Principles of Engineering Economic Analysis, Revised Canadian Edition [rev ed.] 0921332491, 9780921332497

Citation preview

Principles of Engineering Economic Analysis R e v is e d

Andrew J. Szonyi Robert G. Fenton John A. White Marvin H. Agee Kenneth E. Case

C a n a d ia n

E d it io n

Principles of Engineering Economic Analysis Revised Canadian Edition Andrew J. Szonyi Robert G. Fenton John A. White Marvin H. Agee Kenneth E. Case

WE W all & Emerson, Inc. Toronto, O ntario ♦ Dayton, Ohio

Copyright © 1977 by John W iley & Sons, Inc. C opyright © 1982 by John W iley & Sons Canada Limited C opyright © 1 9 8 9 ,2 0 0 0 by Andrew J. Szonyi Cover design by Alexander W all A ll rights reserved. No part o f this publication m ay be reproduced or transm itted in any form o r by any means, electronic or mechanical, In­ cluding photography, recording, o r any Inform ation storage and re­ trieval system, w ith o u t permission In w ritin g from the publisher. Orders fo r this b o o k o r requests fo r permission to make copies of any p a rt o f this w o rk should be sent to: W all & Emerson, Inc. Six O 'C onnor Drive Toronto, O ntario, Canada M 4K 2K1 Telephone: (416) 467-8685 Toll free (outside o f Toronto) 1-877-409-4601 Fax: (416) 35 2-5368 E-mail: w all@ wallbooks.com W ebsite: w w w .w allbooks.com Canadian Cataloguing in Publication Data

M ain entry under title: Principles o f engineering econom ic analysis Rev. Canadian ed. Includes bibliographical references and Index. ISBN 0-921332-49-1 1. Engineering economy. I. Szonyi, Andrew J., 1934TA177.4P74 2 0 00 Printed in Canada. 2nd printing, February 2003.

620 .0068'1

C99-931893-4

Table of Contents Preface to the Revised Canadian Edition.......................................lx

Chapter 1. Introduction 1.1 B ackground................................................................................ 1 1.2 The Problem-Solving Process.................................................. 2 1.3 Cash Flow .................................................................................... 5 1.4 A Fundamental Concept: The Time Valueo f M o n e y .............. 7 1.5 Economic Justification May Not Be E n o u g h ......................... 10 1.6 Non-Monetary C onsiderations...............................................11 1.7 Overview o f the Text................................................................ 14

Chapter 2, Cost Concepts and Cost Estimation 2.1 In tro d u ctio n .............................................................................. 17 2.2 Cost T e rm ino logy.....................................................................18 2.2.1 Life-Cycle Cost, 19 2.2.2 Past and Sunk Costs, 21 2.2.3 Future and Opportunity Costs, 22 2.2.4 Direct Indirect, and Overhead Costs, 24 2.2.5 Fixed and Variable Costs, 26 2.2.6 Average and Marginal Cost, 28

2.3 E stim ation................................................................................ 30 2.3.1 Project Estimation, 32 2.3.2 General Sources of Data, 34

2.4 Capital Cost E stim ation...........................................................35 2.4.1 Price Indexes, 35 2.4.2 Cost-Capacity Relationship, 38

2.5 The Learning C u rv e ................................................................ 40 2.6 Standard Costs, 44

Problems......................................................................................... 45

iv

Chapter 3. Time Value of Money Operations 3.1 Introduction.................................................................................... 4

9

3.2 Interest Calculations...................................................................... 4 9 3.3 Single Sums o f M o n e y .................................................................. 52 3.4 Series o f Cash R ow s...................................................................... 54 3.4.1 Uniform Series of Cash Rows, 58 3.4.2 Gradient Series of Cash Hows, 64 3.4.3 Geometric Series o f Cash Hows, 69

3.5 Multiple Compounding Periods In a Y e a r ................................ 73 3.6 Continuous Compounding............................................................76 3.6.1 Discrete Hows, 77 3.6.2 Continuous How, 80

3.7 Equivalence.................................................................................... 83 3.8 Loan Payments.............................................................................. 9 0 3.9 Special T o p ics................................................................................ 9 3 3.9.1 Changing Interest Rates, 94 3.9.2 End-of-Period Cash Hows and End-of-Period Compounding, 97 3.9.3 Perpetuities and Capitalized Value, 99 3.9.4 Bond Problems, 103 3.9.5 Capital Recovery Cost, 110

3.10 Inflationary Effects...................................................................... 112 3.11 S u m m a ry .................................................................................... 115 Problems...............................................................................................116

Chapter 4. Comparison of Alternatives 4.1 In tro d u ctio n .................................................................................. 137 4.2 Defining M utually Exclusive Alternatives.................................138 4.3 The Planning P e riod.................................................................. ... 4.4 Developing Cash Row P rofiles................................................. 146 4.5 Specifying the Time Value o f M o n e y .......................................148 4.6 Ttie Measures o f M e rit............................................................ 4.6.1 Present Worth Method, 152 4.6.2 Annual Worth Method, 153 4.6.3 Future Worth Method, 154 4.6.4 Payback Period Method, 155 4.6.5 Rate of Return Method, 157

151

Table of Contents

v

4.6.6 Savlngs/lnvestment Ratio Method, 163

4.7 Comparing the Investment A lternatives............................. 164 4.7.1 Ranking Approaches, 165 4.7.2 Incremental Approaches, 167

4.8 Supplementary Analyses..................................................... 172 4.9 Selecting the Preferred Alternative..................................... 172 4.10 Analyzing Alternatives w ith No Positive Cash Rows . . . . 174 4.11 Classical Method of Dealing w ith Unequal L iv e s ............ 179 4.12 Replacement Analysis.......................................................... 183 4.12.1 Cash Row Approach, 184 4.12.2 Classical Approach, 188

4.13 Computer Applications.........................................................195 4.14 S um m ary.............................................................................. 196 Problems........................................................................................ 197

Chapter 5. Depredation and Income Tax Considerations 5.1 Introduction............................................................................ 223 5.2 The Meaning o f Depreciation...............................................223 5.3 Factors Used to Determine D e p re d a tio n ........................... 225 5.4 Methods o f D epredation.......................................................226 5.4.1 Straight-Une Depredation, 226 5.4.2 Sum of the Years' Digits Depredation, 228 5.4.3 Declining Balance Depredation, 229 5.4.4 Capital Cost Allowance, 230 5.4.5 Double Declining Depredation, 232 5.4.6 Sinking Fund Depredation, 235

5.5 Comparison of Depreciation M e th o d s ............................... 238 5.6 Special Provisions fo r Accelerated D epreciation............... 238 5.7 Other Methods of D ep red ation...........................................240 5.7.1 Units of Produdion Method, 240 5.7.2 Operating Day Method, 240 5.7.3 Income Forecast Method, 241 5.7.4 Multiple-Asset Accounts, 241

5.8 ■fex Concepts......................................................................... 242 5.9 Corporate Income Tfcx: Business In c o m e ..........................242 5.9.1 Small Business Thx Credit, 246 5.9.2 M anufadurtng and Processing Profits Deduction, 246

5.9.3 Determining Taxable Income, 247

5.10 After-iax Cash H o w ................................................................ 2

4 8

5.11 Effect of Depreciation M e th o d .............................................. 251 5.12 Effect of Interest on Borrowed M o n e y..................................2 5 4 5.13 Cany-back and Carry-forward R u le s .................................... 2 5 8 5.14 Capital Gains and Losses........................................................ 2 5 9 5.15 Tax ’n-eatment o f Capital Assets............................................ 2 6 0 5.16 Recaptured Capital Cost A llo w a n c e ...................................... 261 5.17 Income Tax Incentives............................................................ 2 6 3 5.18 Lease-Buy Considerations...................................................... 2 6 6 5.19 D epletion.................................................................................. 2 6 9 5.20 Summary.................................................................................. 2 7 0 Problems............................................................................................ 271

Chapter 6. Economic Analysis of Projects in th e Public Sector 6.1 Introduction.................................................................................2 8 3 6.2 The Nature o f Public Projects...................................................2 8 3 6.3 Objectives in Project Evalution.................................................2 8 4 6.4 Benefit-Cost A nalysis.................................................................2 8 5 6.5 Important Considerations in Evaluating Public P ro je c ts . . 2 9 8 6.5.1 Point of View, 298 6.5.2 Selection o f the Interest Rate, 301 6.5.3 Assessment of Benefit-Cost Factors, 305 6.5.4 Overcounting, 308 6.5.5 Unequal Lives, 309 6.5.6 Tolls, Fees, and User Charges, 311

6.6 Multiple-Use Projects..................................................................... 6.7 Problems w ith the B/C R a tio .....................................................314 6.8 Cost-Effectiveness A n a ly s is ....................................................... 315 6.8.1 The Standardized Approach, 317

6.9 Summary................................................. 32o P r O b l e m s .......................................... ■■ ' ^ . ^ ' ' . ' . ' . ^ ' . ' . ^ ' . ' . 3 2 0

Chapter 7. Break-Even, Sensttfvtty, and Risk Analyses 7.1 Introduction....................................

^29

7.2 Unear Break-Even A nalysis.........................

3 3 0

Table of C o n te n ts ________________________________

vi

7.3 Nonlinear Break-Even Analysis............................................335 7.4 Sensitivity A nalysis................................................................ 342 7.5 Risk A n a lysis.......................................................................... 351 7.5.1 Distributions, 352 7.5.2 Risk Aggregation, 354

7.6 Computer S im ulation............................................................ 363 7.7 S um m ary................................................................................ 370 Problems........................................................................................ 371

Chapter 8. Decision Models 8.1 Introduction............................................................................ 381 8.2 The Matrix Decision M odel...................................................385 8.3 Decisions under AssumedC ertainty.................................... 388 8.4 Decisions under R isk............................................................ 390 8.4.1 Dominance, 391 8 .4 2 Expectation-Variance Principle, 392 8.4.3 Most Probable Future Principle, 393 8.4.4 Aspiration-Level Principle, 394

8.5 Decisions under Uncertainty.................................................395 8.5.1 The Laplace Principle, 396 8.5.2 Maximin and M inimax Principles, 396 8.5.3 M aximax and M lnlmin Principles, 397 8.5.4 Hurwicz Principle, 398 8.5.5 Savage Principle (Minimax Regret), 401

8.6 Sequential D ecisions............................................................ 403 8.6.1 Decision Trees, 404 8.6.2 The Conditional Probability Theorem, 417 8.6.3 The Value o f Perfect Information, 421 8.6.4 The Value o f Imperfect Information, 422 8.6.5 Sequential Dedslons-Summary Comments, 435

8.7 Multiple Objectives................................................................ 435 8.7.1 Classifying Objectives According to Importance, 436 8.7.2 Ranking, 436 8.7.3 W eighting Objectives, 438 8.7.4 Determining the Value o f Multiple Objectives, 442

8.8 Sum m ary................................................................................ 444 P roblem s...................................................................................... 446

Chapter 9. Accounting Pr Indples 9.1 Introduction............................................ 9.2 Balance Sheet........................................ 9.3 Income Statem ent................................ 9.4 interpretation o f Financial Statements 9.5 Cost Accounting.................................... Problems.......................................................

