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InputOutput Analysis

The new edition of Ronald Miller and Peter Blairs classictextbook is an essential reference for students and scholarsin the inputoutput research and applications community. Thebook has been fully revised and updated to reflect importantdevelopments in the field since its original publication. Newtopics covered include social accounting matrices (SAMs) (andextended inputoutput models) and their connection to inputoutput data, structural decomposition analysis (SDA), multiplierdecompositions, identifying important coefficients, and interna-tional inputoutput models. A new feature of this edition is thatit is also supported by an accompanying website with solutionsto all problems, a sampling of real-world data sets, and supple-mental appendices with further information for more advancedreaders.

InputOutput Analysis is an ideal introduction to the subjectfor advanced undergraduate and graduate students in a wide vari-ety of fields, including economics, regional science, regionaleconomics, city, regional and urban planning, environmentalplanning, public policy analysis, and public management.

ronald e. miller is Emeritus Professor of Regional Scienceat the University of Pennsylvania, Philadelphia.

peter d. blair is Executive Director of the Division on Engi-neering and Physical Sciences at the National Academy ofSciences, Washington, DC.

InputOutput AnalysisFoundations and Extensions

Second Edition

Ronald E. Miller

and

Peter D. Blair

CAMBRIDGE UNIVERSITY PRESS

Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,

So Paulo, Delhi, Dubai, Tokyo

Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

First published in print format

ISBN-13 978-0-521-51713-3

ISBN-13 978-0-521-73902-3

ISBN-13 978-0-511-65103-8

Ronald E. Miller and Peter D. Blair 2009

2009

Information on this title: www.cambridge.org/9780521517133

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provision of relevant collective licensing agreements, no reproduction of any part

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Published in the United States of America by Cambridge University Press, New York

www.cambridge.org

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http://www.cambridge.org/9780521517133http://www.cambridge.org

Contents

List of Figures page xxii

List of Tables xxiv

Preface xxix

1 Introduction and Overview 11.1 Introduction 11.2 InputOutput Analysis: The Basic Framework 21.3 Outline for this Text 31.4 Internet Website and Text Locations of Real Datasets 8References 9

2 Foundations of InputOutput Analysis 102.1 Introduction 102.2 Notation and Fundamental Relationships 10

2.2.1 InputOutput Transactions and National Accounts 132.2.2 Production Functions and the InputOutput Model 15

2.3 An Illustration of InputOutput Calculations 212.3.1 Numerical Example: Hypothetical Figures Approach I 21

Impacts on Industry Outputs 21Other Impacts 24

2.3.2 Numerical Example: Hypothetical Figures Approach II 262.3.3 Numerical Example: Mathematical Observations 272.3.4 Numerical Example: The US 2003 Data 29

2.4 The Power Series Approximation of (I A)1 312.5 Open Models and Closed Models 342.6 The Price Model 41

2.6.1 Overview 412.6.2 Physical vs. Monetary Transactions 422.6.3 The Price Model based on Monetary Data 43

v

vi Contents

2.6.4 Numerical Examples Using the Price Model basedon Monetary Data 44Example 1: Base Year Prices 44Example 2: Changed Base Year Prices 45

2.6.5 Applications 462.6.6 The Price Model based on Physical Data 47

Introduction of Prices 48Relationship between A and C 49

2.6.7 Numerical Examples Using the Price Model based onPhysical Data 50Example 1: Base Year Prices 50Example 2: Changed Base Year Prices 51

2.6.8 The Quantity Model based on Physical Data 512.6.9 A Basic National Income Identity 53

2.7 Summary 53Appendix 2.1 The Relationship between Approaches I and II 54

A2.1.1 Approach I 54A2.1.2 Approach II 55

Appendix 2.2 The HawkinsSimon Conditions 58Problems 62References 66

3 InputOutput Models at the Regional Level 693.1 Introduction 693.2 Single-Region Models 70

3.2.1 National Coefficients 703.2.2 Regional Coefficients 723.2.3 Closing a Regional Model with respect to Households 74

3.3 Many-Region Models: The Interregional Approach 763.3.1 Basic Structure of Two-Region Interregional InputOutput

Models 773.3.2 Interregional Feedbacks in the Two-Region Model 803.3.3 Numerical Example: Hypothetical Two-Region Interregional

Case 823.3.4 Interregional Models with more than Two Regions 863.3.5 Implementation of the IRIO Model 87

3.4 Many-Region Models: The Multiregional Approach 873.4.1 The Regional Tables 873.4.2 The Interregional Tables 893.4.3 The Multiregional Model 913.4.4 Numerical Example: Hypothetical Two-Region

Multiregional Case 93

Contents vii

3.4.5 The US MRIO Models 963.4.6 Numerical Example: The Chinese Multiregional

Model for 2000 973.5 The Balanced Regional Model 101

3.5.1 Structure of the Balanced Regional Model 1013.5.2 Numerical Example 104

3.6 The Spatial Scale of Regional Models 1053.7 Summary 106Appendix 3.1 Basic Relationships in the Multiregional InputOutput

Model 107Appendix 3.2 Sectoral and Regional Aggregation in the 2000 Chinese

Multiregional Model 109Appendix 3.3 The Balanced Regional Model and the Inverse of a

Partitioned (I A) Matrix 110Problems 111References 115

4 Organization of Basic Data for InputOutput Models 1194.1 Introduction 1194.2 Observations on Ad Hoc Survey-Based InputOutput Tables 1194.3 Observations on Common Methods for Generating

InputOutput Tables 1204.4 A System of National Economic Accounts 121

4.4.1 The Circular Flow of Income and Consumer Expenditure 1224.4.2 Savings and Investment 1234.4.3 Adding Overseas Transactions: Imports, Exports, and Other

Transactions 1264.4.4 The Government Sector 1274.4.5 The Consolidated Balance Statement for National Accounts 1284.4.6 Expressing Net Worth 131

4.5 National Income and Product Accounting Conventions 1334.6 Assembling InputOutput Accounts: The US Case 1344.7 Additional Considerations 137

4.7.1 Secondary Production: Method of Reallocation 140Example 1: Reallocation of Secondary Production 141

4.7.2 Secondary Production: Commodity-by-Industry Accounting 142Example 2: Commodity-by-Industry Accounts 142

4.7.3 Reconciling with the National Accounts 1434.7.4 Producers and Consumers Prices 144

Example 3: Trade and Transportation Margins 1464.7.5 Accounting for Imports and Exports 149

Example 4: Competitive and Noncompetitive Imports 149

viii Contents

4.7.6 Removing Competitive Imports from Total TransactionsTables 150Approximation Method I 151Approximation Method II 151Example 5: Import Scrubbing 152Implications of the Estimating Assumptions 154

4.7.7 Adjustments for Inventory Change 1574.7.8 Adjustments for Scrap 157

4.8 Valuation and Double Deflation 157Example 6: Double Deflation 159

4.9 The Aggregation Problem: Level of Detail in InputOutput Tables 1604.9.1 The Aggregation Matrix 161

Example 7: Sectoral Aggregation 1624.9.2 Measures of Aggregation Bias 165

4.10 Summary 168Appendix 4.1 Spatial Aggregation in IRIO and MRIO Models 168

A4.1.1 Spatial Aggregation of IRIO Models 168A4.1.2 Spatial Aggregation of MRIO Models 172

Problems 176References 180

5 The Commodity-by-Industry Approach in InputOutput Models 1845.1 Introduction 184

5.1.1 The Use Matrix 1855.1.2 The Make Matrix 186

5.2 The Basic Accounting Relationships 1875.3 Technology and Total Requirements Matrices in the

CommodityIndustry Approach 1885.3.1 Industry Source of Commodity Outputs 1895.3.2 Commodity Composition of Industry Outputs 1895.3.3 Generating Total Requirements Matrices 189

Using D 190Using C 191

5.3.4 Industry-Based Technology 1925.3.5 Commodity-Based Technology 1935.3.6 Direct Requirements (Technical Coefficients) Matrices

Derived from Basic Data 1955.3.7 Total Requirements Matrices 196

Approach I: Starting with Technical Coefficients 196Approach II: Avoiding C1 in Commodity Technology Cases 198Is Singularity Likely to be a Problem in Real-World Models? 199

5.4 Numerical Examples of Alternative Direct and Total RequirementsMatrices 201

Contents ix

5.4.1 Direct Requirements Matrices 2025.4.2 Total Requirements Matrices 202

Commodity-Demand Driven Models 202Industry-Demand Driven Models 202

5.5 Negative Elements in the CommodityIndustry Framework 2035.5.1 Commodity Technology 203

Direct Requirements Matrices 203Transactions Matrices 205Total Requirements Matrices 206

5.5.2 Industry Technology 207Direct Requirements Matrices 207Total Requirements Matrices 207

5.5.3 Making a Model Choice 208Which Model to Choose? 208Dealing with Negative Values 209

5.6 Nonsquare CommodityIndustry Systems 2115.6.1 Commodity Technology 2115.6.2 Industry Technology 212

Direct Requirements Matrices 212Total Requirements Matrices 213

5.7 Mixed Technology in the CommodityIndustry Framework 2135.7.1 Commodity Technology in V1 2165.7.2 Industry Technology in V1 2185.7.3 Numerical Examples with Mixed Technology

Assumptions 219Example 1: Commodity Technology in V1 219Example 2: Industry Technology in V1 220

5.7.4 Additional Mixed Technology Variants 2205.8 Summary 222Appendix 5.1 Alternative Approaches to the Derivation of Transactions

Matrices 223A5.1.1 Industry Technology 224

Commodity-by-Commodity Requirements 224Industry-by-Industry Requirements 225

A5.1.2 Commodity Technology 226Commodity-by-Commodity Requirements 226Industry-by-Industry Requirements 228

Appendix 5.2 Elimination of Negatives in Commodity TechnologyModels 229

A5.2.1 The Problem 2293 3 Example 2294 4 Example 2295 5 Example (from Almon, 2000) 230

x Contents

A5.2.2 Approaches to Elimination of Negative Elements 230A5.2.3 Results of the Iterative Procedure 234

