assumption may apply better here because gross estimates of the α terms10 for the different, individual

88 demands are much more stable; they are 0.84, 0.85, 0.70, and 0.52, rather than 1.7, 2.6, 3.4, and 7.6, as estimated for individual-activity (non-tour) demands. This model’s estimates of demand are reasonable predictors of aggregate behavior. The average X i ’s are estimated to be 1.22, 0.57, 0.37, and 0.28, respectively, while the observed per-day average trip-chain rates to the different contours are 1.14, 0.64, 0.38, and 0.32. In Table 4-7, the value of times estimated for this model are negative, though they are not as extreme as those implied by the previous translog-with-constants model of discretionary trip-making. The travel-time elasticity matrix (shown in Table 4-4b) resembles earlier estimates of this matrix, but three of the four income elasticities are now positive.

*

Results of Type 4 Model: Modified Translog with Constants, Using Wage and Total Time Data

As discussed in Chapter 3, the work decision, and thus the income and discretionarytime budgets are likely to be made simultaneously with the discretionary-activity decisions. Thus, a model where these budgets are endogenous may prove useful. The Type 4 model accommodates these decisions by relying on wage and total-time data, rather than income and discretionary-time data, but the demand set analyzed is the same, four-zone discretionary-trip data set analyzed in the first three models discussed here. The results of this analysis are shown in Tables 4-5a and 4-5b.

89 Table 4-5a Parameter Estimates of Modified Translog Model with Intercept Terms and Wage and Total-Time Information, as applied to Discretionary-Activity Participation across Four All-Jobs Iso-Opportunity Contours, using the BATS data set Parameter Final Standard TP-Values Estimates Errors Statistics 0.935 0.020 47 0.000 α 0.387 5.529 0.0 0.944 ?H 3.212 1.537 2.1 0.037 ?1 0.528 0.270 2.0 0.050 ?2 -0.361 0.174 -2.1 0.038 ?3 -0.192 0.117 -1.6 0.100 ?4 3.564 5.456 0.7 0.514 α1 -3.332 6.137 -0.5 0.587 α2 -4.253 5.436 -0.8 0.434 α3 -2.461 11.412 -0.2 0.829 α4 4.211 2.565 1.6 0.101 β11 0.841 0.861 1.0 0.329 β12 -4.060 2.088 -1.9 0.052 β13 1.978 1.231 1.6 0.108 β14 -0.270 2.660 -0.1 0.919 β22 -4.443 2.480 -1.8 0.073 β23 -2.426 1.432 -1.7 0.090 β24 2.300 2.421 1.0 0.342 β33 -6.493 3.401 -1.9 0.056 β34 -2.749 3.779 -0.7 0.467 β44 0.527 0.377 1.4 0.162 γ1W 0.586 0.437 1.3 0.179 γ2W -0.223 0.203 -1.1 0.273 γ3W -2.267 1.132 -2.0 0.045 γ4W 1.000 n/a n/a n/a γ1H (fixed) 3.948 1.520 2.6 0.009 γ2H 5.084 2.238 2.3 0.023 γ3H 4.935 2.174 2.3 0.023 γ4H 4.580 2.318 2.0 0.048 γWH 1.617 0.246 6.6 0.000 γH

L = -46,267 N = 10,834

90 Table 4-5b Economic Implications of Modified Translog Model with Intercept Terms and Wage and Total-Time Information, as applied to Discretionary-Activity Participation across Four All-Jobs Iso-Opportunity Contours, using the BATS data set

VALUE ESTIMATED: Total-Time Elasticity of Demand: Immediate Zone Demand Near Zone Moderate Zone Far Zone Wage Elasticity of Demand: Immediate Zone Demand Near Zone Moderate Zone Far Zone Cross-Time Demand Elasticities: w/r/t Time of: Immediate Zone: Near Zone: Moderate Zone: Far Zone: Median of Sample 0.705 0.554 0.379 0.435 -0.077 -0.079 -0.031 0.104

Immediate 0.850 -0.058 0.297 -0.146

Near -0.100 -0.567 0.291 0.117

Moderate 0.180 0.169 -1.878 0.383

Far -0.181 0.065 0.434 -1.716

Notes: Demands are in trips per day, Discretionary Time is hours, Travel Times are minutes, & Income is before-tax dollars.