459 459 462 462 471 477

Chapter 10. Rmdamental Economic Concepts 10.1 Introduction...................................................... 10.2 Supply and D e m a n d ....................................... 102.1 Demand, 4 8 3 1 0 2 2 Shift in the Demand Curve, 4 8 6 102 .3 Supply, 4 8 8 102 .4 Shift in the Supply Curve, 4 9 0 102 .5 Price, 4 9 0 1 0 2 .6 Elasticity of Demand, 4 9 6 1 0 2 .7 Total Expenditure, 4 9 9 1 0 2 .8 Elasticity of Supply, 5 02 1 0 2 3 Price Control, 5 03 102.10 Price Support, 5 0 6

10.3 Production................................................................................. 5 0 8 103.1 The Production Function, 5 0 8 1 0 3 2 The Univariable Production Function, 5 0 9 10.3.3 Multivariable Production Function, 513 10.3.4 Cost of Production, 515 10.3.5 The Expansion Path, 521

10.4 Summ ary................................................................................... 5 2 4 P roblem s......................................................................................... ...

Appendices A. Discrete C om pounding............................................................. 5 2 9 B. Continuous C o m p o u n d in g ....................................................... 553 C Answers to Even-Numbered P ro b le m s ................................... 583 D. Glossary o f Technical Term s................................................. 589 Index.................................................. 607

Preface to the Revised Canadian Edition To the professional engineer who is daily required to make financial decisions, skill in economic analysis is essential. Development of this skill must therefore be an integral part o f an engineer's training. As we have taught the subject to thousands of students over the past thirty years, we have found that the excellent American text by Professors White, Agee, and Case meets the demands o f such a course. However, Canadian instructors also need a text that provides Canadian informa­ tion and addresses economic topics specific to this country. The con­ cept of producing a Canadian book, originally based on White, Agee, and Case, Principles o f Enginering Economic Analysis emerged In re­ sponse to this need. In this revised edition of the first Canadian edition, several topics have been amended, updated and extended. The section on cost and cost estimation is extended to include learning curves as well as cost­ capacity relations and price indices. The section on depreciation and income taxes are updated to include the latest information about the Canadian Income Tax Act. A new chapter is created to cover the funda­ mentals o f financial and cost accounting. And all other sections o f the book are corrected, simplified and amended. The book includes infor­ mation, Illustration and problems on all traditional topics of engineer­ in g e c o n o m ic a n a ly s is , and m a n a g e m e n t and b u sin e ss decision-making tools. Throughout the text and in the problems, all units and symbols are expressed in SI units (Systeme International d'Unites). While responsibility for the content of this edition must be solely ours, many users o f the American edition and others have contributed to changes through their encouragement, comments, and sugges­ tions. We are indebted to M. L. Bilodeau, McGill U niversity; Philip H. Byer and Scott J. Rogers, University of Toronto; P. S. Chisholm, University o f Guelph; A. Clayton and K. McLachlan, University of Mani­ toba; V. Bruce Irvine, University of Saskatchewan; and C G. Miller,

X

Queen's University; whose reviews o f specific chapters w ere so help­ ful. Thanks are also due to G. Kerl of the National Research Council. A. J. Szonyl R. G. Fenton, Toronto, O n tario July 1 9 9 9

Chapter 1

Introduction 1.1 Background Engineering decisions cover a wide variety o f areas ranging from choosing airport locations to selecting production methods and deter­ mining budget requirements. Since all corporate and government de­ cisions are Influenced by financial considerations, one o f the most effective and Important tools available to engineers Is economic analysis. Here are only a few o f the spheres of activity to which eco­ nomic analysis can be applied: examining design alternatives; decid­ ing the location and size of production plants, railway lines, roads, bridges and tunnels; and installing airports and harbour facilities, dams and Irrigation systems, communication networks, power gener­ ating facilities and power distribution systems. The engineer may also be involved In production decisions such as selecting production methods, assessing w hat quantities of raw materials should be pur­ chased, what types of equipment and machinery should be acquired; determining operating and maintenance procedures, methods fo r the storage, packaging and delivery of goods; and setting down testing and quality-control methods for production. Managerial decisions that engineers may be required to make Include determining capital budget requirements; selecting research and development projects; and establishing production targets. This book is designed to help the engineer learn the rationale and methods of economic analysis and their application to engineering decision making In both the private and public sectors. Traditionally, the application of economic analysis techniques In the comparison of engineering alternatives has been referred to as engi­ neering economy, economic analysis, and economic decision analysis, am ong other names. However, the emergence of a widespread inter­ est In the economic analysis o f engineering decisions In the public sec­ tor has brought about greater use of the more general term economic analysis.

Chapter 1

2

1.2 The Problem-Solving Process Economic analyses are typically performed as a part o f th e o v e ra ll problem-solving process. In deslgnatlnga new or Im proved p ro d u ct a manufacturing process, or a system to provide a des red service, th e •problem solver Is Involved In perform ing the fo llo w in g fiv e steps: 1. Formulation o f the problem. 2. Analysis o f the problem. 3. Search for alternative solutions to the problem . 4. Selection o f the preferred solution. 5. Specification o f the preferred solution. In Step 4 the selection o f the preferred solution Is fre q u e n tly based o n the economic performance o f the alternatives. The formulation o f the problem involves the establishm ent o f its boundaries, and it Is aided by taking a black box approach. A n o rig i­ nating state o f affairs (State A) and a desired state o f affairs (State B) e x­ is t A transformation must take place In going fro m State A to State B, as depicted In Figure 1.1. More than one m ethod o f p e rfo rm in g th e transformation from State A to State B exists, and there is une qual p re f­ erability o f these methods. The solution to the problem is visualized as a black box o f unknown, unspecified contents havin g in p u t A a n d o u t­ put B.

State A

> State B

Fig. 1.1 Black box approach.

p r^

P h a s in g o f t h e n d U d l n g r e s t r l c t i ° n s a nd the crite ria to w S n c l u X T / ^ a f ^ - Considerable fact g a th e rin g is in ­ volved, Including the real restrictions on the problem w h ich m ust be m

“

n S lS tS o f 3

d e ta ile d

Introduction

3

satisfied. Consequently, any budget, quality, safety, personnel, envi­ ronmental, and service-level constraints that may exist are Identified. The search fo r alternative solutions to the problem Involves the use of the engineer's creativity In developing feasible solutions to the prob­ lem. One suggested approach is as follows: 1. Exert the necessary effort. 2. Not get bogged down In details too soon. 3. Make liberal use o f the questioning attitude. 4. Seek many alternatives. 5. Avoid conservatism. 6. Avoid premature rejection. 7. Avoid premature acceptance. 8. Refer to analogous problems for Ideas. 9. Consult others. 10. Attem pt to divorce your thinking from the existing solution. Unfortunately, the natural tendency Is to go on to the evaluation of al­ ternatives at the expense of searching for additional alternative solu­ tions. As a result, the search process often terminates with the development o f the first alternative that can be justified economically. By separating the search process from the selection process, we en­ hance the possibility o f generating a number o f alternatives that can be justified economically. The selection o f the preferred solution consists o f evaluating the alter­ natives, using the appropriate criteria. The alternatives are examined In light o f the constraints, and Infeasible alternatives are eliminated. The benefits produced by the feasible alternatives are then compared. Among the criteria considered for choosing the best alternative Is the economic performance o f each alternative. The specification o f the preferred solution consists of a detailed descrip­ tion of the solution to be Implemented. Predictions of the performance characteristics of the solution to the problem are included In the speci­ fication.

Chapter 1

4

Example 1.1

As an illustration of the problem-solving procedure, Proxax Industries, a leading manufacturer o f automotive brake drums was faced with the need to expand Its distribution operations. A study team was formed to analyze the problem and develop a number of feasible alternative solutions. After analyzing the prob­ lem and projecting future distribution requirements, the fo llo w in g alternatives were reached: 1. Consolidate all distribution activities and expand the existing distribution centre, located In London, Ontario. 2. Consolidate all distribution activities and construct a new distribution centre, location to be determined. 3. Decentralize the distribution function and build several new distribution centres geographically dispersed. After considering the pros and cons of each alternative, the presi­ dent of the company directed the study team to pursue the second. An extensive plant location study was performed. The location study resulted in five candidate locations being selected fo r fin a l consideration. The criteria used to make the final selection in­ cluded: 1. Land cost and availability. 2. Labour availability and cost 3. Proximity to supply and distribution points (present and future). 4. Property taxes and insurance rates. 5. Transportation (access to rail and main highways). 6. Community attitudes. 7. Building costs.

Based on site visits to each location, the director o f engineering o f the company selected a site in Oshawa, Ontario, and directed the study team to develop alternatives for the material handling sys­ tem to be used in the distribution centre. Applying the problem-solving procedure, four alternatives were identified for evaluation. The first involved the use o f pallet racks, lift trucks, and flow racks; the second included the use o f an auto­ mated stacker crane system, lift trucks, conveyors, and flo w racks; the third suggested narrow-aisle, guided picking machines, highrise shelving conveyors, driverless tractor trains, and lift trucks; the fourth consisted of an automated stacker crane system, an automatic guided vehicle system, a conveyor system, and highrise, narrow-aisle lift trucks. A planning period of 10 years was used in perform ing an eco­ nomic analysis of each alternative. The fourth was the m ost eco-

Introduction

5

nomlcal and was recommended to management Based on the detailed presentation and the economics Involved, management approved a budget of $90 million to implement the recommenda­ tions of the study team. As demonstrated by the preceding illustration, engineers must be prepared to defend their solutions to problems. Economic perform­ ance Is among the criteria used to evaluate each alternative. Monetary considerations seldom can be Ignored. If economics are not consid­ ered In the criteria used in the final evaluation o f the alternatives, they are usually involved In an initial screening o f them. In fact, cost can be a lim iting factor on the alternatives that can be considered, as well as the basis fo r the final selection. This textbook treats in detail Step 4 in the problem-solving process, the step that Involves the selection o f the preferred solution. The pro­ cess o f measuring cash flows and benefits, and the consideration of multiple objectives In selecting the preferred alternative are also treated. In comparing alternatives, the differences w ill be emphasized. Conse­ quently, the aspects o f the alternatives that are the same normally w ill not be included in the analysis.