3 3 Example 2344 4 Example 2345 5 Example 235


6 Multipliers in the InputOutput Model 2436.1 Introduction 2436.2 General Structure of Multiplier Analysis 244

6.2.1 Output Multipliers 245Simple Output Multipliers 245Total Output Multipliers 247Example: The US InputOutput Model for 2003 248Output Multipliers in CommodityIndustry Models 249Commodity-Demand-Driven Models 249Industry-Demand-Driven Models 250

6.2.2 Income/Employment Multipliers 250Income Multipliers 250Type I and Type II Income Multipliers 252Relationship Between Simple and Total Income Multipliers orBetween Type I and Type II Income Multipliers 253Which Multiplier to Use? 254Even More Income Multipliers 255Physical Employment Multipliers 255

6.2.3 Value-Added Multipliers 2566.2.4 Matrix Representations 2566.2.5 Summary 257

6.3 Multipliers in Regional Models 2596.3.1 Regional Multipliers 2596.3.2 Interregional InputOutput Multipliers 261

Intraregional Effects 261Interregional Effects 262National Effects 263Sectoral Effects 263More Than Two Regions 264

6.3.3 Multiregional InputOutput Multipliers 264Intraregional Effects 266Interregional Effects 267National Effects 267Sectoral Effects 267

Contents xi

Final Demand for Goods Made in a Particular Region 267More Than Two Regions 268

6.4 Miyazawa Multipliers 2716.4.1 Disaggregated Household Income Groups 2716.4.2 Miyazawas Derivation 2736.4.3 Numerical Example 2756.4.4 Adding a Spatial Dimension 276

6.5 Gross and Net Multipliers in InputOutput Models 2786.5.1 Introduction 2786.5.2 Multipliers in the Net InputOutput Model 278

Numerical Example 2806.5.3 Additional Multiplier Variants 280

(Indirect Effects)/(Direct Effects) 280Growth Equalized Multipliers 281Another Kind of Net Multiplier 282

6.6 Multipliers and Elasticities 2836.6.1 Output Elasticity 2836.6.2 Output-to-Output Multipliers and Elasticities 283

Direct Effects 283Total Effects 284

6.7 Multiplier Decompositions 2856.7.1 Fundamentals 2856.7.2 Decompositions in an Interregional Context 2866.7.3 Stones Additive Decomposition 2886.7.4 A Note on Interregional Feedbacks 2896.7.5 Numerical Illustration 290

6.8 Summary 294Appendix 6.1 The Equivalence of Total Household Income Multipliers

and the Elements in the Bottom Row of (I A)1 295Appendix 6.2 Relationship Between Type I and Type II Income

Multipliers 296Problems 297References 299

7 Nonsurvey and Partial-Survey Methods: Fundamentals 3037.1 Introduction 3037.2 The Question of Stability of InputOutput Data 303

7.2.1 Stability of National Coefficients 304Comparisons of Direct-Input Coefficients 305Comparisons of Leontief Inverse Matrices 305Other Summary Measures 307Data for the US Economy 307

xii Contents

7.2.2 Constant versus Current Prices 3077.2.3 Stability of Regional Coefficients 3097.2.4 Summary 310

7.3 Updating and Projecting Coefficients: Trends, Marginal Coefficientsand Best Practice Methods 3117.3.1 Trends and Extrapolation 3117.3.2 Marginal Input Coefficients 3117.3.3 Best Practice Firms 312

7.4 Updating and Projecting Coefficients: The RAS Approachand Hybrid Methods 3137.4.1 The RAS Technique 3137.4.2 Example of the RAS Procedure 3207.4.3 Updating Coefficients vs. Transactions 327

Numerical Illustration 3277.4.4 An Economic Interpretation of the RAS Procedure 3287.4.5 Incorporating Additional Exogenous Information in an RAS

Calculation 3307.4.6 Modified Example: One Coefficient Known in Advance 3317.4.7 Hybrid Models: RAS with Additional Information 3337.4.8 The Constrained Optimization Context 3347.4.9 Infeasible Problems 335

7.5 Summary 336Appendix 7.1 RAS as a Solution to the Constrained Minimum Information

Distance Problem 337Problems 338References 343

8 Nonsurvey and Partial-Survey Methods: Extensions 3478.1 Introduction 3478.2 Location Quotients and Related Techniques 349

8.2.1 Simple Location Quotients 3498.2.2 Purchases-Only Location Quotients 3538.2.3 Cross-Industry Quotients 3538.2.4 The Semilogarithmic Quotient and its Variants,

FLQ and AFLQ 3548.2.5 SupplyDemand Pool Approaches 3568.2.6 Fabrication Effects 3568.2.7 Regional Purchase Coefficients 3578.2.8 Community InputOutput Models 3588.2.9 Summary 359

8.3 RAS in a Regional Setting 3608.4 Numerical Illustration 361

Contents xiii

8.5 Exchanging Coefficients Matrices 3638.6 Estimating Interregional Flows 364

8.6.1 Gravity Model Formulations 3658.6.2 Two-Region Interregional Models 3668.6.3 Two-Region Logic with more than Two Regions 3678.6.4 Estimating Commodity Inflows to a Substate Region 3698.6.5 Additional Studies 371

Commodity Flows among US States 371An Optimization Model for Interregional Flows 372

8.7 Hybrid Methods 3738.7.1 Generation of Regional InputOutput Tables (GRIT) 3748.7.2 Double-Entry Bi-Regional InputOutput Tables (DEBRIOT) 3758.7.3 The Multiregional InputOutput Model for China, 2000

(CMRIO) 3778.8 International InputOutput Models 378

8.8.1 Introduction 3788.8.2 Asian International InputOutput Tables 3788.8.3 Hybrid Many-Region Models for the EC 3808.8.4 ChinaJapan Transnational Interregional InputOutput

(TIIO) Model, 2000 381Chinese Exports to Japan for Intermediate Demand 381Applications 383

8.8.5 Leontiefs World Model 3838.9 The Reconciliation Issue 3848.10 Summary 386Appendix 8.1 Geographical Classifications in the World InputOutput

Model 387Problems 387References 392

9 Energy InputOutput Analysis 3999.1 Introduction 399

9.1.1 Early Approaches to Energy InputOutput Analysis 4009.1.2 Contemporary Energy InputOutput Analysis 400

9.2 Overview Concepts of Energy InputOutput Analysis 4019.2.1 The Basic Formulation 4039.2.2 The Total Energy Requirements Matrix 404

Example 9.1: Two-Sector Illustration of Hybrid UnitsInputOutput Analysis 407Example 9.2: Generalization to Several Energy Types 408

9.2.3 The Hybrid Units Formulation and Energy ConservationConditions 410

xiv Contents

Example 9.2: Generalization to Several Energy Types(Revisited) 411

9.3 Further Methodological Considerations 4119.3.1 Adjusting for Energy Conversion Efficiencies 412

Example 9.3: Adjusting for Energy Conversion Efficiencies 4129.3.2 Accounting for Imports 4139.3.3 Commodity-by-Industry Energy Models 413

9.4 Applications 4149.4.1 Net Energy Analysis 414

Example 9.4: Net Energy Analysis 4159.4.2 Energy Cost of Goods and Services 4179.4.3 Impacts of New Energy Technologies 4219.4.4 An Energy Tax 4219.4.5 Energy and Structural Change 4219.4.6 Energy InputOutput and Econometrics 4239.4.7 Other Applications 427

9.5 Summary 427Appendix 9.1 Earlier Formulation of Energy InputOutput Models 428

A9.1.1 Introduction 428A9.1.2 Illustration of the Implications of the Traditional Approach 430

Example 9.5: Energy InputOutput Alternative Formulation 430Example 9.6: Energy InputOutput Example (Revised) 431Extensions of Example 9.1 433

A9.1.3 General Limitations of the Alternative Formulation 437Problems 437References 442

10 Environmental InputOutput Analysis 44610.1 Introduction 44610.2 Basic Considerations 44610.3 Generalized InputOutput Analysis: Basic Framework 447

10.3.1 Accounting for Pollution Impacts 44710.3.2 Generalized Impacts 447

Example 10.1: Generalized InputOutput Analysis 44810.3.3 Summary: Generalized InputOutput Formulations 451

Case I: Impact Analysis Form 451Case II: Planning Form 452

10.4 Generalized InputOutput Analysis: Extensions of thePlanning Approach 45210.4.1 Linear Programming: A Brief Introduction by Means of the

Leontief Model 45210.4.2 Multiple Objectives 457

Contents xv

10.4.3 Conflicting Objectives and Linear Goal Programming 45710.4.4 Additional Observations 461

Specifying Objectives 461Tightly Constrained Problems 462Solution Methods 462

10.4.5 Applications to the Generalized InputOutput PlanningProblem 463

10.4.6 Policy Programming 469Impact Analysis Form 470Planning Form 470

10.4.7 Ecological Commodities 47310.5 An Augmented Leontief Model 475

10.5.1 Pollution Generation 47510.5.2 Pollution Elimination 478

Example 10.2: Pollution-Activity-Augmented LeontiefModel 479

10.5.3 Existence of Non-negative Solutions 480Example 10.2 (Revisited): Pollution-Activity-AugmentedLeontief Model 482

10.6 EconomicEcologic Models 48310.6.1 Fully Integrated Models 48310.6.2 Limited EconomicEcologic Models 484

Economic Subsystem 484Ecologic Subsystem 485Commodity-by-Industry Formulation 485Example 10.3: Limited EconomicEcologic Models 485

10.7 Pollution Dispersion 48710.7.1 Gaussian Dispersion Models 48710.7.2 Coupling Pollution Dispersion and InputOutput Models 488

Example 10.4: Coupling InputOutput and PollutionDispersion Models 488

10.8 Other Applications 48910.9 Summary 490Problems 490References 494

11 Social Accounting Matrices 49911.1 Introduction 49911.2 Social Accounting Matrices: Background 49911.3 Social Accounting Matrices: Basic Concepts 50111.4 The Households Account 50211.5 The Value-Added Account 504

xvi Contents

11.6 Interindustry Transactions and the Connection to the InputOutputFramework 504

11.7 Expanding the Social Accounts 50711.8 Additional Social Accounting Variables 50711.9 A Fully Articulated SAM 51011.10 SAM Multipliers 513