Predictions of aggregate behavior using this model do not appear to be as strong as those from the Type 3 model estimates, but they are quite reasonable. The average X i ’s are estimated to be 1.07, 0.62, 0.28, and 0.19, respectively, while the observed per-day average demands are 1.09, 0.62, 0.27, and 0.19. The travel-time elasticity matrix corresponds roughly with those estimated previously and total-time elasticities are positive, as expected. However, the wage elasticities are generally negative and negligible, except for the furthest zone. One might expect more significantly negative wage effects on discretionary trip-making as workers choose to

*

91 work more and engage in fewer discretionary activities (during weekdays at least). However, the act of working more often may add to discretionary activity participation because of added purchasing power and because work travel can put workers in contact with many activity sites (along travel routes to and from work) for lower travel-time costs than home-based trips.11

Comparison of All Model’s Elasticity Estimates

To facilitate comparisons, Table 4-6 provides a summary of elasticities estimated for all five of the model specifications analyzed. The reported values are the median values for the 10,834-household sample, and only the first three models are strictly comparable in terms of all estimates shown, since their response and explanatory variables sets are the same. As described earlier, the fourth model analyzed relies on the same functional form for demand as the third, but it models trip tours to all activity types, rather than individual, discretionary-activity stops. The fifth model allows for work-time (and, therefore, much of discretionary-time) and income endogeneity, so its reported elasticities are with respect to total time and wage variables.

92 Table 4-6 Summary of Elasticity Estimates: Median Values across Households

Model Used: Type 1 Type 2 Type 3 Type 3*

(All Tours)

Type 4**

(Endogen. Work)

Discretionary/Total**-Time Elasticity of Demand: Immediate Zone Demand Near Zone Moderate Zone Far Zone Income/Wage** Elasticity of Demand: Immediate Zone Demand Near Zone Moderate Zone Far Zone Cross-Time Demand Elasticities: Demand for Activities in: Immediate Zone: Near Zone: Moderate Zone: Far Zone: Immediate Zone: Near Zone: Moderate Zone: Far Zone: Immediate Zone: Near Zone: Moderate Zone: Far Zone: Immediate Zone: Near Zone: Moderate Zone: Far Zone:

0.389 0.672 0.572 0.304 0.044 0.163 0.411 0.696

1.028 0.869 0.678 0.706 -0.294 -0.208 0.086 0.122

0.774 0.652 0.440 0.415 -0.206 -0.039 0.144 0.132

0.970 0.956 0.848 0.704 -0.022 0.039 0.123 0.191

0.705 0.554 0.379 0.435 -0.077 -0.079 -0.031 0.104 Type 4** 0.85 -0.06 0.30 -0.15 -0.10 -0.57 0.29 0.12 0.18 0.17 -1.88 0.38 -0.18 0.06 0.43 -1.72

Type 1 Type 2 Type 3 Type 3* w/r/t to Travel Time to Immediate Zone: -0.05 0.02 0.28 -0.22 0.08 -0.73 0.81 -0.66 0.23 0.57 -2.71 -0.21 -0.19 -0.62 1.02 -0.54 0.26 0.10 0.24 0.05 0.06 -0.89 0.83 -0.21 0.10 0.50 -2.95 0.38 -0.03 -0.19 0.44 -1.45 0.57 0.02 0.25 -0.07 -0.01 -0.63 0.69 -0.32 0.12 0.44 -2.61 0.35 -0.14 -0.30 0.38 -1.23 0.36 -0.16 0.05 -0.09 -0.14 -0.40 0.41 -0.26 0.02 0.40 -1.34 0.09 -0.20 -0.43 0.07 -0.71

w/r/t to Travel Time to Near Zone:

w/r/t to Travel Time to Moderate Zone:

w/r/t to Travel Time to Far Zone:

Notes: *The first three models and the fifth model discretionary activity participation; the fourth models trip tours and includes all activities. **The fifth model allows for discretionary-time and income endogeneity, while the others do not. Demands are in trips per day, Discretionary Time is hours, Travel Times are minutes, & Income is before-tax dollars.