1.3 Cash H o w Our approach to the subject o f engineering economic analysis will be a cashflo w approach. A cash flow occurs when money actually changes hands from one individual or organization to another. A single cash flow Item Is either positive, if money is received (revenue), or negative, if money Is dispersed (expenditure). Companies, systems, projects, or any engineering undertaking have cash flows. The cash flow o f a proj­ ect, for example, includes all revenues received and expenditures in­ curred during the life o f the project If economic criteria are used for finding the preferred solution to an engineering problem, the selection is always based on comparing the cash flows. Cash flows are frequently represented graphically as cash flow dia­ grams (see Figure 1.2). Time, measured In periods (years, months, or days), is shown on the horizontal axis o f the diagram. On Figure 1.2, the time period is a year. The numbers Indicate ends o f tim e periods. For example, the end o f the third period Is labelled 3, and It indicates the end o f Year 3. The cash flow Items are represented by vertical ar­ rows. Positive quantities (up arrows) are revenues, and negative quan-

Chapter 1

6

titles (down arrows) are expenditures. The cash flo w d ia g ra m o f Figure 12 shows an expenditure o f $10 000 Incurred at th e b e g in n in g o f the first year, followed by revenue of $8 000 received at th e e n d o f Year 1, and $14 000 received at the end o f Year 5. Cash flow expenditures at any given tim e should Include a ll m onies dispersed, Including Income tax payments made. If Incom e ta x pay­ ments are excluded, the cash flow Is referred to as before-tax cash flow . After-tax cash flow Includes Income tax paym ents In a d d itio n to all other expenditures. In the private sector (business enterprises), after-tax cash flo w should normally be used for selectlngthe preferred alternative. H o w ever, If In­ come tax payments equally affect all alternatives, b e fo re -ta x cash flo w analysis would suffice. Occasionally, when incom e ta x p a y m e n t pre­ diction Is difficult, before-tax cash flow analysis m a y be used fo r o b ­ taining a preliminary solution for the preferred alterna tive. The public sector (federal and provincial governm ents, m u n ic ip a li­ ties, school boards, universities, hospitals, police, arm y, ch a rita b le o r­ ganizations, and government Institutions, etc.) does n o t p a y Incom e tax, therefore In the economic analysis o f public sector projects, the cash flow obviously does not Include Income ta x paym ents. Sound engineering decisions based on econom ic crite ria can o n ly be reached If the cash flows for all alternatives are co m p le te a n d accu­ rate.

▲

(+) H

$14 000

4 $8 000

0

1

2

3

J-----------------------Years 4 5

▼

$10000

R0-1.2 A cash flow diagram.

Introduction

7

1.4 A Fundamental Concept: The Time Value of Money A fu n d a m e n ta l concept underlies m uch o f the m ate rial covered in th e te xt: m oney has a tim e value. T he value o f a given sum o f m o n e y de­ pends o n when th e m o n e y Is received. E xa m p le 1.2 To illustrate the concept o f the time value o f money, suppose you received the follow ing offer for an Invention: you either receive $10 000 now or $X a year from now. Would you choose to receive the $10 000 now or the $X one year from now If $X equaled (1) $10 000, (2) $10 200, (3) $10 800, (4) $12 000? In presenting this situation to numerous students, no students preferred Case 1, very few students preferred Case 2, most stu­ dents preferred Case 3, and all students preferred Case 4. The point is that the value o f $10 000 one year from now was perceived to be less than the value of $10 000 at present For most students, the value o f $10 200 one year from now was believed to be less than the value of $10 000 at present Only a few students felt that $10 800 a year from now was less valuable than $10 000 at pres­ ent. All students believed $12 000 a year from now was more valu­ able than $10 000 at present Thus, for each individual student some value (or range of values) of $X exists for which one would have no preference between receiving $10 000 now versus receiv­ ing $X a year from now. If, for example, one is indifferent for $X equal to $10 600, then we would conclude that $10 600 occurring one year from now has a present value o f $10 000fo r that particular Individual. E xa m p le 1 3 To continue our consideration of different cash flow situations, ex­ amine the tw o cash flow profiles given In Table 1.1. Both alterna­ tives Involve an Investment of $10 000 in ventures that last for four years. Alternative A Involves the purchase of a computer and soft­ ware by a consulting engineer w ho Is planning on providing com­ puterized finite element analysis capability for clients. Since the engineer anticipates that competition w ill develop quickly, a de­ clining revenue profile Is anticipated.

Chapter 1

8_ _ _ _ _ _

Table 1.1

Cash Flow Profiles for TWo Investment Alternatives CF

End of Year

A

B

A -B

-$ 1 0 000

- $10 000

$0

1

+ 7 000

+ 1 000

+ 6 000

2

+ 5 000

+ 3 000

+ 2 0 00

3

+ 3 000

+ 5000

- 2 0 00

4

+ 1 000

+ 7 000

-6 0 0 0

0

Alternative B Involves an Investment In a coal-m lnlng venture by a group of individuals. Increasing quantities o f coal are to be sold over a four-year periold. Consequently, an Increasing revenue profile Is anticipated. The consulting engineer has available funds sufficient to under­ take either investment, but not both. The cash flows show n are af­ ter taxes and other expenses have been ded u cte d . B oth Investments result In $16 000 being received over the four-year period; hence, a net cash flow of $6 000 occurs In both cases. Which would you prefer? If you prefer Alternative B, then you are not acting in a manner consistent with the concept th a t m oney has a time value. The $6 000 difference at the end o f the first year is worth more than the $6 000 difference at the end o f th e fo u rth year. Likewise, the $2 000 difference at the end o f the second year Is worth more than the $2 000 difference at the end o f the th ird year.

Example 1.4 As another Illustration of the Impact o f the tim e value o f m o n ey on the preference between Investment alternatives, consider alterna­ tives c and D, having the cash flow profiles depicted In Figure 1.3. me cash flow diagrams Indicate that the positive cash flo w s fo r Alf o r Alternative D, except th a t the f ^ S V^ a r e W e n t i c a l t 0 te m S X

ye5 b o t h a l t e r n a «ves require an Invest^ t o ^ ^ ^ ^ ^ ^ e s m u s t b e s e l e c t e d , then AlP re fe ,T e d t 0 A " e m a t i v e D ' b a s e d ,h e ,lm e m X?

Introduction

9

(+) 0

— I

1

$3000

$3000

$3000

Jk

Jk

ik

2

▼ $6000

I

3

I

4

$3000

▲

(+) 2

▼ $6000

6

7

Alternative C

$3000

H

5

3

$3000

▲ 4

5

4 6

7

Alternative D

Fig. 1.3 Cash flow diagrams for Alternatives C and D.

Example 1.5 A third Illustration o f the effect of the time value of money on the selection o f the preferred investment Is presented in Figure 1.4. Ei­ ther Alternative E or Alternative F must be selected; the only differ­ ences In the performance characteristics o f the two alternatives are econom ic. As shown in Figure 1.4, the economic differences re­ duce to a situation in which the receipt of $100 Is delayed In order to receive $200 a year later. For this illustration, we would con­ clude that most students polled earlier would prefer Alternative E to Alternative F.

While the previous examples provide perceptual Impressions o f the Importance o f the time value o f money, there are specific mathemati­ cal conversions that may be used to determine exactly the differences between the above alternatives. These conversions, the tim e value o f money operations, are presented In detail In Chapter 3.

Chapter 1 10

$300 $200

$200

$200

I

f

2

3

A

(+ )

(-)

▼

$4 0 0

Alternative E

$300 $200

$200

A

(+) H

▲

0

1

▼

$400

2

$100

3

Alternative F

$200

▲

(+) (-)

0

1

1

1

2

1

3 r $100

4

Difference between Alternatives E and F

Fig. 1.4 Cash flow diagrams for Alternatives E and F.

1.5 Economic Justification M ay N ot Be Enough Even though the text concentrates on the economic aspects o f invest­ ment alternatives, note that in many Instances economic justification may not be enough! Decisions can be quite different from recommen­ dations. Managers typically have multiple criteria to consider in reach­ ing a final decision about the alternative to be adopted. Among the

Introduction

11

factors to be considered are: quality, safety, environmental Impact, community attitudes, labour-management relationships, cash flow position, risks, system reliability, system availability, system maintain­ ability, system operability, system flexibility, Impact on personnel lev­ els, training requirements, comparisons with competitors, impact on the different units w ithin the organization, ego, customers' prefer­ ences, capital requirements, and economic Justification. The list of criteria can be daunting. However, perhaps the most im­ portant ingredient that separates the selected alternative from the runners-up Is the salesmanship of the Individual w ho presents the alter­ native to the manager. It is not unusual for a manager to adopt a weaker alternative because of the persuasive powers o f the person who presented it. A tendency exists for problem solvers to spend too much time trying to develop the recommended solution to the problem and too little time in determining the best way to sell the solution to management. Effective communication Is essential; in particular, it Is important to communicate with management in a way that can be understood and accepted. It is important to write a proposal that includes an executive sum­ mary. The executive summary normally reduces the performance characteristics of the recommendation to a few pages. Technical as­ pects of the recommended solution and the details o f the economic analysis are usually provided in the main report for the manager w ho wishes additional information. A face-to-face presentation is also quite important in achieving effective communication. Communication techniques such as the use of visual aids, voice con­ trol, dress, structure and clarity o f expression are necessary to sell the proposed solution. However, It Is probably more im portant to know your audience well and prepare your proposal using their language. Since the language of managers is often the language of finance, Chapter 9 Introduces the financial terminology used by managers.

1.6 Non-Monetary Considerations The principal emphasis o f this textbook is on the use o f logical meth­ odology to choose an engineering project from among several avail­ able to the decision maker, where the criterion for choice w ill be some economic measure o f effectiveness. However, the decision maker has several economic measures from which to select. For example, be-

12

Chapter 1

cause o f an Increase In market demand, the m anagem ent o f a firm may fare the choice between (1) going to a second shift o f p ro d u c tio n X the same machinery, and (2) Installing m ore high y au to m a te d machinery In order to meet the new production schedules. Clearly, a comparison o f the costs for each o f these alternatives o v e r som e p lan­ ning period is In order, and one objective o f m anagem ent w o u ld n o doubt be to meet production schedules at the lowest possible ann ual cost of production. Although considerable investigation and study are re q u ire d , m any o f the factors involved in a decision th a t m a y a t firs t seem non-economic can be expressed in (or reduced to) m o n e ta ry values. For Instance, In the above example, maintenance personnel m a y have to be trained In the Installation and repair o f the new m a c h in e ry but, most likely, these training costs can be determined and charged to the 'autom ation' alternative. However, other factors Involve d are n o t so easily reduced to monetary values and, Indeed, som e facto rs Involve d are not so easily reduced at all. For example, the Insta lla tion o f the automated machinery would probably reduce the n um be r o f person­ nel required by the present production system, whereas th e second shift alternative would result In additional employees. If persons are laid off or transferred to other Jobs w ithin the firm , th e ir salaries could be considered annual savings accruing to the autom ated m a ch in e ry alternative and, conversely, the salaries o f additional em ployees in te­ gral to the second shift alternative would be annual costs. H ow ever, layoffs or transfers could have a deleterious effect on th e m o ra le o f th e remaining employees; on the other hand, adding n e w personnel could have a beneficial effect on morale. Assessing th e cost o r gain o f these expected changes in morale is virtually Impossible. N everthe­ less, the potential effect on production that m orale changes m a y have must be considered by management. Factors that affect a decision but cannot be expressed in m o n e ta ry terms are often called intangibles or irreducibles. Practically a ll realworld business decisions Involve both m onetary and In ta n g ib le fac­ tors. In the above production example, assume th a t th e a lte rn a tiv e o f a second shift results In a total annual cost o f $100 0 0 0 fo r b o th shifts over a five-year planning period w ith no persons laid o ff o r transferred or new employees hired. For the automated m achinery alternative, n £ Z ° UW annual cost o f $85 0 0 0 o ve r th e five-year s l x p e r s o n s , a l d o f f ‘ T h u s ' t h l s o versim plified d e d m e a s u r e * a m onetary annual cost and th e sin­ gle Intangible factor o f employee morale. If m anagem ent chooses the