11.10.1 SAM Multipliers: Basic Structure 51411.10.2 Decomposition of SAM Multipliers 516

Example 11.1: Reduced Form Case 51811.10.3 Multipliers in an Expanded SAM 522

Example 11.2: The Expanded Case 52411.10.4 Additive Multipliers 528

11.11 The Relationship between InputOutput and SAM Multipliers 53011.12 Balancing SAM Accounts 535

11.12.1 Example: Balancing a SAM 53511.12.2 Example: Balancing a SAM with Additional

Information 53611.13 Some Applications of SAMs 53611.14 Summary 537Problems 537References 541

12 Supply-Side Models, Linkages, and Important Coefficients 54312.1 Supply Side InputOutput Models 543

12.1.1 The Early Interpretation 543Numerical Illustration (Hypothetical Data) 546Numerical Application (US Data) 547

12.1.2 Relationships between A and B and between L and G 54712.1.3 Comments on the Early Interpretation 54812.1.4 Joint Stability 549

The Issue 549Conditions under which both A and B will be Stable 551

12.1.5 Reinterpretation as a Price Model 551Connection to the Leontief Price Model (Algebra) 553Connection to the Leontief Price Model (NumericalIllustration) 553A Ghosh Quantity Model 554

12.2 Linkages in InputOutput Models 55512.2.1 Backward Linkage 55612.2.2 Forward Linkage 55812.2.3 Net Backward Linkage 55812.2.4 Classifying Backward and Forward Linkage Results 559

Contents xvii

12.2.5 Spatial Linkages 56012.2.6 Hypothetical Extraction 563

Backward Linkage 564Forward Linkage 564

12.2.7 Illustration Using US Data 56512.3 Identifying Important Coefficients 567

12.3.1 Mathematical Background 56812.3.2 Relative Sizes of Elements in the Leontief Inverse 569

Observation 1 569Observation 2 570Observation 3 570

12.3.3 Inverse-Important Coefficients 57012.3.4 Numerical Example 57212.3.5 Impacts on Gross Outputs 57312.3.6 Fields of Influence 57812.3.7 Additional Measures of Coefficient Importance 580

Converting Output to Employment, Income, etc. 580Elasticity Coefficient Analysis 581Relative Changes in All Gross Outputs 581Impacts of Changes in more than One Element of theA Matrix 582

12.4 Summary 582Appendix 12.1 The ShermanMorrisonWoodbury Formulation 582

A12.1.1 Introduction 582A12.1.2 Application to Leontief Inverses 585


13 Structural Decomposition, Mixed and Dynamic Models 59313.1 Structural Decomposition Analysis 593

13.1.1 Initial Decompositions: Changes in Gross Outputs 593Numerical Example 596

13.1.2 Next-Level Decompositions: Digging Deeper intof and L 598Additive Decompositions with Products of more than Two Terms 598Changes in Final Demand 599

13.1.3 Numerical Examples 601One Category of Final Demand (p = 1) 601Two Categories of Final Demand (p = 2) 601

13.1.4 Changes in the Direct Inputs Matrix 602Decomposition of L 602Decomposition of A 604Numerical Illustration (continued) 605

xviii Contents

13.1.5 Decompositions of Changes in Some Function of x 60613.1.6 Summary for x 60713.1.7 SDA in a Multiregional InputOutput (MRIO) Model 60713.1.8 Empirical Examples 608

Studies Using National Models 608Studies Using a Single-Region or Connected-RegionModel 615

13.2 Mixed Models 62113.2.1 Exogenous Specification of One Sectors Output 621

Rearranging the Basic Equations 621Extracting the Sector 624

13.2.2 An Alternative Approach When f1, . . ., fn1 and xn AreExogenously Specified 625

13.2.3 Examples with xn Exogenous 626Example 1: f1 = 100, 000, f2 = 200, 000,x3 = 150, 000 626Example 2: f1 = f2 = 0, x3 = 150, 000 627Example 3: f1 = 100, 000, f2 = 200, 000, x3 = 100, 000 628Example 4: The Critical Value of x3 628Multipliers 629

13.2.4 Exogenous Specification of f1, . . ., fk , xk+1, . . ., xn 62913.2.5 An Example with xn1 and xn Exogenous 632

Example 5 (Example 2 expanded) 63213.3 New Industry Impacts in the InputOutput Model 633

13.3.1 New Industry: The Final-Demand Approach 63413.3.2 New Industry: Complete Inclusion in the Technical

Coefficients Matrix 63613.3.3 A New Firm in an Existing Industry 63713.3.4 Other Structural Changes 639

13.4 Dynamic Considerations in InputOutput Models 63913.4.1 General Relationships 63913.4.2 A Three-Period Example 642

Terminal Conditions 643Initial Conditions 644

13.4.3 Numerical Example 1 645Terminal Conditions 646Initial Conditions 648

13.4.4 Numerical Example 2 649Terminal Conditions 649Initial Conditions 649

13.4.5 Dynamic Multipliers 65013.4.6 Turnpike Growth and Dynamic Models 651

Example 653

Contents xix

13.4.7 Alternative InputOutput Dynamics 65313.5 Summary 654Appendix 13.1 Alternative Decompositions of x = LBf 655Appendix 13.2 Exogenous Specification of Some Elements of x 656

A13.2.1 The General Case: An n-sector Model with k EndogenousOutputs 656

A13.2.2 The Output-to-Output Multiplier Matrix 658A13.2.3 The Inverse of a Partitioned (I A(n)) Matrix 659A13.2.4 The Case of k = 2, n = 3 659A13.2.5 The Case of k = 1, n = 3 660A13.2.6 Extracting the Last (n k) Sectors 662


14 Additional Topics 66914.1 Introduction 66914.2 InputOutput and Measuring Economic Productivity 670

14.2.1 Total Factor Productivity 67014.2.2 Numerical Example: Total Factor Productivity 67214.2.3 Accounting for Prices 67314.2.4 References for Section 14.2 674

14.3 Graph Theory, Structural Path Analysis, and QualitativeInputOutput Analysis (QIOA) 67414.3.1 References for Section 14.3 677

14.4 Fundamental Economic Structure (FES) 67814.4.1 References for Section 14.4 679

14.5 InputOutput, Econometrics, and Computable General EquilibriumModels 67914.5.1 The Variable InputOutput Model 68014.5.2 Regional InputOutput Econometric Models 68114.5.3 Computable General Equilibrium Models 68114.5.4 References for Section 14.5 682

14.6 Additional Resources for InputOutput Extensions andApplications 68314.6.1 Edited Collections 68414.6.2 Journal Special Issues 68514.6.3 Collections of Reprints 68614.6.4 References for Section 14.6 686

14.7 Some Concluding Reflections 686

xx Contents

Appendix A Matrix Algebra for InputOutput Models 688A.1 Introduction 688A.2 Matrix Operations: Addition and Subtraction 689

A.2.1 Addition 689A.2.2 Subtraction 689A.2.3 Equality 689A.2.4 The Null Matrix 689

A.3 Matrix Operations: Multiplication 689A.3.1 Multiplication of a Matrix by a Number 689A.3.2 Multiplication of a Matrix by Another Matrix 689A.3.3 The Identity Matrix 690

A.4 Matrix Operations: Transposition 691A.5 Representation of Linear Equation Systems 691A.6 Matrix Operations: Division 693A.7 Diagonal Matrices 696A.8 Summation Vectors 698A.9 Matrix Inequalities 698A.10 Partitioned Matrices 699

A.10.1 Multiplying Partitioned Matrices 699A.10.2 The Inverse of a Partitioned Matrix 700

References 701

Appendix B Reference InputOutput Tables for the United States(19192006) 702

B.1 Introduction 702B.2 Transactions Accounts 703B.3 Matrices of Technical Coefficients and Total Requirements 715References for US InputOutput Tables (19192006) 722

Appendix C Historical Notes on the Development of LeontiefsInputOutput Analysis 724

C.1 Conceptual Foundations 724C.2 Quesnay and the Physiocrats 725C.3 Mathematical Formalization 728C.4 Leontief and the Economy as a Circular Flow 729C.5 Development of InputOutput Analysis 731References 735

Contents xxi

Author Index 738

Subject Index 746

The Following Supplementary Appendices are Available Online atwww.cambridge.org/millerandblairWeb Appendix 5W.1 Left and Right Inverses in Nonsquare InputOutput

SystemsA5W.1.1 IntroductionA5W.1.2 More Commodities than Industries (m > n)

Numerical IllustrationCommodity TechnologyIndustry Technology

A5W.1.3 Fewer Commodities than Industries (m < n)Numerical IllustrationCommodity TechnologyIndustry Technology

Web Appendix 8W.1 Detailed Results for the Numerical Illustration inSection 8.4

Web Appendix 12W.1 Hypothetical Extractions with Partitioned MatricesA12W.1.1 Case 1: Complete Extraction of Sector 1A12W.1.2 Case 2: Extraction of Sector 1s Intersectoral RelationsA12W.1.3 Case 3: Extraction of Sector 1s Intermediate PurchasesA12W.1.4 Case 4: Extraction of Sector 1s Intermediate SalesA12W.1.5 Case 5: Extraction of Sector 1s Intersectoral Intermediate

PurchasesA12W.1.6 Case 6: Extraction of Sector 1s Intersectoral Intermediate

SalesA12W.1.7 The Ghosh Model and Some Comparisons

Web Appendix 12W.2 Brief History of Leontief Inverses with Errors in theCoefficients of AA12W.2.1 Mathematical BackgroundA12W.2.2 Application of Leontief Inverses

Dwyer and Waugh (1953)Evans (1954)West (1982)Sherman-Morrison and Sonis-Hewings

Figures

1.1 InputOutput Transactions Table 32.1ad Production Functions in Input Space. (a) Linear production function.

(b) Classical production function. (c) Leontief production function.(d) Activity analysis production function. 18

A2.2.1a Solution Space Representation of (A2.2.2); a12 > 0 and a21 > 0 59A2.2.1b Solution Space Representation of (A2.2.2); a21 = 0 60A2.2.1c Solution Space Representation of (A2.2.2); a12 = 0 613.1 Increases in Washington Final Demands Affecting Washington Outputs

via Connecticut 80A3.2.1 Regional Aggregation in the 2000 Chinese Multiregional Model 1094.1 The Circular Flow of Income and Expenditures 1224.2 Circular Flow Example: Point of Departure 1234.3 Introduction of Savings and Investment into the Circular Flow of