Of the first three, comparable models, signs on estimates are strongly consistent between the second and third models; but the first model, which relies on a modified version of the linear expenditure functional form, is not so consistent with these two. As described briefly in Chapter Three and earlier in this chapter, the first model suffers from

93 several constraints on its predictions – and the second and third specifications have their own inflexibilities. Despite their differences, all models predict strong elasticities of demand with respect to time budgets; however, only the second model produces an estimate exceeding one. The own-travel-time elasticities of demand tend to be significantly negative, with elasticities for demand for activities in the moderately distant contour estimated to be the most notably inelastic. Additionally, though generally positive, many cross-time effects on demand to the furthest iso-opportunity contour are estimated to be negative. All five models predict rather negligible elasticities of discretionary-activity and tour demands with respect to income and wages. This effect may be due to a lack of identification of all income contributions to indirect utility (as discussed earlier), but, also, it may be that additional money does not lead to additional activity participation, everything else constant. For example, the quality of activities and the amount of money spent on them may be substantially affected by income and/or wages, but rates of activity participation may not change much, given fixed time constraints. More detailed data sets, including expenditure and price information, may resolve this question.

Discussion of Value of Time Estimates

In contrast to the reasonable and rather stable elasticity results, the value-of-time estimates vary substantially across the five models. The quartiles of the sample’s valueof-time estimates are provided in Table 4-7.

94 Table 4-7 Summary of Value-of-Time Estimates12: Quartiles across Households

Model Used: Type 1 Type 2 Type 3 Type 3

(All Tours)

Type 4**

(Endogen. Work)

Value of Discretionary/Total** Time: Minimum First Quartile Median Third Quartile Maximum

$ (10.73) $ 0.22 $(7.67E+6) $ (82.11) $ 0 $ (8.92) $ 7.94 $ (187.62) $ (13.92) $ 11.72 $ (8.06) $ 13.13 $ (102.80) $ (9.09) $ 26.17 $ (6.22) $ 21.47 $ (58.95) $ (5.61) $ 42.66 $ (1.98) $ 151.80 $ 2.42E+6 $ (0.16) $ 384.90

All values are before-tax, 1990 dollars per hour. **The fifth model allows for discretionary-time and income endogeneity, while the others do not. Thus, the fifth model’s value of time elasticities are with respect to total time and wage.

While of the expected order of magnitude, the signs are opposite of one’s expectations in three of the five models! As mentioned in Chapter Three’s specification of the first model, the modified linear expenditure system’s structure is so limiting that its value-of-time estimates depend only on travel times – rather than income and discretionary time levels. This reasoning may explain a large part of the unexpected results for the first of the five models. But what about the other negative estimates? Isn’t a modified translog form flexible enough to provide a first-order estimate of the value of time? A Functional Conflict between Behavioral Indicators One interesting but restrictive consequence of the modified-translog specification is that there may be some conflict between the income elasticity signs and those on the marginal utility of income (dv/dY); both depend on the γ iY and γ

TY

terms, and for one to

be positive the other is likely to wind up negative. Equation 4-2 illustrates the conflicting incorporation of these parameters for the Type 3 model specification. In this set of equations, note how the final term of the income elasticity is implicitly negative and the

95 first term has a negative denominator; therefore, it is just the γ iY parameter that has flexibility to determine the sign on income elasticity for the ith demand.

? ? ?1 ? ? ? ∑ γ jT ln(t j ) + γ TY ln(Y )? ?o + ? T ? d ?? j ? = ? ? 1 ? ∑ γ ln(t ) + γ ln(T )? jY j TY d Y ? j ?