Introduction

13

second shift alternative, this would Imply that management places a higher subjective utility value on 'good employee morale' than on the $15 000 difference In annual costs between the tw o alternatives. That Is, the objective of good employee morale is more Important than the objective of lowest annual cost (at least for this difference of $15 000). Instances o f decision situations where intangible considerations out­ weigh monetary ones are frequent, and the engineer should not be­ come distraught at this fa c t W ithin the hierarchy of management responsibilities fo r a firm, the higher the level o f management, the more likely it Is that Intangible considerations will be given greater subjective w eigh t Engineering project proposals should therefore re­ flect a knowledge o f Intangibles that management will wish to con­ sider. Miller and Starr* have made Interesting observations In regard to the decision-making process. 1. Being unable to satisfactorily describe goals In terms of one objective, people customarily maintain various objectives. 2. Multiple objectives are frequently In conflict with each other, and when they are, a suboptimization problem exists. 3. At best, we can only optimize as of that time when the decision Is made. This will frequently produce a suboptimization when viewed in subsequent times. 4. Typically, decision problems are so complex that any attempt to discover the set of optimal actions Is useless. Instead, people set their goals In terms o f outcomes that are good enough. 5. Granted all the difficulties, human beings make every effort to be rational in resolving their decision problems. Accepting the premise that multiple objectives are Involved In a de­ cision, It follows that different measures fo r these objectives may arise. For Instance, tw o or more engineering project alternatives could each involve annual costs In dollars, mass in kilograms, repairability values on an arbitrary index from 1 to 10, and so forth. The decision problem would be greatly simplified if the various measures could be trans-

1 David W. M iller and M. K. Starr, Executive Decisions and Operations Research, 2d. Ed., Prentice-Hall, 1969, pp. 52-53.

14

Chapter 1

v e rtrt to a utility v a lu T the new utility values could be w e ig h te d by toelrrelatlve Importance, and the weighted values could be aggreX t an appropriate functional form . A single u tility v a lu e w o uld thus result for each alternative, and a selection w o u ld be m ade by choosing the alternative having the m axim um u tility value. This area of study is the study of utility theory and value m easurem ents, a con­ troversial subject outside the scope o f this book. There will be some additional discussion on m u ltip le objectives in Chapter 8, but otherwise It will be assumed th ro u g h o u t th is b o o k that only a single economic measure o f effectiveness is re le va n t In com par­ ing the alternative projects.

1.7 Overview of the Text As an overview, we have organized the m aterial in a m a n n e r consis­ tent with the logical sequence o f steps follow ed in p e rfo rm in g an eco­ nomic evaluation of Investment alternatives. Chapter 2 provides a discussion o f cost concepts and Includes discussions o f elem ents of costs, measurement ofcash flows, incremental costs, fu tu re costs, sunk costs, intangibles, irredudbles, nonquantlfiables, standard costs, direct costs, indirect costs, fixed costs, variable costs, average costs, m arginal costs, and opportunity costs. Given a feel fo r h o w th e data are to be ob­ tained for an economic analysis, Chapter 3 presents som e Im p o rta n t fundamental concepts Involving the time value o f money. In fact, Chap­ ter 3 provides the foundation for the rem ainder o f th e b o o k ; therefo re the reader must understand the tim e value o f m o n e y o p e ra tio n s pre­ sented in this chapter. Chapter 4 uses the time value o f m oney operations In com paring In­ vestment alternatives. Measures o f econom ic w o rth such as present worth, annual worth, future worth, payback period, rate o f return, and savings/investment ratio are used in com paring m u tu a lly exclusive in­ vestment alternatives. Chapter 5 addresses the Issue o f Income taxes and th e ir Incorporaim,n i e c. ° 2 ° m ,< ?i a n a , y s e s -' Chapter 6 treats benefit-cost analysis. Those l n t h e p r , v a t e s e c t o r W |H P robably choose to study t .h ° s e *Pv o l v e d Primarily In the public sector w ill proba bly < = h a p t e r 6 ’ W h e n o n e ls l n v o l v e d in b o th th e public and the private sectors, both chapters are appropriate!

Introduction_____________________________________________________________ _______________________________ 15

Chapter 7 presents supplementary analysis techniques. In particu­ lar, the effects of risk and uncertainty on the analysis of economic in­ vestment alternatives are considered. In modelling the effects o f risk and uncertainty mathematically, tw o approaches are taken: a pre­ scriptive (normative) approach and a descriptive approach. A prescrip­ tive model prescribes the action that, in some sense, is optimal; a descriptive model describes the behaviour of the situation modelled. Chapter 7 provides a descriptive treatment o f the effects of risk and un­ certainty by presenting the subjects o f break-even, sensitivity, and risk analyses. Prescriptive models are presented in Chapter 8. Subjects con­ sidered include decision making under risk and uncertainty. Chapter 9 gives an Introduction to financial and cost accounting. Chapter 10 covers the basic economic principles that are essential to the understanding of engineering economic analysis. The concept of supply and demand Interaction and the theory o f production are out­ lined in some detail, especially fo r the benefit of readers w ithout any previous exposure to economic theory. Readers familiar with these fundamentals may om it Chapter 10. However, those who have no background in economic theory should read Chapter 10 first, before reading any other chapter o f this book.

Chapter 1

Chapter 2

Cost Concepts and Cost Estimation 2.1 Introduction Engineering eco n o m ic analysis is p rim a rily concerned w ith co m p a rin g alternative projects o n th e basis o f an e co n o m ic m easure o f effe ctive­ ness. This com pariso n process has a v a rie ty o f cost te rm in o lo g ie s and cost concepts, a n d It w ill be h e lp fu l to present th e m p rio r to th e discus­ sion In Chapter 4 o f th e e co n o m ic m easures o f effectiveness fo r com ­ paring a lte rn a tive projects. T o Im p le m e n t th e discussion o f cost te rm in o lo g y, a typ ica l p ro d u c tio n situ a tio n w ill n o w be described. Let us assume th a t th e business o f a sm all m anufacturing firm is Job­ shop m achining. T ha t is, the firm produces a variety o f products and com ­ ponent parts according to custom er order. A n y given order m ay be fo r quantities o f as fe w as five parts o r as m any as several hundred parts. The firm has periodically received orders to m anufacture a p a rt w hich w e w ill identify as Part No. 163H, fo r the M ontreal Gear W orks Ltd. The part has been m anufactured In a four-stage production sequence consisting o f (1) cutting bar stock to length on a horizontal band saw, (2) m achining o n an engine lathe, (3) d rillin g on a vertical drill press, and (4) packaging. The unit cost o f producing Part No. 163H b y this sequence is $30, w here the unit cost consists o f th e cost o f direct labour, materials, and overhead (prorated cost o f adm inistrative and m anagem ent expenses, building and m aintenance costs, utilities, etc.) The Ann Is n o w In tire process o f ne­ gotiations w ith the M ontreal Gear W orks to ob ta in a contract fo r produc­ ing 10 0 0 0 o f these parts over a period o f fo u r years. A contract fo r this volum e o f parts is highly desirable but, In order to obtain the co n tra ct the flrm m ust low er the unit cost. An e ngineer fo r th e firm has been assigned to d e te rm in e a lte rn a tive produ ction m eth ods to lo w e r th e u n it cost. A fte r study, th e e n g in e e r recom m ends th e purchase o f a tu rre t lathe. W ith th e tu rre t lathe, th e processing sequence o f Part No. 163H w o u ld consist essentially o f (1) m achin ing bar stock o n th e tu rre t la th e and (2) packaging. T he esti­ m ated u n it cost fo r Part No. 163H b y th is p ro d u c tio n m e th o d w o u ld o n ly be $20. Furtherm ore, th e p ro d u c tio n rate b y th e n e w m e th o d w o uld be Increased o v e r th e o ld m eth od.

Chapter 2

18

If the turret lathe Is purchased, the saw, engine lathe, and d rill press would not be sold but would be kept for other Jobs fo r w h ic h th e firm mav receive orders. The turret lathe would be reserved fo r th e produc­ tion o f Part No. 163H, and Its excess production capacity could be de­ voted to other Jobs. The investment required to purchase the turret lathe and th e new tooling required, as well as Installing the machine, Is $180 000 . The physical life o f the turret lathe is about 25 years, but th e firm believes that the useful economic life o f the machine w ill be o n ly 10 years. The estimated salvage value o f the turret lathe at the end o f th e 10-year pe­ riod is $60 000. If the maximum unit price th a t the M o n tre a l Gear Works will pay for Part No. 163H is $28, should the firm accept th e con­ tract for 10 000 parts and then purchase the turret la the In o rd e r to execute the contract? We w ill not answer the question In this chapter, but have cited the example to illustrate one type o f decision w ith w hich this te x t Is con­ cerned. In fact, additional Information and the m e th o d o lo g y pre­ sented in Chapter 4 would have to be used to find a com plete answ er to the question. However, the reader can readily appreciate th a t cer­ tain cost figures in the above example must be determ ined o r esti­ mated before any rational decision can be reached as to th e purchase o f the turret lathe. The cost elements contributing to th e $180 0 0 0 m a­ chine cost, the unit production cost o f $20, and salvage va lu e o f $60 000 would be determined or estimated fro m In fo rm a tio n ob­ tained from a variety o f sources. These sources m ight ty p ic a lly be pro­ duction records, accountants' records, m anufacturers' catalogues, publications from Statistics Canada, or other governm ent sources, etc. The engineer should therefore be fam iliar w ith cost te rm in o lo g y , cost factors, and cost concepts as used by different specialists if effective economic comparison and intelligent recom m endations are to be made.

2.2 Cost Terminology C°m

? e f l n l t i o n s a n d cost concepts are Included In this section, clarity w ill be achieved by the use o f six categories: 1. Life-cycle costs. 2. Past and sunk costs. 3. Future and opportunity costs.