Income and Expenditures 1244.4 Introduction of Depreciation into the Circular Flow of Income

and Expenditures 1244.5 Addition of the Rest of World Account 1264.6 Addition of the Government Account 1284.7 Net Worth 1319.1 US Energy Use for 2006 4029.2 Net Energy Analysis 4149.3 Changes in US Energy Consumption: 19721985 4239.4 HudsonJorgenson Model 42410.1 Two-Sector Leontief Model 45310.2 InputOutput and Linear Programming: Example 10.1 45410.3 Different Values of GNP for Example 10.1 45510.4 Linear Programming Solution 45810.5 Goal Programming Solution: Objective 1 46010.6 Goal Programming Solution: Objective 2 46010.7 Goal Programming Solution: Objective 3 461

xxii

List of Figures xxiii

10.8 Goal Programming Solution: Objective 4 46210.9 Goal Programming Generalized InputOutput Initial Solution 46410.10 Generalized InputOutput Goal Programming: Example 10.1

(Objective 1) 46510.11 Generalized InputOutput Goal Programming: Example 10.1





(Objective 6) 46810.16 Location of Air Pollution Sources and Receptor Points 48911.1 Circular Flow of Income, Expenditure, and Markets 50011.2 Sample Macroeconomy: Problem 11.1 537C.1 Franois Quesnays Tableau conomique 727

Tables

1.1 Illustrative Real InputOutput Data Locations 82.1 InputOutput Table of Interindustry Flows of Goods 132.2 Expanded Flow Table for a Two-Sector Economy 142.3 Flows (zij) for the Hypothetical Example 222.4 Technical Coefficients (the A Matrix) for the Hypothetical Example 222.5 Flows (zij) for the Hypothetical Example Associated with xnew 242.6 Round-by-Round Impacts (in dollars) of f 11 = $600 and f 12 = $1500 272.7 The 2003 US Domestic Direct Requirements Matrix, A 292.8 The 2003 US Domestic Total Requirements Matrix, L = (I A)1 292.9 InputOutput Table of Interindustry Flows with Households Endogenous 352.10 Flows (zij) for Hypothetical Example, with Households Endogenous 382.11 Transactions in Physical Units 422.12 Transactions in Monetary Units 422.13 Transactions in Revised Physical Units 422.14 Transactions in Monetary Terms 432.15 The Leontief Quantity and Price Models 452.16 Transactions for Hypothetical Example with One Primary Input 452.17 Flows in Physical Units 472.18 Transactions in Physical Terms (Germany, 1990) (millions of tons) 522.19 Alternative InputOutput Price and Quantity Models 543.1 Interindustry, Interregional Flows of Goods 773.2 Flow Data for a Hypothetical Two-Region Interregional Case 823.3 Data Needed for Conversion of National to Regional Coefficients via the

Product-Mix Approach 893.4 Interregional Shipments of Commodity i 903.5 Flow Data for a Hypothetical Two-Region Multiregional Case 933.6 Interregional Commodity Shipments for the Hypothetical Two-Region

Multiregional Case 943.7 Chinese Interregional and Intraregional Transactions, 2000 (in 10,000) 983.8 Direct Input Coefficients for the Chinese Multiregional Economy, 2000 99

xxiv

List of Tables xxv

3.9 Leontief Inverse Matrix for the Chinese Multiregional Economy, 2000 1003.10 Region- and Sector-Specific Effects (in 1000) of a 100,000

Increase in Final Demand for Manufacturing Goods, China, 2000 101A3.2.1 Regional Classifications in the 2000 Chinese Multiregional Model 109A3.2.2 Sectoral Aggregation in the 2000 Chinese Multiregional Model 1104.1 Basic National Accounts: Example Economy 1254.2 Basic National Accounts Including Rest of World 1274.3 Basic National Accounts Including the Government Sector 1294.4 Balance Statement for the Basic National Accounts 1304.5 The Basic National Accounts Balance Statement in Matrix Form 1304.6 Matrix of National Accounts Including Net Worth Calculations 1324.7 The Commodity-by-Industry Use Table 1364.8 The Industry-by-Commodity Make Table 1384.9 Consolidated Make and Use Accounts 1394.10 InputOutput Transactions: Example 1 1414.11 Consolidated InputOutput Accounts: Example 2 1434.12 Production Account Allocated to Individual Products and Sectors 1454.13 Industry-by-Commodity Make Matrix: Running Example 1464.14 Consolidated Commodity-by-Industry InputOutput Accounts: Running

Example 1474.15 Example Trade and Transportation Margins: Example 2 1484.16 Domestic Interindustry Transactions: Example 4 1504.17 Modified Interindustry Transactions: Example 4 1504.18 Approximation Methods for Scrubbing Interindustry Transactions of

Competitive Imports: Example 5 1554.19 Double Deflation: Example 6 159A4.1.1 Input Coefficients for the Five-Sector, Three-Region Interregional

InputOutput Table for Japan (1965) 169A4.1.2 Spatial Aggregation of IRIO Models: Results for Japanese IRIO Table 171A4.1.3 Five-Sector, Three-Region Multiregional InputOutput Tables for the

United States (1963) 173A4.1.4 Spatial Aggregation of MRIO Models: Results for US MRIO Model 1755.1 The Use Matrix (U) and Other Data for a Two-Commodity,

Two-Industry Hypothetical Example (in Dollars) 1865.2 The Make Matrix (V) and Other Data for a Two-Commodity,

Two-Industry Hypothetical Example (in Dollars) 1875.3 The Complete Set of CommodityIndustry Data 1875.4 Total Requirements Matrices, Commodity-Demand Driven Models 1975.5 Total Requirements Matrices, Industry-Demand Driven Models 1975.6 Rewritten Forms of Total Requirements Matrices 1995.7 Alternative Classifications, Total Requirements Matrices,

Commodity-Demand Driven Models 209

xxvi List of Tables

5.8 Examples of Negative Elements in Real-World Commodity-TechnologyDirect Requirements Matrices [ AC

(cc)= BC1] 210

5.9 A Three-Commodity, Two-Industry Example 2125.10 Share of Secondary Product Output in Total Industry Output (European

Union Countries, 60-sector level) 223A5.2.1 Summary of Two Commodity/Two Industry Results 230A5.2.2 Steps in the Iterative Procedure for the 3 3 Example 2355.11 Commodity Final Demands for US 2003 InputOutput Tables 2396.1 Total Requirements Matrices in CommodityIndustry Models 2496.2 Model Closures with Respect to Households 2566.3 InputOutput Multipliers 2586.4 General Multiplier Formulas 2596.5 Leontief Inverse Matrix, L, for the Chinese Multiregional

Economy, 2000 2696.6 Simple Intra- and Interregional Output Multipliers for the Chinese

Multiregional InputOutput System, 2000 2706.7 Sector-Specific Simple Output Multipliers for the Chinese Multiregional

InputOutput System, 2000 2716.8 Interrelational Interregional Income Multipliers 2777.1 Values of a11 and a23 at Each Step in the RAS Adjustment Procedure 3237.2 Differences from Row and Column Margins at Each Step in the RAS

Adjustment Procedure 3237.3 Elements in the Diagonal Matrices rk and sk , for k = 1, . . . , 7 3247.4 MAD and MAPE when One Coefficient is Known in Advance in an RAS

Estimate 3328.1 Total Intraregional Intermediate Inputs and Intraregional Output

Multipliers for Region 1 (North China) Calculated from SeveralRegionalization Techniques 362

8.2 Components in the DEBRIOT Approach 3758.3 Structure of the TIIO Model 3818.4 China-to-Japan Intermediate Transactions in TIIO 3829.1 Energy and Dollar Flows: Example 9.1 4079.2 Interindustry Economic Transactions: Example 9.2 (millions of dollars) 4099.3 Energy Flows: Example 9.2 (1015 BTUs) 4099.4 Interindustry Transactions in Hybrid Units: Example 9.3 4129.5 InputOutput Transactions for the US Economy in Hybrid Units (1967) 4169.6 Technical Coefficients: Example 9.4 4189.7 Leontief Inverse: Example 9.4 4199.8 Total Primary Energy Intensities: Example 9.4 4209.9 Power Plant Inputs: Example 9.4 4209.10 Summary of Energy InputOutput Relationships: Initial Formulation 428A9.1.1 Dollar Transactions for Example 9.5 (millions of dollars) 431

List of Tables xxvii

A9.1.2 Energy Flows for Example 9.5 (1015 BTUs) 431A9.1.3 Energy Flows for Example 1 Revised (1015 BTUs) 432A9.1.4 Summary of Energy InputOutput Relationships 433A9.1.5 Energy and Dollar Flows for Example 9.1 (Revised) 435A9.1.6 Implied Energy Prices for Example 9.1 (Revised) 435A9.1.7 Results for Example 9.1 (Revised) 43510.1 InputOutput Transactions (millions of dollars) 44810.2 Direct Impact Coefficients 44810.3 Policy Programming: Composite Scenario Weights 47210.4 EconomicEcologic Commodity Flows 47410.5 EconomicEcologic Commodity Flows: Matrix Definitions 47610.6 Pollution-Generation Example: Dollar Transactions 47610.7 InputOutput Transactions: Pollution-Expanded Model Example 47910.8 Basic Structure of EconomicEcologic Models 48310.9 Limited Commodity-by-Industry EconomicEcologic Model 48510.10 EconomicEcologic Models: Example 10.3 48611.1 The Basic National Accounts Balance Statement in Matrix Form 50111.2 (Table 4.5 Revisited) The Basic National Accounts Balance Statement in

Matrix Form: Example 50211.3 The Basic National Accounts Balance Statement in Matrix

Form Expanded to Include the Households Account 50311.4 The Basic National Accounts Balance Statement in Matrix

Form: Example, Expanded to Include the Households Account 50411.5 The Basic National Accounts Balance Statement in Matrix Form Expanded

to Include the Value-Added Account 50511.6 The Basic National Accounts Balance Statement in Matrix Form:

