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TRANSPORTATION MODELS

OF TIME ALLOCATION

A Contribution to Objective Consumption Theory

A the s i s i n partial fulfilment o f the requirements for the degr e e of Doctor of Phi l o sophy in Economic s at Mas s ey University .

IAN THOMAS MAHON

1990

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To my Parents of happy memory

ABSTRACT

This the sis inve s tigates the o ptimal all ocation of time by a rational agent in

terms of his behaviour settings and social requirements . Time is considered

as a scarce resourc e and as an obj e c tive measure of activities . Conceptually

the models of time allocation are transportation models and share the same

mathematical structure .

The findings of e c o - behavioural scienc e suggest that the behaviour of an

agent , as an individual de cision make r , will be shaped by environments .

Behaviour settings , corresponding t o s ources in the t ransportation mode l s , are

us ed t o define environment s . As a member of society the agent is required t o

mee t parameters o f social position , a s e t of r equirements corresponding t o

sinks in the transportation model s . Time use s tudies provide quantitative

measures of the agent ' s activitie s . Hence the model is abl e to specify

cons traints on the agent ' s time use in terms of behaviour settings and social

relations .

The c o re model shows the relationship between groups, or classe s , of agents

and their lifestyles . The a gent as rational decision maker is faced with the

choice of mee ting the demands of social position by activities in s e l e c ted

environments , while minimizing the total cost of the life styl e . Each ac tivity

use s up time and incurs a money c o s t . The optimal s olutions spe cify both the

type and l evel of the activities which the agent undertake s in order t o mee t

the_ parameters o f social position . An equivalent program ( the dual) exist s .

The agent is faced with the choice of maximising the net imputed value of time

use , so l ong as the net value of a uni t of time is less than or e qual to the

per uni t cost .

Conceptually there are two transportatition model s . Both are concerned with

the particular case of a s tudent as a rational dec ision maker . In the slack

model the focus is on the activi ties of a particular student . By way of

contras t the focus in the t ight model is on the activities of the average

student , and there is a t ime distribution not only at source s but al s o at

sinks . Thi s model i s useful to soc ial account s . Three e quivalent

formul ations of the transportation model are outlined .

iv

A technol ogy matrix , defined as the agent ' s soc i o - economic production

funct i on , denotes the set of product i on proces ses available to the agent ,

given behaviour settings ( environment s ) and parameters of soc ial pos ition . An

element of the soc i o - economic production funct i on i s termed an activity . The

choice of c ertain activi t i e s by the agent repre sents a particular l ifestyl e

described by a specific time distribution . Social income , defined as the

value of soc ial pos it ion plus net earned income i s a scalar measure ( in

doll ars ) of the agent's l ife styl e .

To show that the models are operat ional , simple 2 x 2 and 3 x 3 models are

introduced and extended in the final three chapters . A methodol ogy i s

devel oped f or obtaining per uni t cos t s. A step - by - step approach i s used to

derive a 5 x 5 cost matrix from two sets of actual data , obtained

independently . The effects of change s in the parameters of the time

all ocation models are analyzed .

PREFACE

Economic investigat ions of t ime al location can be regarded as a venture into

relat ively unexplored territory . When S oul e ( 19 5 5 ) stated that time was the

scarce s t resource , and proposed�hat t ime should be regarded as coordinate

with land , labour and capital , he was breaking fresh ground .

Whil e the transportation model s of t ime allocation devel oped in this the s i s

repre sent a compl etely different approach from that of Soul e , hi s que stions

rai s e s ome fundamental issue s . From a wider perspective , so too do the

questions sugges ted by Braudel in his masterly survey of the rise of

capital ism . He pointed to the soc ial dimension in economics and revealed the

impact of capital ism on patterns of human activity .

The first part of the the s i s introduces the research program and outl ines the

soc ial and historical factors that shaped the environment in which workers

carr i ed out the ir activities . The section concludes with an outline of two

s ignif icant , but different , models o f t ime allocation . Each extends the

boundaries of economics .

The transportati on model s of time all ocation are developed in the second part

of the thes is . The models owe much to the insights of the pathfinders , and

are the outcome of wre stl ing with unanswered questions and answers

unques t ioned .

The s tarred sections ( ** ) which begin in Chapter 4 provide a formulation of

the models within the framework of act ivity analysis . This mode sheds l ight

on the agent's production function . These more technical sections can be

omitted in a f irst reading without loss of continui ty .

vi

ACKNOWLEDGEMENTS

I wi sh t o thank my supervisors for stimulating discuss ions and for their

encouragement . The chief supervi sor Professor Paul van Moeseke provided a

rigorous cri tique of each draft . Mrs . Jean Goldsmith made practical

sugges t i ons about presentation . I n their different ways both emphasized the

need to complement theory with numerical exampl es .