Cost Concepts and Cost Estimation

19

4. Direct, Indirect, and overhead costs. 5. Fixed and variable costs. 6. Average and marginal costs.

2.2.1 Life-CydeCost The life-cycle cost o f an Item Is the sum o f all expenditures associated with the Item during Its entire service life. The term item should be In­ terpreted In the general sense as a machine, a unit o f equipment, a building, a project, a system, and so forth. Life-cyde costs may Indude engineering design and development costs; building, manufacturing, fabrication, and testing costs; shipping and Installation costs; operat­ ing and maintenance costs; and disposal costs. Life-cyde costs may also be expressed as the summation of acquisition, operating, mainte­ nance, and disposal costs. Thus, life-cyde cost terminology may vary from author to author, but the basic meaning o f the term is clear. This textbook Is predominantly concerned with the economic justifi­ cation of engineering projects, the replacement o f existing projects or capital assets, and the economic comparison of alternative projects. For the purpose of these types of analyses, we will define life-cyde costs as consisting of: 1. First cost (Initial Investment or capital cost). 2. Operating and maintenance costs. 3. Disposal costs. Thefirst cost of an Item Is the total Initial investment required to get the Item ready fo r service; such costs are nonrecurring during the life o f the Item. For example, for the purchase of a numerically controlled machine tool, the first cost may consist o f the follow ing major ele­ ments: (1) the basic machine cost; (2) Installation cost (Including: cost of foundation, vibration and noise insulation, temperature and humid­ ity control, heat, light, compressed air, and power supply, and cable connection to computers, etc.); (3) cost o f testing the machine; (4) cost of training personnel; (5) cost o f special tooling, Jigs, and fixtures; (6) cost of all supporting equipment (computer hardware, software, etc.). For some other Item, a different set of first-cost elements may be appropriate, but the first cost o f an Item normally involves more cost elements than Just the basic purchase price.

Chapter 2

20

Operating and maintenance costs are recurring costs th a t are necessary to operate and maintain an item during Its useful life. O perating co+ IV P(l+I?l P(l+I?l

P d + if-’ ?(i+ /r

P(1 +/)"■'1

p + pi=p(i+n P(1 + /) + P(1 + i)l = P(1 + If Pd+fl? + p(i+/f/=p(i+/F pci+iy + p(i+/?i= p(i+ n 4 P(1 + I f ’ + P(i + I f ’I = P(1 + IT

* Notice, the value In column (C) for the end o f period (n —1) provides the value In column (A) for the end o f period n.

Chapter 3

52

Example 3.1 Person A borrows $4 000 from the Royal Bank and agrees to pay 51 000 plus accrued Interest at the end of the first year and $3 000 plus the accrued Interest at the end o f the fourth year. W h a t are the amounts for the two payments If 18% annual sim ple Interest ap­ plies? For the first year, the payment Is $1000+(0.18X$4 000) = $1720 For the fourth year, the payment Is $3 000+(0.18X$4 0 0 0 - $1000>3) = $4 620

3.3 Single Sums of Money To illustrate the mathematical operations Involved In m o d e llin g cash flow profiles using compound Interest, first consider th e investm ent of a single sum of money, P, In a savings account fo r n Interest periods. Let the interest rate per interest period be denoted b y I and le t th e accu­ mulated total in the fund n periods in the fu tu re be d e n o te d b y F. As shown In Table 3.1, assuming no monies are w ith d ra w n d u rin g the In­ terim, the amount In the fund after n periods equals P(1 +•/)". As a con­ venience in computing values o f F (the fu tu re w o rth ) w h e n given values of P (the present worth), the q u a ntity (1 + /)" Is ta b u la te d In Ap­ pendix A for various values o f/a n d n. The q u a n tity (1 + l) n is referred to as the single sum, future worth factor and is den ote d (F| P l,n). The ex­ pression (F| P l,n) is read as the F, given P fa cto r a t /% fo r n periods. The above discussion is summarized as follow s: Let

the equivalent value o f an am ount o f m oney at tim e zero, or present worth f = t h e equivalent value of an am ount o f m oney at tim e n, or future worth / = the Interest rate per Interest period

n = the number of interest periods Thus, the future worth Is related to the present w o rth as fo llo w s:

Time Value of Money Operator™

53

F=p4 /,12) + $950(P|F M 2 ) It Is then necessary to solve for the unknown / either analytically or by trial and error as follows: F0r/ = 1%, $ 1 0 2 0 * $20(11.255l)+$950(0B 874) $1020 *$ 1 0 6 8 .1 3

Chapters

I $1020

Fig. 3.25 Cash flow fol a $1000 b o n d -^ e te a n ln e yield.

Fori = 1.5%, $1020 * $20(105075)+5950(0.8364) $1020*51012.73 Then, interpolating gives 0 5 1 -0 5 1 5 __________0.01- X $1068.13-51012.73“ $ 1 0 6 8.13-5102 0 X =051434

or

1434% per quarter

The effective annual yield Is [(1+051434)* - l ] l 0 0 % = 5 3 6 1 %

3 3 5 Capital Recovery Cost In the engineering economy literature th e re Is fre q u e n t reference to the term capital recovery cost If an Investm ent o f $P Is m a d e In an as . the asset Is used for n years, and disposed o f fo r salvage value ot > , then the capital recovery cost, CR, Is defined as CR = P ( 4 | P I, n ) - F ( 4 |F I ,n )

(3.68)

Time Value of Money Operations

111

However, since ( A \P I,n ) = (A \F I,n ) + i

(3.69)

then on substituting Equation 3.69 Into Equatton 3.68 we obtain CR = (P -F )(A \F I,n )+ P i

(3.70)

As will be noted In Section 5.4.4, the first term on the right hand side of Equation 3.70 is the annual sinking fund deposit fo r sinking fund de­ preciation; the second term is the opportunity cost due to $P being tied up in the asset. On the basis of Equation 3.70, capital recovery cost Is often defined as the cost of depreciation plus a minimum return on the Investment. Alternately, Equation 3.68 can be given as CR = (P -F )(A \P I,n ) + Fi

(3.71)

due to the relationship between the factors (A |P i,n ) and (A |F /,n). Among the three methods o f computing the capital recovery cost, Equation 3.71 appears to be the most popular. We tend to use Equa­ tion 3.68, since It Is a direct application o f the cash flow approach. Example 3.39 Consider an Investment of $10 000 in a unit of equipment which lasts for 10 years and is sold for $1 000. An interest rate o f 10% Is used. Determine the capital recovery cost using the three methods presented. C R = $10000(A |P 10% ,10)-$1000 (A |F 10%,10) = $ 1 0 0 0 0 (0 .1 6 2 7 )-$ ! 000(0.0627) = $1 56430

per year

CR= ($10 000 - $1000X

F l 0%,10) + $10 000(0.10)

» $9 00 0 (0 .0 6 2 7 )+ $ 1 000 = $1 5 6 4 3 0

per year

Chapters

112

CR=($10000-$1000XX»|P 10%,10) + $1000(0.10) = $9 000(0.1627)+ $100 = $1 56430

per year

3.10 Inflationary Effects The dynamic nature of the econom y in th e past has focused consider­ able attention on inflation and Its effect on e co n o m ic decision making. Alternative approaches that are typically used to acco u n t fo r Inflation­ ary effects Include: 1. Express all cash flows in terms o f 'th e n -c u rre n t' d o lla r amounts and combine the Inflation rate w ith th e Interest rate. 2. Express all cash flows In terms o f 'c o n s ta n t w o r th ' dollar amounts and use an interest rate w ith o u t an In fla tio n rate component. The latter approach appears to be the m eth od m o st preferred by prac­ titioners. However, it is not uncom m on to e n co u n te r claim s th a t the in­ terest or discount rate used In tim e value o f m o n e y computations Includes a component for Inflation. If care Is n o t ta k e n to Insure a proper accounting for inflation, one m ig h t p a rtia lly account for Infla­ tion, rather than account for it In both the disco u n t rate used and the cash flow estimates. Letting) be the Inflation rate, Ck be th e 'c o n s ta n t w o r th ' value of a cash flow at the end o f period k, and Tk be th e 'th e n -c u rre n t' value of a cash flow at the end o f period k, the fo llo w in g re la tio n holds:

L=Q(1 + J)k

(3.63)

Thus, when the set of constant w o rth cash flo w s constitutes a uniform senes, (l.e., C* = A, k = 1,... ,n), the set o f th e n -cu rre n t cash flows con­ stitutes a geometric series, l.e., Tk = A 3(1 + ) ) k _ 1 , w h e re A, = A(1 + J). Using the first approach to account fo r In fla tio n a ry effects, the presT * « s h flo w s Is com puted as folk)wsW ° r t h e q u l v a , e n t o f a s e r , e s

Time Value of Money Operations

113

/’=£t.( W k -0

( i +')'‘

fc-0

(364) w here d Is a discount rate equal to I + j+ IJ . The second approach com ­ putes th e present w o rth e q u iva le n t using constant w o rth d o lla r am ounts b y n o tin g th a t

k-Q

can be given as

p = £ c ,( i+ o "

(3.65)

In the preceding sections o f th e ch ap ter w e Im p licitly assum ed th a t ei­ th e r the cash flo w s w e re In th e fo rm o f co n stant w o rth dolla rs o r th e discount rate used w ith th e n -cu rre n t dolla rs Included a co m p o n e n t fo r inflation.

Example 3.40 To illustrate the tw o approaches in dealing with inflation, suppose j = 5% / = 10% To = -$ 1 0 0 0 0 T, = $1000 T2 = $3 000 T4 = $7 000 As shown in Table 3.5, the present w orth equivalent for the situa­ tion is —$356. In this Instance, d = 0.05 + 0.10 + 0.005 = 0.155, or 15.5%. If the present w orth were computed using d = 10% and then-current dollar estimates, the present worth would have been $1175 Instead o f th e -$356 obtained considering Inflation. Conse­ quently, what appears to have been a profitable investment with-

114

__________________________________________________ Chapter

out considering Inflation has a negative present w orth when the effects of inflation are Included In the analysis. •ftble 3.5 Present Worth Calculations Under Inflation k

(1+ «/)’ *

T*(1+d)'*

G*

(1 + D-*

Ck (1+|)-*

0

-$10000

1.0000

-$10000

-$100 00

1.0000

-$10000

1

1 000

0.8658

866

952

0.9091

866

2

3000

0.7496

2249

2 721

0.8264

2249

3

4000

0.6490

2596

3 455

0.7513

2596

4

7000

0.5619

3933

5 759

0.6830

3933

-3 5 6

-356

•C, =1,(105)*

Although the second approach, In v o lv in g c o n sta n t w orth dollar amounts, appears to be the simplest m e th o d o f In co rp o ra tin g Inflation In the analysis, estimating cash flow s In term s o f co n sta n t w orth dollar amounts is not a trivial exercise. W e m e n tio n e d p re vio u sly that a uni­ form series of constant w orth cash flow s co n ve rts to a geom etric series of then-current cash flows. You expect cash flo w s fo r, say, labour costs to Increase over time; some portion, but n o t all, o f th e Increase is due to Inflation. Thus, It Is difficult to fa cto r o u t th e p o rtio n o f the Increase that is due to inflation in order to provide in fla tio n -fre e estimates of fu­ ture cash flows. Another aspect of Inflation th a t tends to co m p lica te the analysis is the differences that may exist in In fla tio n rates fo r n o t o nly various types of cash flows, but also fo r d iffe re n t regio ns o f th e country and world. The firm that has num erous plants scattered throughout not only the same country, but also the w o rld , m u st cope w ith the eco­ nomic differences that exist am ong th e locations, as w e ll as the differ­ ences in Inflation rates fo r labour, equipm ent, m aterials, utilities, and supplies. As a further complication in dealing w ith in fla tio n , th e inflation rate tends to change from one tim e period to th e ne xt. If L denotes the Inatlon rate for period t, and l t denotes th e Interest rate fo r period t, the present worth can be expressed In th e n -cu rre n t d o lla rs as