Example, Expanded to Include the Value-Added Account 50511.7 SAM Framework Example: InputOutput Representation 50611.8 SAM Framework Example Using Social Accounting Conventions 50611.9 InputOutput Accounts for Table 11.6 Revisited 50811.10 Expanded Value-Added Accounts 50911.11 Expanded Final-Demand Accounts 50911.12 Sources of Value-Added Income 50911.13 Expanded InputOutput Accounts 51111.14 SAM Representation of Expanded InputOutput Accounts 51211.15 The Basic National Accounts Balance Statement in Matrix

Form Expanded to Include Additional Macro Transactions 51311.16 Expanded National Accounts Balance Statement in Matrix

Form: Example, Expanded to Include Additional Macro Transactions 51311.17 Reduced Form Fully Articulated SAM: Example 11.1 51911.18 Expanded Form Fully Articulated SAM: Example 11.1 52511.19 SAM Framework Example Using Social Accounting Conventions

(revised to include endogenous final demand) 532

xxviii List of Tables

11.20 Comparative InputOutput and SAM Multipliers 53511.21 Unbalanced SAM: Examples 11.12.1 and 11.12.2 53611.22 SAM with Expanded Interindustry Detail for United States, 1988 54012.1 Overview of the Leontief and Ghosh Quantity and Price Models 55512.2 Linkage Measures 55912.3 Classification of Backward and Forward Linkage Results 56012.4 Summary of Spatial/Sectoral Linkage Measures (Two-Region Example) 56212.5 Hypothetical Extraction Linkages 56512.6 Linkage Results, US 2003 Data 56612.7 Classification of Linkage Results, US 2003 Data 56612.8 Hypothetical Extraction Results, US 2003 Data 56612.9 Classification of Hypothetical Extraction Results, US 2003 Data 56712.10 Number of Important Transactions in the 2000 China MRIO Model 56812.11 (lri/xr) 106 for the 2003 US Seven-Sector Model 57512.12 Percentage Change in x Resulting from aij = (0.2)aij 57512.13 Upper Threshold on aij/aij for = 1 Percent 57612.14 Average Values in US Total Requirements Matrices 57712.15 Column Sums of |F[i, j]| for Numerical Example 58012.16 Sum of all Elements in |F[i, j]| (F =

ij

fij) 58013.1 Alternative Structural Decompositions 59713.2 Sector-Specific and Economy-Wide Decomposition Results [Equation

(13.7)] 59713.3 Sector-Specific and Economy-Wide Decomposition Results (with

Two-Factor Final-Demand Decomposition Detail) 60213.4 Sector-Specific and Economy-Wide Decomposition Results (with

Three-Factor Final-Demand Decomposition Detail) 60313.5 Sector-Specific and Economy-Wide Decomposition Results (with

Additional Technology and Final-Demand Decomposition Detail) 60613.6 Selected Empirical Structural Decomposition Studies 61213.7 Selected Empirical Structural Decompositions of Changes in Energy

Use or Pollution Emissions 61613.8 SDA Percentage Change Sensitivities 61713.9 Selected Empirical Structural Decompositions at a Regional,

Interregional or Multiregional Level 620A13.1.1 Alternative Decompositions of x = LBf 657B.2 Transactions Accounts 703B.3 Matrices of Technical Coefficients and Total Requirements 715C.1 Selected International Conferences on InputOutput Analysis 733

Preface

We started working on the first edition of this book (Miller and Blair, 1985) in the late1970s. At that time, inputoutput as an academic topic (outside of Wassily LeontiefsHarvard research group) was a little more than 25 years old 19521979, give ortake a year at either end. We use 1952 because that was when the first author wasintroduced to inputoutput analysis in a sophomore-year economics class at Harvardtaught by Robert Kuenne, who later claimed that was the first time inputoutput hadbeen included (anywhere) in an undergraduate economics course.

In 1962, the first author joined the faculty of the Regional Science Department atthe University of Pennsylvania. He was asked by then department chair Walter Isardto teach the graduate course in linear models for regional analysis; this was to includea strong inputoutput component. At that time coverage of the topic in texts was to befound primarily in two chapters of Dorfman, Samuelson and Solow (1958), in Cheneryand Clark (1959), in Stone (1961) and in a long chapter on inputoutput at the regionallevel in Isard et al. (1960); later there were texts by Miernyk (1965), Yan (1969), andRichardson (1972).

The second author of the current text began teaching an applied course coveringextensions of the inputoutput approach to energy, environmental, and other contem-porary policy issues of the time in that same regional science program at Penn in theearly 1970s, and by the end of that decade the need for a comprehensive and up-to-datetextbook became apparent to us. So the first edition of this book very much reflectedour shared experiences with students (primarily graduate or undergraduate submatric-ulants) in mostly regional science and public policy courses at Penn during the 1960sand 1970s. In addition to the basics (foundations), many of the additional topicswe included (extensions) reflected our research interests at that time interregionalfeedbacks for one of us, energy and environmental applications for the other and spatialaggregation in many-region models as a joint interest.

Over the past decade or so it became increasingly and abundantly clear that thetime was ripe for an update/revision. We began to take this notion seriously around20002001 almost an additional 25 years further into the inputoutput timeline, so

xxix

xxx Preface

the subject was essentially twice as old as when we wrote the first edition. Activity inthe field during that quarter century seems to have exploded. For example:

The International InputOutput Association (IIOA) was founded (1988). The IIOAs journal, Economic Systems Research, began publication (1989). International conferences were held with increasing frequency and drawing increas-

ing numbers of participants (summarized at the end of Appendix C) and startingrecently, intermediate inputoutput meetings are held in nonconference years,co-organized by the IIOA.

In 1998 Heinz Kurz, Erik Dietzenbacher and Christian Lager published an editedthree-volume set of almost 1,500 pages that reproduces some 85 significant inputoutput papers, along with extensive and detailed introductions to each of the volumes(Kurz, Dietzenbacher and Lager, 1998).

These activities are in part a reflection of enormous increases in computer speed andcapacity since the 1950s. The net result is that there is now considerable new materialto be examined, digested and considered for inclusion and explanation.

Accordingly, around the end of 2000 we communicated with about 30 of our inputoutput colleagues throughout the world, asking for help in finding our way throughthis maze of material. We listed some new topics that we thought should be included(e.g., social accounting matrices or SAMs), some that we might emphasize more (e.g.,commodity-by-industry models), some less (e.g., detailed numerical interregional ormultiregional examples), and we asked for reactions and suggestions. Additionally, wetook into account what we knew of the uses to which the first edition had been put, e.g.,as a text for teaching purposes or desk reference for practitioners and researchers.

As a result, we have added some discussion of:

SAMs (and extended inputoutput models) and their connection to inputoutput data; Structural decomposition analysis (SDA); Multiplier decompositions [Miyazawa, additive (Stone), multiplicative (Pyatt and

Round)]; Identifying important coefficients; International inputoutput models.We have expanded discussions of:

The historical background and context for Leontiefs work; The connection of inputoutput accounts and national income and product accounts

(NIPAs); Commodity-by-industry accounting and models; Multipliers, including Miyazawa multipliers, net multipliers, elasticity measures, and

output-to-output multipliers; Location quotients and related techniques for estimating regional technology,

including numerical examples and real-world illustrations; Energy inputoutput analysis to include references to econometric extensions;

Preface xxxi

Environmental applications to include linear programming and multiobjectiveprogramming extensions;

The hypothetical extraction approach to linkage analysis; The Ghosh (supply-side) model; The Leontief price model; Estimating interregional flows; Hybrid methods; Mixed exogenous/endogenous models.

In order to keep the new edition to manageable length, there are topics that had tobe excluded or treated only very briefly; these include:

Econometric/inputoutput model connections; Qualitative inputoutput analysis; Recent developments in dynamic inputoutput modeling; Discussions and comparisons of alternative working models (e.g., REMI and

IMPLAN in the USA and others elsewhere); The role and interpretation of eigenvectors and eigenvalues in inputoutput models.

The historical material on US inputoutput data has been reworked and updated,especially to reflect the international movement toward commodity-by-industry for-mulations. With the ready accessibility of computing capabilities, we have greatlyexpanded the end-of-chapter problems to include many more realistic examples aswell as some real-world examples and applications. Because of the higher level ofmathematical competence that we see in our potential readers as compared with 20+years ago, we have tried to use more compact matrix representations more extensivelyand whenever possible.

With appreciation we acknowledge many helpful conversations, face-to-face andelectronic, with Takahiro Akita, William Beyers, Faye Duchin, Geoffrey Hewings,Takeo Ihara, Andrew Isserman, Randall Jackson, Louis de Mesnard, Jan Oosterhaven,Karen Polenske, Jeffery Round and Guy West. Anne Carter and Joseph Richter helpedus fill in the historical record of IIOA meetings. Sandra Svaljek and Ivan Rusan atThe Institute of Economics, Zagreb, Croatia, kindly supplied us with a copy of MijoSekulics important 1968 article in Ekonomska Analiza, a publication of their Institute.We are grateful to IDE/JETRO (Tokyo), especially Satoshi Inomata, who provided uswith many of their important inputoutput tables and studies using those tables and toleadership and staff of the US Department of Commerce, Bureau of EconomicAnalysis,especially Mark Planting, who helped us navigate the US inputoutput tables. Finally,we single out two colleagues with whom we have had almost continuous interaction foryears: Erik Dietzenbacher, with whom we have had literally hundreds of discussionsand from whom we have had as many suggestions, and Michael Lahr who has beena constant source of critical observations and has recommended and helped us trackdown countless important references.

xxxii Preface

References

Chenery, Hollis B. and Paul G. Clark. 1959. Interindustry Economics. New York: John Wiley andSons.

Dorfman, Robert, PaulA. Samuelson and Robert M. Solow. 1958. Linear Programming and EconomicAnalysis. New York: McGraw-Hill.

Isard, Walter, David F. Bramhall, Gerald A. P. Carrothers, John H. Cumberland, Leon N. Moses,Daniel O. Price and Eugene W. Schooler. 1960. Methods of Regional Analysis: An Introductionto Regional Science. New York: The Technology Press of MIT and John Wiley and Sons.

Kurz, Heinz D., Erik Dietzenbacher and Christian Lager (eds.). 1998. Input-Output Analysis. ThreeVolumes. The International Library of Critical Writings in Economics, No. 92. Cheltenham, UK:Edward Elgar.