Of tho s e kind enough to read or to check sect ions of the the s i s I would l ike

to thank the following : Susan Guthrie for commenting on early versions of the

theoret i cal model; Lorraine S tachurski for verifying most of the soluti ons to

the numerical exampl es , and for reading the final draft; Soh Yong - Chuan for

independently verifying most of the solut i ons , and also for checking the

calculations in Chapter 7 ; Hugo Val era -Alvarez for verifying solutions of

several numerical examples .

Thanks al s o go to Bruce Baker , S tuart Birks, Maurice Birchler and John

Mor iarty for providing access to , or advice on , the use of appropriate

softwar e . I would al s o l ike to thank Toni Snowball and Just ine Douthett for

undertaking most of the typing , and Gwen Flavell and Wendy King for kindly

providing secretarial assistance .

To the Head , and to the Act ing Head , of the Economics Department my thanks for

pract ical support . Last , but by no means least , I wish to express gratitude

to my family , and thanks to the f riends and colleagues , too many to menti on by

nam� , who have given encouragement along the way. The ir important

contributi on is gratefully acknowl edged .

TABLE OF CONTENTS

Abstract

Preface

Acknowl edgements

List of Tables

PART 1 BACKGROUND

CHAPTER 1 INTRODUCTION - THE RESEARCH PROGRAM

1 . 1 Outl ine of the Probl em

1 . 2 Aims of the Re search

1 . 3 Me thodology

1 . 4 Formulation of the Problem

1 . 5 S igni f icance of the Study

1 . 6 As sumptions and L imi tati ons

CHAPTER 2 MEASUREMENT AND VALUATION OF TIME USE

2 . 1 Outl ine

2 . 2 Time use in Produc tion

2 . 3 Summary

iii

V

vii

xiii

1

1

5

6

7 7 8

10

1 0

10

15

CHAPTER 3 SOME MODELS OF TIME ALLOCATI ON

3 . 1 Paradigms

3.2 The Becker model

3 . 3 The Fox-Moeseke model

3 . 4 The Moeseke model

3 . 5 Comparisons and Constrasts

3.6 Transportati on mode l s of Time Al location

PART 2 TRANSPORTATION MODELS OF TIME ALLOCATION

CHAPTER 4 BEHAVI OURAL BACKGROUND TO THE MODELS

4 . 1 Introduction

4.2 Integration of Economic s with other Discipl ines

4 . 3 Behaviour and Environment

4.4 T ime Use S tudies

4.5 Requirements of S o c ial Position

4.6 The Cost of an Act ivity

** 4. 7 Ac tivity Analysi s

ix

16

16

17

22

26

35

38

39

39

40

40

44

48

53

55

CHAPTER 5 FORMULATION

5.1 Outl ine

5.2 Numerical Exampl e

5 . 3 Core Model

5.4 Slack and Tight Model s

5.5 The Transportation Probl em

5 . 6 Standard Linear Programming Model

** 5 . 7 Activity Analysis

** 5. 8 Changes in Technol ogy

CHAPTER 6 ECONOMIC INTERPRETATION

6 . 1 Introduc tion

6 . 2 Outl ine

6 . 3 Transportation Model s - Economic Interpretations

6.4 Soc ial Income and Savings

**6 . 5 Ac tivity Analys is - Economic Interpretati ons

X

57

57

57

64

66

70

78

80

88

92

92

92

94

104

107

CHAPTER 7 NUMERICAL EXAMPLES ( 1)

7 . 1 Introduction

7 . 2 The Slack Model

7.3 The Tight Model

7 . 4 Dimensionality Problem

7 . 5 The Per Unit Cost Matrix

** 7 . 6 Activity Analys is

7 . 7 Summary

CHAPTER 8 NUMERICAL EXAMPLES ( 2)

8 . 1 Introduction

8 . 2 5x5 and 5x6 Slack Models

8 . 3 5x5 and 5x6 Tight Models

8 . 4 Derivation of the Cost Matrix

** 8 . 5 Changes in Technology

CHAPTER 9 NUMERICAL EXAMPLES ( 3)