Time Value of Money Operations

115

k -0

t-0

(3.66)

In the case of discrete compounding and as

(3.67) In the case of continuous compounding, where c t is the nominal infla­ tion rate for period t, and r, Is the nominal Interest rate for period t; In the above, c 0 , r 0 , Jo , and /0 are defined to be equal to zero. Inflation Is a much discussed subject In the area of economic Invest­ ment alternatives. Some argue that inflation effects can be Ignored, since Inflation will affect all Investment alternatives in roughly the same way. Thus, It is argued, the relative differences In the alternatives will be approximately the same with or w ithout inflation considered. Others argue that the inflation rate In the past has been so dynamic that an accurate prediction of the true Inflation rate and Its impact on future cash flows Is not possible. Another argument for ignoring ex­ plicitly the effects of inflation Is that It is accounted for Implicitly, since cash flow estimates fo r the future are made by Individuals conditioned by an Inflationary economy. Pius, it Is argued, any estimates of future cash flows probably Incorporate Implicitly inflationary effects. A final argument for Ignoring Inflation In comparing investment alternatives Involving only negative cash flows Is that an alternative that Is pre­ ferred by ignoring Inflation effects will be even more attractive when effects o f Inflation are Incorporated in the analysis. The above arguments are certainly valid in some instances. How­ ever, counter-arguments can be given for each. Consequently, before dismissing Inflation effects as unnecessary In performing economic analysis, the individual situation should be considered closely.

3.11 Summary In this chapter we developed the concept of the time value of money and defined a number o f mathematical operations consistent with that concept. The principles Introduced here from the viewpoint o f per­ sonal financing w ill be applied In subsequent chapters to the study of Investment alternatives from the viewpoint o f an ongoing enterprise.

116

Problems 1 Person A sells Person B a used a u to m o b ile fo r $2 000. Person B pays $500 cash down and gives Person A tw o personal notes for the remainder due. The principal o f each n o te Is therefore $750. One note Is due at the end o f th e firs t year; th e other note is due at the end o f the second year. T he a n n u a l simple Interest rate agreed on Is 8.5%. H ow m uch to ta l Interest will Person B pay Person A? (3.2) 2. A debt of $1 000 Is Incurred at t = 0. A n a n n u a l sim ple Interest rate of 8% on the unpaid balance Is agreed o n. Three equal payments of $388 each at t = 1 ,2 , 3 w ill p a y o ff th is debt and the relevant Interest due. For each p aym en t, w h a t Is (a) the payment on the principal and (b) th e in terest a m o u n t paid? (3.2) 3. If $5 000 is deposited at t = 0 Into a fu n d p a y in g 6% compounded per period, w hat sum w ill be accum ulated at the end of eight periods, or f = 8 ? W h a t w o u ld be th e sum accumulated If the fund paid 5% co m p o u n d e d p e r period? (3.3) 4. If a deposit of $1 000 at t - 0 am ounts to $4 3 0 0 a t the end of the eighth compounding period, w h a t va lu e o f / Is involved? (3.3) 5. How long does It take a deposit In a 6 % fu n d to trip le In value? (3.3) 6. If a fund pays 6% com pounded ann ually, w h a t single deposit is required at t ~ 0 In order to accum ulate $8 0 0 0 in the fund at the end of the tenth year (f = 10)? (3.3) 7. How much money today is equivalent to $1 0 0 0 in fiv e years, with interest at 7% com pounded ann ually? (3.3) 8. A person deposits $500, $1 200, and $2 0 0 0 a t t = 0 ,1 ,2 , respectively, if the fund pays 6% co m p o u n d e d per period, what sum will be accumulated In the fu n d a t (a) t = 2 a n d (b) t = 3? 9

'

m u c h s h o u ld

iuvo

be deposited at t = 0 , In to a fu n d paying

compounded per period, in o rd e r to w ith d ra w $700 at

Time Value of Money Operations_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 117

t =1, $1 500 at t = 3, and $2 000 a tt = 7 and the fund be depleted? 10. A person deposits $4 000 In a savings account that pays 6% compounded annually. Three years later he deposits $4 500. TWO years after the $4 500 deposit $2 500 Is deposited. Four years after the $2 500 deposit, half of the accumulated funds is transferred to a fund that pays 7% compounded annually. How much money w ill be In each fund six years after the transfer? (3.4) 11. A woman annually deposits $A k In an account at time k, k =1,..., 20, where Ak = 1 0 k (1 .0 5 )*' 1. If the fund pays 5%, how much Is In the fund Immediately after the seventeenth deposit? (3.4) Note:

12. What equal, annual deposits must be made at t = 1 ,2 ,3 ,4 , 5, 6 In order to accumulate $10 000 at t = 6 If money Is 'w o rth ' 10% compounded annually? (3.4.1) 13. A debt o f $1 000 Is Incurred at t = 0. What is the amount of three equal payments at t = 1, 2, 3 that will repay the debt If money is 'w o rth ' 8% compounded per period? (3.4.1) 14. Five deposits of $300 each are made at t » 1 ,2 , 3, 4, 5 Into a fund paying 8% compounded per period. How much will be accumulated In the fund at (a) t = 5 and (b) t = 9? (3.4.1) 15. If you know the values o f the (F|P 6 ,n ) factor for n =1,2,... ,10, show how you would determine the value for th e (A |P 6,10) factor. (3.4.1) 16. A deposit of $X Is placed Into a fund paying 10% compounded annually at t « 0. If withdrawals o f $3154.67 can be made at t = 1,2, 3, and 4 such that the fund Is depleted w ith the last withdrawal, show that the value of X Is $10 000 by (a)

------ --------------- - ----------------------------- Chapterj

118

use of the Interest factor (P | A 1 0 ,4 ) and (b) use o f the Interest factors: (P|F 10,1), (P |F 10,2), (P |F 1 0 ,3 ), a n d ( P |F 10,4). (3.4.1) 17 A person deposits $1 000 In a savings accou nt; fiv e years after the deposit, half of the account balance Is w ith d ra w n ; $2 000 Is deposited annually for five m ore years, w ith th e to ta l balance withdrawn at the end o f the fiftee nth year. If th e account earns interest at a rate of 5%, how m uch Is w ith d ra w n (a) at the end of five years and (b) at the end o f 15 years? (3.4.1) 18. With 5% interest com pounded ann ually, h o w m uch money will be accumulated in a fund im m e d ia te ly a fte r th e fifth deposit, if equal deposits o f $521.08 are m ade annually? (3.4.1) 19. A person borrows $10 000 at 8% c o m p o u n d e d annually and wishes to pay the loan back over a five -ye a r pe rio d w ith annual payment. However, the second p a y m e n t Is to be $500 greater than the first paym ent; the th ird p a y m e n t is to be $500 greater than the second paym ent; th e fo u rth p a y m e n t is to be $500 greater than the third paym ent; and th e fifth paym ent Is to be $500 greater than the fo u rth p aym en t. D e term ine the size of the first payment. (3.3) 20. A debt of $X is Incurred at t = 0 (purchase o f land). It Is agreed that payments of $5 000, $4 000, $3 0 0 0 , and $2 0 0 0 at t= 4 , 5,6, and 7, respectively, w ill satisfy th e d e b t If 10% compounded per period is the Interest rate. D e term ine the amount of the debt, $X. (3.4.2) 21. Determine the future w orth, F, at t = 15 o f th e fo llo w in g deposits: $1 000 at t = 8, $900 at t = 9, $ 8 0 0 a t t - 1 0 , and $700 at t = 11. Assume the fund pays 2 0 % com po u n d e d per period. (3.4.2) 22. Mr. Jones receives an annual bonus fro m his em ployer. He wishes to deposit the bonus in a fu n d th a t pays Interest at a rate of 6% compounded continuously. His firs t bon u s Is $1 000. The size of his bonus Is expected to increase a t a rate o f 4% per year. How much m oney w ill be in th e fu n d Im m ediately before the tenth deposit? (3.6) 23. An individual works fo r a com pany th a t pays an annual onus, the size of which is based on experience w ith the

Time Value of Money Operations

119

company. After one year with the company, the bonus equals $500. The size of the bonus thereafter compounds at an annual rate o f 4%. The Individual decides to place half the bonus in a fund that pays 5% compounded annually. (a) How much money will be in the fund immediately prior to the sixth deposit? (b) What Is the answer to (a) if the fund compounds at 4% annually? (3.4.3) 24. A man invests $10 000 In a venture that returns him $500(0 .SO)*"1 at the end o f year k fo r k - 1,..., 20. With an Interest rate of 10%, what is the equivalent uniform annual profit (cost) fo r the venture? (3.4.3) 25. A man places $1 000 in a fund at the end of 1999. He places end-of-year deposits In the fund until the end of 2018, when his last deposit Is made. The fund pays 5% compounded annually. If the size of a deposit, A k , at the end o f year k, equals 0.90 A*.,, how much will be in the fund immediately after the last deposit? (3.4.3) 26. Ms. Smith deposits $1 000 in a savings account in her local bank. The bank pays interest at a rate of 6% compounded annually. Three years after making the single deposit she withdraws half the accumulated money In her account. Hve years after the Initial deposit, she withdraws all of the accumulated money remaining in the account How much does she withdraw five years after her Initial deposit? (3.5) 27. A person wishes to make a single deposit P at t = 0 into a fund paying 8% compounded quarterly such that $1 000 payments are received at t = 1,2, 3, and 4 (periods are three-month Intervals) and a single payment of $5 000 Is received at t - 12. W hat single deposit Is required? (3.5) 28. A man borrows $20 000 at 8% compounded quarterly. He wishes to repay the money w ith 10 equal semiannual installments. W hat must be the size o f the payment if the first payment Is made one year after obtaining the $20 000? (3.5) 29. A woman borrows $2 000 at 8% per year compounded monthly. She wishes to repay the loan with 12 end-of-month