Miernyk, William. 1965. The Elements of Input-Output Analysis. New York: Random House.Miller, Ronald E. and Peter D. Blair. 1985. Input-Output Analysis: Foundations and Extensions.

Englewood Cliffs, NJ: Prentice-Hall.Richardson, Harry W. 1972. Input-Output and Regional Economics. New York: John Wiley and Sons

(Halsted Press).Stone, Richard. 1961. Input-Output and National Accounts. Paris: Organization for Economic

Cooperation and Development.Yan, Chiou-Shuang. 1969. Introduction to Input-Output Economics. New York: Holt, Rinehart and

Winston.

1 Introduction and Overview

1.1 Introduction

Inputoutput analysis is the name given to an analytical framework developed byProfessor Wassily Leontief in the late 1930s, in recognition of which he received theNobel Prize in Economic Science in 1973 (Leontief, 1936, 1941). One often speaksof a Leontief model when referring to inputoutput. The term interindustry analysis isalso used, since the fundamental purpose of the inputoutput framework is to analyzethe interdependence of industries in an economy. Today the basic concepts set forth byLeontief are key components of many types of economic analysis and, indeed, inputoutput analysis is one of the most widely applied methods in economics (Baumol,2000). This book develops the framework set forth by Leontief and explores the manyextensions that have been developed over the last nearly three quarters of a century.

In its most basic form, an inputoutput model consists of a system of linear equations,each one of which describes the distribution of an industrys product throughout theeconomy. Most of the extensions to the basic inputoutput framework are introducedto incorporate additional detail of economic activity, such as over time or space, toaccommodate limitations of available data or to connect inputoutput models to otherkinds of economic analysis tools. This book is an updated and considerably expandededition of our 1985 textbook (Miller and Blair, 1985).

In this chapter we introduce the basic inputoutput analysis framework and outlinethe topics to be covered in the balance of the text. Appendix C provides a historicalaccount of the work leading up to Leontiefs formulation and its subsequent devel-opment and refinement. More detailed historical accounts of the early developmentof inputoutput analysis and inputoutput accounts are given in Polenske and Skolka(1976, Chapter 1) and Stone (1984). A fairly complete history of applications of inputoutput analysis since Leontiefs introduction of it is provided in Rose and Miernyk(1989). In the present text we cover many of the developments in inputoutput since itswidespread application as an analysis tool began in the early 1950s. Leontief himselfparticipated in a number of these developments and applications, as will be evidentthroughout this text (see also Polenske, 1999, 2004).

1


The widespread availability of high-speed digital computers has made Leontiefsinputoutput analysis a widely applied and useful tool for economic analysis at manygeographic levels local, regional, national, and even international. Prior to the appear-ance of modern computers, the computational requirements of inputoutput modelsmade them very difficult and even impractical to implement. Today, in the USA alone,inputoutput is routinely applied in national economic analysis by the US Departmentof Commerce, and in regional economic planning and analysis by states, industry, andthe research community. The model is widely applied throughout the world; the UnitedNations has promoted inputoutput as a practical planning tool for developing coun-tries and has sponsored a standardized system of economic accounts for constructinginputoutput tables.

Inputoutput has been also extended to be part of an integrated framework of employ-ment and social accounting metrics associated with industrial production and othereconomic activity, as well as to accommodate more explicitly such topics as inter-national and interregional flows of products and services or accounting for energyconsumption and environmental pollution associated with interindustry activity. In thistext, we present the foundations of the inputoutput model as originally developedby Leontief, as well as the evolution of many methodological extensions to the basicframework. In addition, we illustrate many of the applications of inputoutput and itsusefulness for practical policy questions. Throughout the text, we will review some ofthe current research frontiers.

1.2 InputOutput Analysis: The Basic Framework

The basic Leontief inputoutput model is generally constructed from observed eco-nomic data for a specific geographic region (nation, state, county, etc.). One is concernedwith the activity of a group of industries that both produce goods (outputs) and consumegoods from other industries (inputs) in the process of producing each industrys ownoutput. In practice, the number of industries considered may vary from only a few tohundreds or even thousands. For instance, an industrial sector title might read man-ufactured products, or that same sector might be broken down into many differentspecific products.

The fundamental information used in inputoutput analysis concerns the flows ofproducts from each industrial sector, considered as a producer, to each of the sectors,itself and others, considered as consumers. This basic information from which an inputoutput model is developed is contained in an interindustry transactions table. The rowsof such a table describe the distribution of a producers output throughout the economy.The columns describe the composition of inputs required by a particular industry toproduce its output. These interindustry exchanges of goods constitute the shaded portionof the table depicted in Figure 1.1. The additional columns, labeled Final Demand ,record the sales by each sector to final markets for their production, such as personalconsumption purchases and sales to the federal government. For example, electricity issold to businesses in other sectors as an input to production (an interindustry transaction)

1.3 Outline for this Text 3

Figure 1.1 InputOutput Transactions Table

and also to residential consumers (a final-demand sale). The additional rows, labeledValue Added , account for the other (non-industrial) inputs to production, such as labor,depreciation of capital, indirect business taxes, and imports.

The formulation of analytical models using the basic inputoutput data as justdescribed is the principal purpose of this text. There is a considerable literature devotedto assembling the basic data used in inputoutput models from surveys or interpretationof other primary and secondary sources of economic data. Some of this literature isreferenced in Chapter 4, but, for the most part, in this text we focus on the formulationof models using available data or on methods to compensate for the lack of availabledata.

1.3 Outline for this Text

This text is organized into 14 chapters, beginning with the theory and assumptions ofthe basic inputoutput framework, then exploring many of the extensions developedover the last half century. The text deals mostly with methodological developments,but also covers some of the practical issues associated with implementation of inputoutput models, including many references to the applied literature. Chapters 26cover the main methodological considerations in inputoutput analysis. Chapters 713cover many issues associated with the application of inputoutput analysis to prac-tical problems. The concluding chapter, Chapter 14, sketches a number of relevanttopics for which available space did not permit a more detailed treatment or that werebeyond the scope of this text. The following describes the main topics covered in eachchapter:

Chapter 2 introduces Leontiefs conceptual inputoutput framework and explainshow to develop the fundamental mathematical relationships from the interindustry


transactions table. The key assumptions associated with the basic Leontief modeland implications of those assumptions are recounted and the economic interpretationof the basic framework is explored. The basic framework is illustrated with a highlyaggregated model of the US economy. In addition, the price model formulationof the inputoutput framework is introduced to explore the role of prices in inputoutput models. Appendices to this chapter include a fundamental set of mathematicalconditions for inputoutput models, known as the HawkinsSimon conditions.

Chapter 3 extends the basic inputoutput framework to analysis of regions and therelationships between regions. First, single-region models are presented and thevarious assumptions employed in formulating regional models versus national mod-els are explored. Next, the structure of an interregional inputoutput (IRIO) model,designed to expand the basic inputoutput framework to capture transactions betweenindustrial sectors in regions, is presented. An important simplification of the IRIOmodel designed to deal with the most common of data limitations in constructingsuch models is known as the multiregional inputoutput (MRIO) model. The basicMRIO formulation is presented and the implications of the simplifying assumptionsexplored. Next the balanced regional model is presented, which is mathematicallyidentical to the IRIO framework, but is designed conceptually to capture the distinc-tion between industrial production for regional versus national markets as opposedto delivery to specific regions as in the IRIO framework. In the final section anumber of applied studies are cited in order to illustrate the extraordinary rangeof geographic scale reflected in real-world studies from sub-city neighborhoodsto so-called world models. Appendices to this chapter provide additional develop-ment of mathematical tools helpful for conceptualizing and implementing regionalmodels.

Chapter 4 deals with the construction of inputoutput tables from standardized con-ventions of national economic accounts, such as the widely used System of NationalAccounts (SNA) promoted by the United Nations, including a basic introduction tothe so-called commodity-by-industry or supply-use inputoutput framework devel-oped in additional detail in Chapter 5. A simplified SNA is derived from fundamentaleconomic concepts of the circular flow of income and expenditure, that, as additionalsectoral details are defined for businesses, households, government, foreign trade,and capital formation, ultimately result in the basic commodity-by-industry formu-lation of inputoutput accounts. The process is illustrated with the US inputoutputmodel and some of the key traditional conventions widely applied for such consid-erations as secondary production (multiple products or commodities produced by abusiness), competitive imports (commodities that are also produced domestically)versus non-competitive imports (commodities not produced domestically), trade andtransportation margins on interindustry transactions, or the treatment of scrap andsecondhand goods. Finally, the chapter concludes with an examination of issuesassociated with the level of sectoral and spatial detail in inputoutput models, e.g.,the potential bias introduced by the level of aggregation of industries or regions.


The appendices illustrate the implications of aggregation bias using IRIO and MRIOmodels for Japan and the USA.

Chapter 5 explores variations to the commodity-by-industry inputoutput frame-work introduced in Chapter 4, expanding the basic inputoutput framework toinclude distinguishing between commodities and industries, i.e., the supply ofspecific commodities in the economy and the use of those commodities by col-lections of businesses defined as industries. The chapter introduces the fundamentalcommodity-by-industry accounting relationships and how they relate to the basicinputoutput framework. Alternative assumptions are defined for handling the com-mon accounting issue of secondary production, and economic interpretations ofthose alternative assumptions are presented. The formulations of commodity-drivenand industry-driven models are also presented along with illustrations of vari-ants on combining alternative assumptions for secondary production. Finally, thechapter illustrates a variety of special circumstances encountered with commodity-by-industry models, such as nonsquare commodityindustry systems or the inter-pretation of negative elements. Appendices to this chapter provide some alternativederivations of commodity-by-industry transactions matrices, methods for eliminatingnegative entries in specific types of commodity-by-industry models where appear-ance of such entries is most common, and additional observations on nonsquarecommodity-by-industry systems are provided in an appendix on this texts website(www.cambridge.org/millerandblair).