9 . 1 Introduction

9 . 2 Effects of Changes in the Vector c

9 . 3 Effects of Changes in the Vector b

9 . 4 The Addition of a New Constraint

2 X 2, 2 X 3 AND 3 X 3 MODELS

5 X 5 AND 5 X 6 MODELS

EFFECTS OF CHANGES IN THE PARAMETERS

**9 . 5 Effects of Changes in the Technology Matrix A

9 . 6 Summary

xi

115

115

116

132

147

150

154

1 5 7

159

159

160

171

182

197

200

200

200

205

210

213

224

xi i

SUMMARY AND CONCLUS IONS 227

LIST OF SYMBOLS AND NOTATI ON 231

REFERENCES 234

3-1

4-1

4-2

4-3

S-1

S-2

S-3

S-4

5-S

S-6

7-1

7-2

7-3

7-4

7-5

7-6

7- 7

7 - 8

7 - 9

7 - 10

7 - 1 1

7-12 -

7-13

LIST OF TABLES

De scription

Comparison of the Becker model and the Moe seke model

Environments clas s ified by behaviour settings

Bas i c categories of time use

Cla s s ificati on of social requirements

Act ivity matrix

Cost matrix

Row and column arrays

S tandard l inear programming model

Technology matrix , A

The activity analysis model

2 x 2 act ivi ty matrix - slack model

2 x 2 cost matrix - slack model

2 x 2 new cost matrix ( 1 )

2 x 3 activity matrix - slack model

2 x 3 cost matrix - sl ack model

3 x 3 activity matrix - slack model

3 x 3 new cost matrix ( 1 )

2 x 2 activity matrix - tight model

2 x 2 cost matrix - tight model

2 x 2 new cost matrix ( 1 ) - tight model

2 x 2 new cost matrix ( 2 ) - tight model

Le Chatel i er princ ipl e - summary of re sul ts

3 x 3 ac tivity matrix - tight model

35

43

47

so

62

63

7 9

7 9

8 7

8 7

117

118

121

124

124

126

1 2 7

135

135

136

139

142

142

7-14

7-15

7-16

7-17

7-18

7 - 19

7-20

7-21

7-22

7-23

7-24

8-1

8-2

8-3

8-4

8-5

8-6

8-7

8-8

8-9

8-10

8-11

8 - 12

8-13

8 - 14

3 x 3 cost matrix - tight model

3 x 3 new cost matrix ( 1)

Space - t ime behaviour pattern

3 x 3 new cost matrix - tight model

Example 3 as a standard linear problem

Estimated dollar costs for the academic year

Estimated dollar costs for the academic year by behaviour settings

Minimum e stimated costs for academic year

2 x 2 matrix of daily e s timated dollar c osts

2 x 2 matrix of estimated time spent on student activities in minute s

2 x 2 cost matrix i n c ents per minute

Statement of problem - slack model

5 x 5 activity matrix

5 x 5 cost matrix

Solutions to exampl e 1

5 x 6 activi ty matrix

5 x 6 cost matrix

xiv

142

144

145

146

147

150

151

152

153

153

154

160

161

161

162

167

167

Changes in the cost of study on campus - solutions 170

Statement of problem - 5 x 5 tight model 175

5 x 6 activity matrix - tight model 181

Students ' daily activity patterns 18 2

Classification of daily activities 183

Daily activi t i e s class ified by soc ial requirement categories 184

Estimates of proporti on of soc ial requirement all oc ated to behaviour settings 185

Matrix of daily estimated average time in minutes for activities 186

8 - 1 5

8 - 1 6

8 - 1 7

8 - 18

8 - 19

8 - 20

8 - 2 1

9 - 1

9 - 2

9 - 3

9 -4

9-5

9 - 6

9 - 7

9 - 8

9 - 9

9 - 10

9 - 11

9 - 12

Minimum estimated dollar c osts for the academic year

Minimum estimated dollar c o sts for the academic year by behaviour settings

Matrix of minimum estimated dollar costs for the academic year by social requirements and expenditure classifications

Matrix of minimum es timate d dollar costs for the academi c year by behaviour setting and social requirement

Components of dollar costs for the academic year

Matrix of daily es timated dollar costs

5 x 5 cost matrix

5 x 5 activity matrix

5 x 5 cost matrix

Effects of increases in state behaviour setting costs - opt imal soluti ons

Effects of increases in s tate behaviour setting costs on shadow prices

Effects of increases in state behaviour setting costs on soc ial income

Effects of changes in the distributi on of endowment s c52 - - 11 . 0 centsjmin

Effects of change s in the distributi on of endowments - optimal solut i ons c 52 - - 1 1.0 c entsjmin

Effects of changes in the distribution of endowments c52 - - 8 . 5 centsjmin

Effects of changes in the distributi on of endowments - optimal soluti ons c 52 - - 8 . 5 cents/min

The additi on of a new constraint - optimal solutions

XV

1 8 8

189

192

193

195

196

1 9 7

202

202

204

204

205

207

208

208

209

212

The addit i on of a new constraint - effects on social income 213

Change s in technology - solutions 216

9- 13

9-14

9-15

Effects of a n increase in the eff i c iency of meeting the academic social requir ement

Effects of an increase in the eff i c i ency of mee ting the academic social requirement on shadow pric e s .

Changes in technology - no excess capaci ty

xvi

216

217

2 24