Chapter 3

120

payments. She wishes to m ake her first p a y m e n t th re e months after receiving the $2 000. She also w ishes th a t, a fte r the first payment, the size o f her paym ent be 10% g re a te r th a n the previous payment. W hat Is the size o f her sixth paym en t? (3.5) 30. Monthly deposits o f $100 are m ade In to an a ccou nt paying 4% compounded quarterly. Ten m o n th ly deposits are made. Determine how much w ill be accum ulated in th e account two months after the last deposit. (3.8.3) 31. A person makes four consecutive sem ian nual deposits o f $1 000 in a savings account th a t pays in terest a t a rate o f 6% compounded semiannually. H ow m uch m o n e y w ill be In the account two years after the last deposit? (3.5) 32. An Individual borrows $5 000 at an Interest rate o f 8% per year compounded semiannually and desires to re p a y th e money with five equal end-of-year paym ents, w ith th e first payment made tw o years after receiving th e $5 0 0 0 . W hat should be the size o f the annual paym ent? (3.5) 33. What is the effective interest rate fo r 6 % co m p o u n d e d monthly? (3.5) 34. What monthly deposits must be m ade in o rd e r to accumulate $4 000 in five years If a fund pays 9% co m p o u n d e d m onthly? (Assume the first deposit is m ade at th e end o f th e firs t month, and the final deposit is made at the end o f th e fifth year.) (3.5) 35. What monthly payments are required in o rd e r to p a y o ff a $20 000 debt in five years if the nom inal rate o f Interest charged is 9% compounded annually? (3.8.3) 36. A man borrows $20 000 at 8% com pou nded sem iannually. H ep a ys back the loan w ith fo u r equal s e m ia n n u a l payments, with the first payments made one year a fte r re ce ivin g th e $20 000. What should be the size o f each o f th e fo u r payments? (3.5) 0 0 0 and pays the lo a n o ff, w ith th e ° rro w s Mterest, after tw o years. He pays back $ 1 1 5 0 . W h a t is th e effective Interest rate fo r this transaction? (3.5) b

Time Value of Money Operations

121

38. A woman borrows $5 000 at 1% per month. She desires to repay the money using equal end-of-month payments for 10 months. The woman makes four such payments and decides to pay off the remaining debt with one lump sum payment at the time for the fifth payment. What should the size o f this payment be If Interest is truly compounded at a rate of 1% per month? 39. Mr. W right borrows $8 000 from a bank that charges interest at 6% compounded semiannually. Mr. W right is to pay the money back with six equal payments. However, the first payment Is to be made Immediately on receipt of the $8 000. Successive payments are spaced one year apart (a) Determine the size of the equal annual paym ent (b) At the time of the fourth payment, suppose Mr. W right decides to pay off the loan with one lump sum payment. How much should be paid? Include the fourth payment. (3.5) 40. Approximately how long will It take a deposit to triple In value if money is worth 8% compounded semiannually? (3.5) 41. Assume a person deposits $2 000 now, $1 000 tw o years from now, and $5 000 five years from now Into a fund paying 8% compounded semiannually. (a) What sum of money will have accumulated In the fund at the end o f the sixth year? (b) What equal deposits of size A, made every six months (with the first deposit at f = 0 and the last deposit at the end o f the sixth year), are equivalent to the three deposits stated above? (3.5) 42. A man borrows $1 000 from the Shady Deal Finance Company. He Is told the Interest rate Is merely 1.7% per month, and his payment Is computed as follows: Payback period = 30 months Interest - 30(0.017)($1 000) - $510 Credit Investigation and Insurance = $20 Total amount owed = $1 530 Payment size = $1 530/30 = $51 per month

Chapter 3

122

What Is the approximate Interest rate fo r th is tra n sa ctio n ? (3.5) 43 Operating and maintenance costs fo r a p ro d u c tio n m achine occur continuously during the year. If th e to ta l a n n u a l operating and maintenance cost Is $8 0 0 0 a n d m o n e y Is worth 15% compounded continuously, w h a t single sum o f m oney at the present Is equivalent to five years o f o p e ra tin g and maintenance costs? (3.6) 44. Labour costs occur continuously d u rin g th e year. A t th e end of each year a new labour contract becom es e ffe ctive fo r the following year. Let At denote the cu m u la tive la b o u r cost occurring uniformly during year j, w h ere A j = 1.05A ; _r If money Is worth 10% com pounded c o n tin u o u sly and 4 i = $25 000, determine the present w o rth e q u iv a le n t fo r five years of labour costs. (3.6) 45. A person borrows $10 000 and wishes to p a y It ba ck w ith 10 equal annual payments. W hat w ill th e size o f th e paym ents be if the interest charge is 10% com pounded (a) a n n u a lly, (b) semiannually, and (c) continuously? (3.6.1) 46. Operating and maintenance costs fo r ye a r k are g iv e n as C4 = ( e 0 0 5 JC([_v k = 2 ,3 ,...,1 5, w ith C, = $ 1 0 0 0 . D eterm ine the equivalent uniform annual operating and m a in te n a n ce cost based on continuous com pounding w ith a n o m in a l Interest rate of 10%. (3.6.1) 47. Four equal, quarterly deposits o f $1 0 0 0 each are m ade at t = 0 ,1 ,2 , and 3 (time periods are th re e -m o n th Intervals) Into a fund that pays 8% compounded continuously. Then, a t t = 7 and t = 10, withdrawals o f size $4 are m ade so th a t th e fund is depleted at t = 10. What is the size o f th e w ith d ra w a ls? (3.6.1) Se m ' a n n u a l deposits o f $500 are m ade In to a fu n d paying 8% compounded continuously. W hat is th e accum ulated value in the fund after 10 such deposits? (3.6.1) 49. A firm buys a new computer system th a t costs $100 000. It may either pay cash now or pay $25 0 0 0 d o w n a n d $10 000 P®r y®a r forJO years. If the firm can earn 5% o n Investments, which would you suggest? (3.7)

Time Value of Money Operation*

123

50. An Individual is considering tw o investment alternatives. Alternative A requires an initial Investment of $10 000; it will yield Incomes o f $2 500, $3 000, $3 500, and $4 000 over its 4-year life. Alternative B requires an initial Investment of $12 000; it is anticipated that the revenue received w ill Increase at a compound rate o f 6% per year. Based on an Interest rate o f 6% compounded annually, what must be the revenue the first year fo r B In order fo r A and B to be equivalent? (3.7) 51. It Is desired to determine the size o f the uniform series over the time period [2,6] that Is equivalent to the cash flow profile shown below using an Interest rate of 10%. (3.7) CM

6

$3 000

52. Given the follow ing cash flows, what single sum at t = 6 Is equivalent to the given data. Assume i = 8 % . (3.7)

t I

Chapter 3

124

53 A machine Is purchased at t - 0 fo r $20 0 0 0 (Including Installation costs). Net annual revenues re s u ltin g fro m operating the machine are $6 000. The m a ch in e Is sold at the end of 10 years, t =10, for $2 000. The cash flo w series fo r the machine Is equivalent to w h at single sum, $X, a t f = 6 If money is worth 10% compounded continuously? (3.7) 54. Examine the follow ing tw o Investm ent plans. (a) Purchase for $4 386.68 (a negative cash flo w ) a n d receive $400 at the end o f each six m onths fo r fo u r years ( t =1,2,... ,8), and a single paym ent o f $2 0 0 0 a t th e end of the fourth year. (b) Purchase for $4 091.14 and receive $ 9 0 0 a t th e beginning of each year for the four-year period, and a sin gle paym en t of $X at the end of the fourth year. If money is worth 6% com pounded se m ia n n u a lly, w h a t is the value of X so that the tw o investm ents are e q u iva le n t? (3.7) 55. A man Invests $5 000 and receives $ 5 0 0 each y e a r fo r 10 years, at w h id i time he sells o ut fo r $1 0 0 0 . W ith a 10% Interest rate, what equal annual cost (profit) is e q u iv a le n t to the venture? (3.7) 56. Consider the following cash flo w series.

EOY 0 2 3 4 5 6 7 8

CF - $10 000 3 000 3 500 4 000 4 500 5 000 5 500 6 000 6 500

Time Value of Money Operations

125

57. A person borrows $1 000 from a bank at t = 0 at 8% simple interest for tw o years. He pays the total interest due for the two-year period at t = 0 and thus receives $840 at t = 0. If he pays back $1 000 at t = 2, the person Is, In effect, paying an Interest rate of X% compounded annually. Solve for X. (3.7) 58. Assume payments o f $2 000, $5 000, and $3 000 are received at t = 3 ,4 , and 5, respectively. W hat five equal payments occurring at f = 1 ,2 ,3 ,4 , and 5, respectively, are equivalent if /=10% compounded per period? (3.7) 59. What single deposit o f size $X into a fund paying 10% compounded annually is required at t = 0 In order to make withdrawals of $500 each at t — 4, 5,6, and 7 and a single withdrawal of $1 000 at t = 20? If the above withdrawals are immediately placed Into another fund paying 6% compounded annually, w hat amount will be accumulated in this fund at t = 20? (3.7) 60. A college student borrows $4 000 and repays the loan with four quarterly payments o f $400 during the first year and four quarterly payments o f $1 000 during the second year after receiving the $4 000 loan. Determine the effective interest rate for the loan transaction. (3.7) 61. Quarterly deposits of $500 are made at t = 1,2, 3,4, 5,6, 7. Then withdrawals o f size A are made at t = 12,13,14,15 and the fund is depleted with the last withdrawal. If the fund pays 6% compounded quarterly, w hat is the value of A? (3.7) 62. Determine the value o f X so that the follow ing cash flow series are equivalent at 8% Interest. (3.7) EOY 0 1 2 3 4 5 6

CF(A) -$8000 6000 5 000 4 000 5 000 6000 5 000

CF(B) -$15000 4 000 3 000+X 2 000+2X 3 000+3X 4 000+4X 3 000+5X

Chapter

126

63. Given the cash flow diagram show n b e lo w a n d an Interest rate o f 8% per period, solve fo r th e v a lu e o f a n e qu ivalent amount at (a) t = 5, (b) t = 12, and (c) t - 1 5 . (R em em ber that upward arrows represent cash Inflow s and d o w n w a rd arrows represent cash outflows.)