Chapter 6 examines a number of key summary analytical measures known as multipli-ers that can be derived from inputoutput models to estimate the effects of exogenouschanges on (1) new outputs of economic sectors, (2) income earned by householdsresulting from new outputs, and (3) employment generated from new outputs or (4)value-added generated by production. The general structure of multiplier analysisand special considerations associated with regional, IRIO, and MRIO models aredeveloped. Extensions to capture the effects of income generation for various house-hold groups are explored, as well as additional multiplier variants and decompositioninto meaningful economic components. Chapter appendices expand on a number ofmathematical formulations of household and income multipliers.

Chapter 7 introduces approaches designed to deal with the major challenge in inputoutput analysis that the kinds of information-gathering surveys needed to collectinputoutput data for an economy can be expensive and very time consuming, result-ing in tables of inputoutput coefficients that are outdated before they are produced.These techniques, known as partial survey and nonsurvey approaches to inputoutputtable construction, are central to modern applications of inputoutput analysis. Thechapter begins by reviewing the basic factors contributing to the stability of inputoutput data over time, such as changing technology, prices, and the scale and scopeof business enterprises. Several techniques for updating inputoutput data are devel-oped and the economic implications of each described. The bulk of the chapter isconcerned with the biproportional scaling (or RAS) technique and some hybridmodel variants.


Chapter 8 surveys a range of partial survey and nonsurvey estimation approachesfor creating inputoutput tables at the regional level. Variants of the commonly usedclass of estimating procedures using location quotients are reviewed, which presumea regional estimate of inputoutput data can be derived using some information abouta target region. The RAS technique developed in Chapter 7 is applied to developingregional inputoutput tables using a base national table or a table for another regionand some available data for the target region. These are illustrated using data from athree-region model for China. Techniques for partial survey estimation of commodityflows between regions are also presented along with discussions of several real-worldmultinational applications, including the ChinaJapan Transnational InterregionalModel and Leontiefs World Model.

Chapter 9 explores the extension of the inputoutput framework to more detailedanalysis of energy consumption associated with industrial production, includingsome of the complications that can arise when measuring inputoutput transac-tions in physical units of production rather than in monetary terms of the value ofproduction. Early approaches to energy inputoutput analysis are reviewed and com-pared with contemporary approaches and the strengths and limitations of alternativeapproaches are examined. Special methodological considerations such as adjustingfor energy conversion efficiencies are developed and a number of illustrative appli-cations are presented, including estimation of the energy costs of goods and services,impacts of new energy technologies, and energy taxes. Finally, the role of structuralchange of an inputoutput economy associated with changing patterns of energy useis introduced (more general approaches to structural decomposition analysis usinginputoutput models are covered in Chapter 13). The appendix to this chapter devel-ops more formally the strengths and limitations of alternative energy inputoutputformulations.

Chapter 10 reviews the extensions of the inputoutput framework to incorporateactivities of environmental pollution and elimination associated with economic activ-ities as well as the linkages of inputoutput to models of ecosystems. The chapterbegins with a generalized inputoutput framework which assumes that pollutiongeneration (as well as other measurable factors associated with industrial production,such as energy or material consumption measured in physical units or employmentmeasured in person-years) simply vary in direct proportion to the level of industrialproduction. Applications are presented of the generalized inputoutput formulationto measuring impacts of specified changes to industrial activity and to planningproblems where the objective is to seek an optimal mix of industrial productionsubject to inputoutput relationships between industrial sectors and to constraintson factors associated with industrial production, such as pollution, energy use andemployment. In exploring the application of the generalized inputoutput frameworkto planning problems, basic concepts of linear and multiobjective programming areintroduced. The chapter also explores augmenting a basic Leontief inputoutputmodel with pollution generation and elimination sectors. Finally, expansion of theinputoutput framework to include ecologic sectors to more comprehensively trace


economicecosystem relationships is presented along with a variety of illustrativeapplications.

Chapter 11 expands the inputoutput framework to a broader class of economic anal-ysis tools known as social accounting matrices (SAM) and other so-called extendedinputoutput models to capture activities of income distribution in the economy ina more comprehensive and integrated way, including especially employment andsocial welfare features of an economy. The basic concepts of SAMs are explored andderived from the SNA introduced in Chapters 4 and 5, and the relationships betweenSAMs and inputoutput accounts are presented. The concept of SAM multipliers aswell as the decomposition of SAM multipliers into components with specific eco-nomic interpretations are introduced and illustrated. Finally, techniques for balancingSAM accounts for internal accounting consistency are discussed and a number ofillustrative applications of the use of SAMs are presented.

Chapter 12 presents the so-called supply side inputoutput model, with which thename Ghosh is most often associated. It is discussed both as a quantity model (theearly interpretation) and as a price model (the more modern interpretation). Rela-tionships to the standard Leontief quantity and price models are also explored. Inaddition, the fast growing literature on quantification of economic linkages and analy-sis of the overall structure of economies using inputoutput data is examined. Finally,approaches for identifying key or important coefficients in inputoutput models andalternative measures of coefficient importance are presented.

Chapter 13 introduces and illustrates the basic concepts of structural decompositionanalysis (SDA) within an inputoutput framework. The concept of decompositionof multipliers introduced in Chapter 6 and in Chapter 10 as applied to SAMs isrevisited as a way to analyze economic structure. The application of SDA to MRIOis developed to introduce a spatial context, many applications are cited and sum-maries of their results are presented. Next, mixed endogenousexogenous models areexplored. These models expand upon the standard inputoutput model by allowingfor exogenous specification of both (some) final demands and (some) outputs. Thischapter also introduces dynamic inputoutput models that more explicitly capturethe role of capital investment and utilization in the production process. Appen-dices develop extended presentations of additional decomposition and mixed-modelresults.

Chapter 14 briefly describes some additional extensions to inputoutput analysisfor which space does not permit a detailed treatment, including linkages to econo-metric models, computable general equilibrium models, and measuring economicproductivity.

Appendix A is an introductory review of matrix algebra concepts and methods usedthroughout this text.

Appendix B presents a highly aggregated series of the US inputoutput tablesreferenced and used in end-of-chapter problems in a number of chapters or in sup-plementary problems included on the Internet website associated with this book(www.cambridge.org/millerandblair).


Table 1.1 Illustrative Real InputOutput Data Locations

Data Location

US Domestic Direct Requirements Matrix, 2003 Table 2.7US Domestic Total Requirements Matrix, 2003 Table 2.8Chinese Interregional and Intraregional Transactions, 2000 Table 3.7Direct Input Coefficients for the Chinese Multiregional Economy, 2000 Table 3.8Leontief Inverse Matrix for the Chinese Multiregional Economy, 2000 Table 3.9Four-Region, Three-Sector IRIO Model for the USA and Asia Prob. 3.9Three-Region, Five-Sector IRIO Model for Japan, 1965 Table A4.1.1Three-Region, Five-Sector MRIO Model for the USA, 1963 Table A4.1.3Components of US Total Commodity Final Demand, 2003 Table 5.11Seven-Sector US InputOutput Tables for 1997, 2003, and 2005 Prob. 7.1Seven-Sector Direct Input Coefficients Outputs for Washington State, 1997 Prob. 8.10InputOutput Transactions for the US Economy in Hybrid Units, 1967 Table 9.5Technical Coefficients for the US Economy in Hybrid Units, 1967 Table 9.6Leontief Inverse for the US Economy in Hybrid Units, 1967 Table 9.7Nine-Sector Hybrid Units US Technical Coefficients, 1963 and 1980 Prob. 9.10Macro SAM for Sri Lanka, 1970 Prob. 11.5Macro SAM for the US Economy, 1988 Prob. 11.8SAM with Expanded Interindustry Detail for the USA, 1988 Table 11.22Selected US InputOutput Tables, 19192006 Appendix B

Appendix C provides an historical account of the early development of inputoutputanalysis, including a pre-history of the concepts that led to Leontiefs work as wellas the many methodological developments and applications since.

1.4 Internet Website and Text Locations of Real Datasets

A website associated with this text, www.cambridge.org/millerandblair, includes sup-plementary information in three general areas: (1) additional text (appendices) inselected areas that were not possible to include in the printed text for a variety ofreasons, (2) solutions to end-of-chapter problems as well as supplementary problems,case studies, and suggested inputoutput analysis experiments and study projects and(3) downloadable datasets of many of the examples and problems printed in the text aswell as a library of supplementary real-world datasets and references to additional datathat have come to our attention.

Throughout this text, in various illustrative examples and problems, we employ realbut highly aggregated inputoutput related data for various regions and nations aswell as illustrative interregional inputoutput (IRIO) and multiregional inputoutput(MRIO) data and social accounting matrices (SAM). For convenience, Table 1.1 showsa listing of these sets of data and their locations in this text.

References 9

References

Baumol, William. 2000. Leontiefs Great Leap Forward, Economic Systems Research, 12,141152.

Leontief, Wassily. 1936. Quantitative Input-Output Relations in the Economic System of the UnitedStates, Review of Economics and Statistics, 18, 105125.

1941. The Structure of American Economy 19191939. New York: Oxford University Press.Miller, Ronald E. and Peter D. Blair. 1985. Input-Output Analysis: Foundations and Extensions,

Englewood Cliffs, NJ: Prentice-Hall.Polenske, Karen R. 1999. Wassily W. Leontief, 19051999, Economic Systems Research, 11,

341348.2004. Leontiefs Magnificent Machine and Other Contributions to Applied Economics, in Erik

Dietzenbacher and Michael L. Lahr (eds.), Wassily Leontief and Input-Output Economics. NewYork: Cambridge University Press, pp. 929.

Polenske, Karen R. and Jir V. Skolka (eds.). 1976. Advances in Input-Output Analysis. Proceedingsof the Sixth International Conference on Input-Output Techniques. Vienna, April 2226, 1974.Cambridge, MA: Ballinger.

Rose, Adam and William Miernyk. 1989. Input-Output Analysis: The First Fifty Years, EconomicSystems Research, 1, 229271.

Stone, Richard. 1984. Where Are We Now? A Short Account of Input-Output Studies and TheirPresent Trends, in United Nations Industrial Development Organization (UNIDO), Proceedingsof the Seventh International Conference on Input-Output Techniques. New York: United Nations,pp. 439459. [Reprinted in Ira Sohn (ed.). 1986. Readings in Input-Output Analysis. New York:Oxford University Press, pp. 1331.]