Value of TimeType 3

( )

=

v Td Marginal Utility of Income

,

(Chapter

dX i * γ γ Y Income Elasticity of Demand i = × * = iY ? TY , dY t i v ti Td v Td Xi where Td > 0, t i > 0, γ TY = + 1 ( fixed , a priori , for model identifiability ), v ti = Marginal Utility of Travel Time < 0, and v Td = Marginal Utility of Discretionary Time > 0.

Four:-2) The same conflict holds true for the less flexible, Type 2 Model. However, its valueof-time results happen to be much more in line with expectations here13. One should be wary of these ostensibly flexible specifications, and further functional flexibility may be sought where practical14. But something more fundamental to the structure of the models may be causing the unanticipated results. Identification of All Income/Wage Terms in Indirect Utility In fact, the primary reason for a negative marginal-utility-of-time result may be that the indirect utility functions underlying the estimated models are limited in their representation of income (or wage) effects. Everywhere one finds an income term (Y) [or wage term (w)] in the different model specifications, it is interacted with either a travel time or available-time-budget term, allowing immediate estimation of the assumed indirect utility function from the results of the system-of-demands estimation. However,

96 if there are other, isolated income (or wage) effects, in the form of g(Y) (for example, log(Y ) 2 ), these will impact the marginal-utility-of-income estimates and thus the value-of-time estimates. If one is relying on a system of demand equations derived from the application of Roy’s Identity in a time environment, one can only identify the magnitude of the effects that are available from derivatives of indirect utility with respect to time and available time. In order to identify the purely income (or wage) effects (or these effects interacted with the vector of prices, which are assumed not to vary across households and thus show up as constants or concealed within fundamental parameters in the regression equations), one needs observable information based on these effects. Essentially then, one needs a system of demands derived from application of Roy’s Identity in a money/price environment so that parameters characterizing the derivatives of indirect with respect to income (or wage) are all present. Assume then that one has the system of demand equations as developed from the negative ratios of the derivatives of indirect utility with respect to (invariable) prices and income. The entire model should be estimated in a simultaneous fashion, so that the estimates of optimal demand levels developed in the time setting equal those developed in the price/money setting. One can impose equality across the two demand systems by substituting rather complicated functions of explanatory variables and parameters for several of the constant terms (e.g., the ? i ’s). Given this imposed equality, one can then maximize the likelihood of the sample observations using this single set of significantly more complicated demand equations and one should have access to all parameters of interest.

97 There is a different way to assess the magnitude of income effects which do not appear in derivatives of indirect utility with respect to travel times and available time; however, it is not as elegant and may not produce consistent estimators. It requires taking the results of the existing models and regressing these estimates of optimal activity participation on a system of demands developed in a price environment. This method was used with the third model to take a closer look at the marginal utility of income, and it consistently produced positive marginal utilities, thanks to the incorporation of Y, log(Y ) and log(Y ) 2 effects. The indirect utility specification used is the following: Indirect Utility = v v v v = v1 (t , Td , P (implicitly ), Y ) + v 2 ( P, Y ) =

i i

α o ? ∑ ? i t i + ? o Td + ∑ α i ln(t i ) + ∑ (1 2)β ij ln(t i ) ln(t j ) +

ij

(Chapter

∑γ ∑δ

i i

i

ln(Td ) ln(t i ) + ∑ γ iY ln(Y ) ln(t i ) + γ TY ln(Td ) ln(Y ) + ln( Pi ) + ∑ δ iPY ln( Pi ) ln(Y ) + δ Y Y + δ YY ln(Y ) 2

i i

iP

Four:-3) The optimal demand levels which result from application of Roy’s Identity in a price environment to the above formulation are the following:

98 ? Xi =

*

δ iP δ iPY ? ln(Y ) Pi Pi

? ? δ Y + 1Y ? ∑ γ iY ln(t i ) + γ TY ln(Td ) + ∑ δ jPY ln( Pj ) + 2 δ YY ln(Y )? ? i ? j ′ ′ ?δ iP ? δ iPY ln(Y ) = , ? ? δ Y + 1Y ? ∑ γ iY ln(t i ) + γ TY ln(Td ) + δ PYT + 2 δ YY ln(Y )? ? ? i where t i = Travel Time to Activity i , Pi = Price for Activity i , Y = Income, Td = Discretionary Time Available, ′ δ ′ δ and δ iP = iP , δ iPY = iPY , δ PYT = ∑ δ jPY ln( Pj ), Pi Pi j

( )

( )

(Chapter

& γ TY = +1 ( for identifiability of parameters). Four:-4) Note that assumption of constant price levels leads to non-identification of the fundamental price-interacted parameters but produces a similar, estimable functional form, where the price-interacted parameters are subsumed into identifiable parameters. This implicit incorporation of the price-specific parameters can be thought of as having occurred in all model specifications used here, particularly those with the constant terms (which thereby allow for isolated-price effects). Using this specification, a solution was sought which minimized the squared difference between earlier estimates of X i ’s (derived in a time environment and constructed using Table 4-2a’s parameter estimates) and the estimates arising from Equation 4-4’s demand specification (with γ TY = +1 and the already-estimated γ iY ’s substituted in directly). This method produced estimates of the marginal utility of income (i.e., the denominator in the demand equation of Equation 4-4) which are dominated by a positive δ Y term. The negative terms in the marginal utility of income which come from

dv1 dY are negligible in comparison with the highly positive estimate of δ Y . The new

*

99 value-of-time estimates are all positive, but their magnitude is too low by several orders (e.g., the median value is $.00045/hour). Moreover, the second set of estimates relies so substantially on the constant terms in the demand equations that estimates are predicted to vary little across households. Apparently then, this expanded indirect utility specification and/or the methods used to estimate this system’s parameters (including simply minimizing the sum of squared differences over all demands and strong assumptions like price invariability) remain lacking. These modeling complexities are a prime area for additional research. Other Reasons for Incorrect Marginal Utility of Income Estimates In addition to full identification of the indirect utility function and flexibility in functional form, there are other issues to consider in the estimation of the value of time. For example, in the fifth model estimated here, which comes from the Type 4 Model specification, the assumption of exogenously determined income and discretionary time budgets is dropped, theoretically removing any unmodeled dependencies across these explanatory variables and demand which may have caused erroneous results. The loss of this implicit and strong exogeneity assumption – which is present in the previously estimated models – may be what gives this final model its reasonable value-of-time estimates.15 Another reason for a negative marginal utility of income (and thus negative value of time) results may be that high income households are able to live in lower travel-time environments, so the parameters affecting the marginal utility of travel times might pick up an income effect, leaving the final income effect rather ambiguous and ostensibly negative in many models. For example, after normalizing income for total-time

100 availability, H, one finds that average travel times to each of the four all-job isoopportunity contours gradually fall as normalized income increases; the mean travel times between the first and fourth quartiles of Y H fall from 11.5 to 11.0 minutes to reach the first contour, 22.1 to 20.6 to reach the second, 33.5 to 31.3 to reach the third, and 49.4 to 47.2 to reach the fourth. It appears that high-income/low-time households are residing in locations that better fit their constraints, as one would expect; thus, if household location choice were made endogenous to the model, one could avoid some of the biases this dependence may create.

Further Qualifications

While the results of this research are interesting, one should recognize that the data are imperfect and the model assumptions are strong. For example, the travel-time data are measured with some error, thanks to zonal aggregation and reliance on free-flow, automobile travel times (– and due to the chaining of trips, as discussed at the end of Chapter Three). And the income and wage variables either come from survey-bin midpoints, in the case of income-reporting households, or have been estimated, for the nonreporting households. Simply the use of a model with one or more explanatory variables measured with error leads to highly uncertain impacts on estimates. (Greene 1993) Unfortunately, most models of travel behavior are subject to such deficiencies in the data set, since income tends to be reported by ranges and/or travel times come from a time-ofday-independent data base. There also is the concern that cross-sectional data do not provide the necessary variation to discern heterogeneity from state dependence in the unobserved information which influences decisions. Meurs (1990) recommends use of panel data for estimation

101 of trip generation due to difficulties arising in cross-sectional models from omitted timeinvariant/fixed effects across individuals; for example, Meurs’s models using the Dutch Mobility Panel data set indicate that cross-sectional income elasticities of demand tend to be biased high. Kitamura (1988) uses a three-year panel data set to study trip generation rates and finds the serial correlation to be substantial “suggest(ing) that important determinants of trip generation lie outside the set of variables that have traditionally been considered in travel behavior analysis.” Kitamura et al. discuss the need for longitudinal calibration to avoid a “longitudinal extrapolation of cross-section variations” (Kitamura et al. 1996). In other words, crosssectional elasticities are observed over different individuals yet often “applied as if they represent longitudinal elasticities that capture the change in behavior that follows a change in a contributing factor within each behavioral unit.” (Kitamura et al. 1996, pg. 274). The use of cross-sectional elasticities for estimation of longitudinal behavior is only rigorously valid under restrictive conditions, such as response being immediate and its magnitude being independent of past behaviors, according to Goodwin et al. (1990). In some cases, the greater the amount of time between a change in an independent variable and measurement of behavioral response, the higher the likelihood that crosssectional estimates apply. Becker (1965) voices some concern about the interpretation of cross-sectional elasticities for a different reason. His primary thesis is that the true cost of “commodity” consumption involves a time cost, not just a monetary cost for the non-time factors/goods used to produce commodities. Thus, he argues that “traditional cross-sectional estimates of income elasticity (which) do not hold either factor or commodity prices constant...” are

102 “...biased downward for time-intensive commodities, and give a misleading impression of the effect of income on the quality of commodities consumed.” (1965, pg. 517) Unfortunately, without adequate longitudinal data sets, Kitamura’s comments can only be used to qualify the results of this research, in the estimates of such elasticities. Accommodation of Becker’s fully general model requires information on the production technology of commodities (e.g., the combinations of time and money that produce a dining-out experience), so Becker’s concerns may can only be stated as qualifications here. The globalness of the likelihood’s maxima used to estimate parameter values and assess covariance also represents an assumption of these results. While the global maximum is a consistent estimate of the true parameter values – assuming the model and its distribution have been correctly specified, there is no guarantee that the search routine has converged upon the function’s global maximum16. This particular model’s requirement of positive activity-participation rate-parameter estimates for a calculable likelihood value17 often makes the acquisition of feasible starting parameter values a significant chore, particularly for the most functionally flexible models; thus, it is not easy to try starting at a wide variety of highly distinct parameter sets and comparing final points of convergence in an effort to avoid local maxima. However, as long as the results seem reasonable (e.g., as long as estimates of the means and proportions of trip-making correspond well with observed values), one may expect that one’s results are not a local maximum of poor prediction quality. And, as long as the convergent set seems robust to some changes in starting values, one m

赞助商链接

- An Introduction to Utility Theory
- The Household Registration System Computer Software for the Rapid Dissemination of Demograp
- How do households choose quality and time to replacement for a rapidly improving
- Labor allocation in transition Evidence from Chinese rural households
- 经济学说史——5第四章 边际效用学派
- Self-consistent system of equations for a kinetic description of the low-pressure discharge
- 77.A self-consistent theory for graphene transport
- Consistent Enterprise Software System Architecture for the CIO – A utility-Cost Approach
- Consistent cosmological modifications to the Einstein equations
- Hydrodynamic Self-Consistent Field Theory for Inhomogeneous Polymer Melts
- 1 Performance of Consistent Checkpointing in a Modular Operating System Results of the FTM
- Comment on ``Consistent Sets Yield Contrary Inferences in Quantum Theory''
- A FDR-consistent field theory for the stochastic dynamic density functional model
- A system of reaction diffusion equations arising in the theory of reinforced random walks
- Self-Consistent Approximations to Non-Equilibrium Many-Body Theory

更多相关文章：
更多相关标签：