2 Foundations of InputOutputAnalysis

2.1 Introduction

In this chapter we begin to explore the fundamental structure of the inputoutput model,the assumptions behind it, and also some of the simplest kinds of problems to whichit is applied. Later chapters will examine the special features that are associated withregional models and some of the extensions that are necessary for particular kinds ofproblems for example, in energy or environmental studies or as part of a broadersystem of social accounts.

The mathematical structure of an inputoutput system consists of a set of n linearequations with n unknowns; therefore, matrix representations can readily be used. Inthis chapter we will start with more detailed algebraic statements of the fundamen-tal relationships and then go on to use matrix notation and manipulations more andmore frequently. Appendix A contains a review of matrix algebra definitions and oper-ations that are essential for inputoutput models. While solutions to the inputoutputequation system, via an inverse matrix, are straightforward mathematically, we willdiscover that there are interesting economic interpretations to some of the algebraicresults.

2.2 Notation and Fundamental Relationships

An inputoutput model is constructed from observed data for a particular economicarea a nation, a region (however defined), a state, etc. In the beginning, we willassume (for reasons that will become clear in the next chapter) that the economic areais a country. The economic activity in the area must be able to be separated into anumber of segments or producing sectors. These may be industries in the usual sense(e.g., steel) or they may be much smaller categories (e.g., steel nails and spikes) or muchlarger ones (e.g., manufacturing). The necessary data are the flows of products fromeach of the sectors (as a producer/seller) to each of the sectors (as a purchaser/buyer);these interindustry flows, or transactions (or intersectoral flows the terms industryand sector are often used interchangeably in inputoutput analysis) are measured for a

10

2.2 Notation and Fundamental Relationships 11

particular time period (usually a year) and in monetary terms for example, the dollarvalue of steel sold to automobile manufacturers last year.1

The exchanges of goods between sectors are, ultimately, sales and purchases of phys-ical goods tons of steel bought by automobile manufacturers last year. In accountingfor transactions between and among all sectors, it is possible in principle to recordall exchanges either in physical or in monetary terms. While the physical measure isperhaps a better reflection of one sectors use of another sectors product, there aresubstantial measurement problems when sectors actually sell more than one good(a Cadillac CTS and a Ford Focus are distinctly different products with differentprices; in physical units, however, both are cars). For these and other reasons, then,accounts are generally kept in monetary terms, even though this introduces problemsdue to changes in prices that do not reflect changes in the use of physical inputs.(In section 2.6 we will explore the implications of a data set in which transactions areexpressed in physical units for example, tons of steel sold to the automobile sector lastyear.)

One essential set of data for an inputoutput model are monetary values of thetransactions between pairs of sectors (from each sector i to each sector j); these areusually designated as zij. Sector js demand for inputs from other sectors during theyear will have been related to the amount of goods produced by sector j over that sameperiod. For example, the demand from the automobile sector for the output of the steelsector is very closely related to the output of automobiles, the demand for leather bythe shoe-producing sector depends on the number of shoes being produced, etc.

In addition, in any country there are sales to purchasers who are more external orexogenous to the industrial sectors that constitute the producers in the economy forexample, households, government, and foreign trade. The demands of these units and hence the magnitudes of their purchases from each of the industrial sectors aregenerally determined by considerations that are relatively unrelated to the amount beingproduced. For example, government demand for aircraft is related to broad changesin national policy, budget levels, or defense needs; consumer demand for small cars isrelated to gasoline availability, and so on. The demand of these external units, since ittends to be much more for goods to be used as such and not to be used as an input toan industrial production process, is generally referred to as final demand.

Assume that the economy can be categorized into n sectors. If we denote by xi thetotal output (production) of sector i and by fi the total final demand for sector is product,we may write a simple equation accounting for the way in which sector i distributes itsproduct through sales to other sectors and to final demand:

xi = zi1 + + zij + + zin + fi =n

j=1zij + fi (2.1)

1 In Chapters 4 and 5 we will explore more recent distinctions between commodities and industries and seehow these observations lead to alternative representations of the inputoutput model.

12 Foundations of InputOutput Analysis

The zij terms represent interindustry sales by sector i (also known as intermediate sales)to all sectors j (including itself, when j = i). Equation (2.1) represents the distributionof sector i output. There will be an equation like this that identifies sales of the outputof each of the n sectors:

x1 = z11 + + z1j + + z1n + f1...

xi = zi1 + + zij + + zin + fi (2.2)...

xn = zn1 + + znj + + znn + fnLet

x = x1...

xn

, Z = z11 z1n... . . . ...

zn1 znn

and f = f1...

fn

(2.3)Here and throughout this text we use lower-case bold letters for (column) vectors, as inf and x (so x is the corresponding row vector) and upper case bold letters for matrices,as in Z. With this notation, the information in (2.2) on the distribution of each sectorssales can be compactly summarized in matrix notation as

x = Zi + f (2.4)We use i to represent a column vector of 1s (of appropriate dimension here n). This isknown as a summation vector (Section A.8, Appendix A). The important observationis that post-multiplication of a matrix by i creates a column vector whose elements arethe row sums of the matrix. Similarly, i is a row vector of 1s, and premultiplication ofa matrix by i creates a row vector whose elements are the column sums of the matrix.We will use summation vectors often in this and subsequent chapters.

Consider the information in the jth column of zs on the right-hand side:z1j...

zij...

znj

These elements are sales to sector j js purchases of the products of the variousproducing sectors in the country; the column thus represents the sources and magnitudesof sector js inputs. Clearly, in engaging in production, a sector also pays for other items for example, labor and capital and uses other inputs as well, such as inventoried items.


Table 2.1 InputOutput Table of InterindustryFlows of Goods

Buying Sector

1 j nSelling Sector 1 z11 z1j z1n

......

......

i zi1 zij zin...

......

...n zn1 znj znn

All of these primary inputs together are termed the value added in sector j. In addition,imported goods may be purchased as inputs by sector j. All of these inputs (value addedand imports) are often lumped together as purchases from what is called the paymentssector, whereas the zs on the right-hand side of (2.2) serve to record the purchasesfrom the processing sector, the interindustry inputs (or intermediate inputs). Sinceeach equation in (2.2) includes the possibility of purchases by a sector of its own outputas an input to production, these interindustry inputs include intraindustry transactionsas well.

The magnitudes of these interindustry flows can be recorded in a table, with sectors oforigin (producers) listed on the left and the same sectors, now destinations (purchasers),listed across the top. From the column point of view, these show each sectors inputs;from the row point of view the figures are each sectors outputs; hence the nameinputoutput table. These figures are the core of inputoutput analysis.

2.2.1 InputOutput Transactions and National AccountsAs was suggested by Table 1.1, an inputoutput transactions (flow) table, such as thatshown in Table 2.1, constitutes part of a complete set of income and product accountsfor an economy. To emphasize the other elements in a full set of accounts, we considera small, two-sector economy. We present an expanded flow table for this extremelysimple economy in Table 2.2. (We examine more of the details of a system of nationalaccounts in Chapter 4.)

The component parts of the final demand vector for sectors 1 and 2 represent, respec-tively, consumer (household) purchases, purchases for (private) investment purposes,government (federal, state, and local) purchases, and sales abroad (exports). These areoften grouped into domestic final demand (C+I+G) and foreign final demand (exports,E). Then f1 = c1 + i1 + g1 + e1 and similarly f2 = c2 + i2 + g2 + e2.

The component parts of the payments sector are payments by sectors 1 and 2 foremployee compensation (labor services, l1 and l2) and for all other value-added items for example, government services (paid for in taxes), capital (interest payments), land

14 Foundations of InputOutput Analysis

Table 2.2 Expanded Flow Table for a Two-Sector Economy

ProcessingSectors

Final Total1 2 Demand Output (x)

Processing 1 z11 z12 c1 i1 g1 e1 x1Sectors 2 z21 z22 c2 i2 g2 e2 x2

Payments Value Added (v) l1 l2 lC lI lG lE LSectors n1 n2 nC nI nG nE N

Imports m1 m2 mC mI mG mE M

TotalOutlays (x) x1 x2 C I G E X

(rental payments), entrepreneurship (profit), and so on. Denote these other value-addedpayments by n1 and n2; then total value-added payments are v1 = l1+n1, and v2 = l2+n2,for the two sectors.

Finally, assume that some (or perhaps all) sectors use imported goods in producingtheir outputs. One approach is to record these import amounts in an imports row in thepayments sector as m1 and m2.2 Total expenditures in the payments sector by sectors 1and 2 are l1 + n1 + m1 = v1 + m1 and l2 + n2 + m2 = v2 + m2, respectively. However,it is often the case that the exports part of the final demand column is expressed as netexports so that the sum of all final demands is equal to traditional definitions of grossdomestic product, i.e., net of imports. In that case a distinction is often made betweenimports of goods that are also domestically produced (competitive imports) and thosefor which there is no domestic source (noncompetitive imports), and all the competitiveimports in the imports row will have been netted out of the appropriate elements in agross exports column. Under these circumstances it is possible for one or more elementsin the net export column to be negative, if the value of imports of those goods exceedsthe value of exports. (For example, if an economy exported d300 million of agriculturalproducts last year but imported d350 million, the net exports figure for the agriculturalsector would be d50 million.) Also, if the federal government sells more of a stockpileditem (e.g., wheat) than it buys, a negative entry in the government column of the finaldemand part of the table could result. If the negative number is large enough, it couldswamp the other (positive) final demand purchases of that good, leaving a negativetotal final demand figure.

The elements in the intersection of the value-added rows and the final demandcolumns represent payments by final consumers for labor services (for example,lC includes household payments for, say, domestic help; lG represents payments to

2 The treatment of imports in inputoutput accounts is much more complicated than this, but for the present weprefer to concentrate on the overall structure of a transactions table. We return to imports in section 2.3.4 below,and in more detail in Chapter 4.


government workers) and for other value added (for example, nC includes tax paymentsby households). In the imports row and final demand columns are, for example, mG,which represents government purchases of imported items, and mE , which representsimported items that are re-exported.

Summing down the total output column, total gross output throughout the economy,X , is found as

X = x1 + x2 + L + N + MThis same value can be found by summing across the total outlays row;

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