0 ▼

1r

2

3

i

1r

4

5

ti

1’

8

9

f

1f

f

▼

64. Given the cash flo w profiles show n be lo w , d e te rm in e the value of X so that the tw o cash flo w p ro file s are e q u iva le n t at a 10% Interest rate. (3.7) EOY

CF(A)

CRB)

1

-$12 000

-$10 000

2

1 000

7 000

3

3000

6 000+0.5X

4

5000

5 000+1.0X

5

7000

4 000+1.5X

6

900 0

65. What single sum o f m oney at f = 4 Is e q u iv a le n t t o th e cash flow profile shown below? Use a 6% interest ra te In y o u r analysis. (3.7)

Time Value of Money Operations

0

127

2

3

4

5

6

66. Given the tw o cash flow profiles shown below, for what value of X are the tw o series equivalent using an Interest rate of 10%? (3.7) EOY

CRA)

CHB)

0

- $200 000

- $140 000

1

24 000

16 000

2

32 000

16 000

3

40 000

16 000+X

4

48 000

16 000+2X

5

56 000

16 000+3X

6

64 000

16 000+4X

7

72 000

16 000+5X

8

80 000

16 000+6X

67. At what Interest rate Is $1 000 today equivalent to $1 967 In 10 years? (3.7) 68. Given the cash flow profiles shown below, determine the value o f X such that the tw o cash flow profiles are equivalent at 8% compounded annually. (3.7)

Chapter j

128

CF(A)

EOY

-$12 000

1

CRB) $ -x

1 000

7000

3

4000

9 000

4

6000

10 000

5

7000

100 00

6

5000

7 000

2

I

69. If a machine costs $10 000 and lasts fo r 10 years, a t which time It is sold for $2 000, w h a t equal a n n u a l cost o v e r its life is equivalent to these tw o cash flow s w ith a 10% a n n u a l interest rate? (3.7) 70. Given the two cash flo w profiles show n b e lo w , fo r w h a t value of X are the tw o series equivalent using an in te re st rate o f 10%? (3.7) EOY 0

CHA) -$100 000

CF(B) -$ 7 0 000

1

12 000

8000

2

16 000

8000

3

20 000

8000

4

24 000

8000+X

5

28 000

8000+2X

6

32 000

8000+3X

7

36 000

8000+4X

8

40 000

8000+5X

71. Given the three cash flo w profiles show n b e lo w , determ ine the values o f X and Y so th a t all three cash flo w pro file s are equivalent at an annual Interest rate o f 8% . S h ow all w o rk.

Time Value of Money Operations

EOY 0

129

CF(A) -$ 1 000

CF(B)

CF(Q

- $ 2 500

$ Y

1

X

3 000

Y

2

1.5X

2 500

Y

3

2.0X

2000

2Y

4

2.5X

1 500

2Y

5

3.0X

1 000

2Y

72. An investment opportunity offered by commercial banks is guaranteed investment certificates. For example, a guaranteed Investment certificate (GIC) having a four-year maturity date may have a stated interest rate o f '7.5%, payable quarterly.' This statement means that every three months, the issuing bank would pay the GIC holder an amount o f (0.075X1/4) times the face value (purchase price) of the GIC Assuming a $5 000, four-year, 7.5% GIC were purchased at t = 0, quarterly payments o f R0.075)(0.25)($5 000)1 = $93 75 would be received at f - 1 , 2 , 3, ...,16, and at t = 16, the GIC would be redeemed by the bank for the original purchase price, or $5000. (This payment series is an application of simple Interest on a quarterly basis.) It is also possible for the purchaser of the GIC to arrange with the bank for each Interest payment to be deposited Into a regular savings account that pays, say, 5.5% compounded quarterly. Let it be assumed that this is done; the first deposit of $93.75 occurs at t = 1, and the last deposit is made at t - 16. (a) For the Initial Investment of $5 000, what total sum o f money will be received by the purchaser at the end of the fourth year (GIC redemption value plus savings account balance)? (b) Now consider only the Initial Investment, P = $5 000, and the F value calculated from (a). W hat effective annual compound Interest rate relates the tw o single sums of money? (c) For P and F from (b) above, what annual simple Interest rate relates the tw o single sums of money? (3.7)

Chaplet 3

130

73. Assume the follow ing tw o Investm ent plans. Plan A-Purchase for $4 3 86 and receive th e fo llo w in g : (1) $400 at the end of each six months for four years. (2) A single payment of $2 000 at the end o f the fourth year. Plan B-Purchase for $3 302 and receive th e fo llo w in g : (1) $900 at the beginning of each year fo r four years. (2) A single payment of $X at the end o f the fourth year. If money Is worth 6% com pounded se m ian nually, w h a t Is the value of X such that the tw o plans are e q u iva le n t? (3.7) 74. A person borrows $5 000 at 6% co m p o u n d e d annually. Five equal annual payments are used to re p a y th e lo a n , w ith the first payment occurring one year a fte r re ce ivin g th e $5 000. Determine the Interest am ount included In each paym ent. (3.8) 75. An Individual borrows $8 0 0 0 at 7% c o m p o u n d e d annually. Six equal annual payments are used to re p a y th e lo an, w ith the first payment occurring tw o years afte r re ce ivin g th e $8 000. Determine the payment on principal in clud ed In each paym ent (3.8) 76. A man borrows $10 000 at 8% co m p o u n d e d m o n th ly . He is to pay off the loan w ith 60 m o n th ly paym ents. O ne m onth after making the thirtieth paym ent, he elects to p a y o ff the unpaid balance on the note. H ow m uch s h o u ld he repay? (3.8) 77. An Individual makes five annual deposits o f $1 0 0 0 In a savings account that pays interest a t a rate o f 4 % com pounded annually. One year after m aking th e last deposit, th e Interest rate changes to 5% com pounded annually. Five years after the last deposit the accumulated m oney Is w ith d ra w n fro m the account. How much Is w ithdraw n? (3.9.1) 78. A person deposits $ io 000 In a savings acco u n t p a y in g 5% compounded annually fo r the first tw o years a n d 6 % compounded annually fo r the next tw o years. Four annual withdrawals are made from the savings account. T he size of , e.withdrawal Increases by $1 0 0 0 per year, w ith th e first withdrawal occurring one year after th e deposit. D eterm ine the e of the last withdrawal, w hich depletes th e balance o f the account (3.9.1)

Time Value of Money Operations

131

79. Assume that six equal deposits o f $500 are made at t = 1,2, 3,4, 5, and 6 (three-month periods) into a fund paying 6% compounded quarterly. This interest applies until t = 16. The accumulated sum is withdrawn at this time and immediately deposited into a fund paying 8% compounded continuously. Beginning with t =17, determine the amount o f three equal quarterly payments, A, that may be withdrawn such that the fund Is depleted at f =19. (3.9.1) 80. A man borrows $10 000 and repays the loan with four equal annual payments. The interest rate for the first tw o years of the loan Is 5% compounded continuously, and for the third and fourth years of the loan It Is 6% compounded continuously. Determine the size o f the annual payment. (3.9.1) 81. A person deposits $5 000 In a savings account. One year after the initial deposit, $1 000 is withdrawn. TWo years after the first withdrawal, $4 000 Is deposited in the account Three years after the second deposit, $2 000 is withdrawn from the account Four years after the second withdrawal, all funds are withdrawn from the account During the period of time the savings account was In use, the bank paid 4% compounded continuously for the first tw o years, 5% compounded annually for the next four years, and 6% compounded quarterly for the remainder of the tim e the account was In use. Determine the amount of the final withdrawal. (3.9.1) 82. A man borrows $5 000 and repays the loan with three equal annual payments. The Interest rate for the first year o f the loan Is 5% compounded annually, for the second year of the loan is 6% compounded annually, and for the third year of the loan is 7% compounded annually. Determine the size o f the equal annual payment. (3.9.1) 83. A person deposits $1 000 in a fund that pays Interest at a continuous compound rate of r t for the fth year after the Initial deposit, where _ f 0.04+0.005t

r * "(0 .0 7

1=12,3,4,5,6 t = 7,8,9,10,...

Thus, a continuous compound rate of 4.5% Is earned during the first year, 5% Is earned during the second year, 5.5% Is

132

earned during the third year, etc. A m a x im u m o f 7% compounded continuously Is earned d u rin g th e sixth and each successive year. If the person w ith dra w s $ 5 0 0 tw o years after the Initial deposit, how much m oney w ill be In th e fu n d six years after the Initial deposit? {3.9.1) 84. A person deposits $2 500 In a savings acco u n t th a t pays Interest at a rate of 5% com pounded ann ually. TWo years after the deposit the savings account begins p a y in g in terest at a rate of 5% compounded continuously. Five years a fte r the deposit, the savings account begins p a y in g Interest a t a rate of 6% compounded semiannually. (a) How much money should be In th e savings a ccou nt 10 years after the Initial deposit? (b) What annual compound interest rate is e q u iv a le n t to the interest pattern o f the savings account o v e r th e 10-year period? (3.9.1) 85. A man deposits $1 000 In a fund each ye a r fo r a 10-year period. The fund initially pays 4% co m p o u n d e d annually. Immediately after the m an m akes his sixth deposit, th e fund begins paying 5% com pounded annually. T he m a n removes his money from the fund three years afte r his last deposit. How much should he be able to w ith d ra w a t th a t tim e ? (3.9.1) 86. Based on discrete cash flow s and discrete co m p o u n d in g , compute the present w orth and ann ual w o rth fo r th e follow ing situations: (3.9.1) EOP t

CF

Interest Rate During Period

0

- $10 000

1

20 0 0

2

0.05

4000

3

0.06

60 0 0

4

0.07

800 0

0.08

10 000

0.09

5

0

Time Value of Money Operations

87. Solve problem 86 for the case of continuous compounding with the Interest rates shown interpreted as nominal rates. (3.9.1) 88. A person deposits $2 000 in a savings account that pays 7% compounded annually; tw o years after the deposit, the interest rate increases to 8% compounded annually. A second deposit of $2 000 is made immediately after the interest rate changes to 8%. How much w ill be In the fund five years after the second deposit? (3.9.1) 89. An individual makes m onthly deposits o f $100 in a savings account that pays interest at a rate equivalent to 6% compounded quarterly. How much money should be In the account Immediately after the sixtieth deposit? If no Interest Is earned on money deposited during a quarter and the first deposit coincides with the beginning of a quarter, what will be the account balance Immediately after the sixtieth deposit? (3.9.2) 90. An individual makes semiannual deposits of $500 into an account that pays interest equivalent to 7% compounded quarterly. Determine the account balance immediately before the tenth deposit (3.9.2) 91. A rental contract consists of annual charges payable In advance. The first charge is $1 000 at t - 0 and then decreases by $200 each year. If five annual payments are made and money is worth 15% compounded annually, what Is the present worth at t = 0 o f ail payments by using the uniform gradient series formula? (3.9.2) 92. If a fund pays 6% compounded annually, what deposit Is required today such that $1 000 can be withdrawn every five years forever? (Ignore any tax considerations.) (3.9.3) 93. Maintenance on a reservoir Is cyclical with the following costs occurring over a five-year period: $3 000, $2 000, $5 000, $0, and $1 000. It Is anticipated that the sequence w ill repeat Itself every five years forever. Determine the capitalized cost for the maintenance costs based on a time value o f money of 8%. (3.9.3)

133

Chapter 3

134

94 A bond is purchased for $900 and kept for 10 years, at which time It matures at a face value of $1 000. During the 10-year period $60 is received every six months (l.e„ 20 receipts of $60 each). What Is the rate of return for the Investment? (3.9.4) 95. A person buys a $2 000 bond for $1 800. The bond has a bond rate of 8% with bond premiums paid annually. If the bond Is kept for 10 years and sold for par value, determine the equivalent annual return (rate of return) for the bond Investment. (3.9.4) 96. A person is considering purchasing a bond having a face value of $2 500 and a bond rate of 8% payable semiannually. The bond has a remaining life of 15 years. How much should be paid for the bond In order to eam a rate o f return o f 10% compounded semiannually? Assume the bond w ill be redeemed for face value. (3.9.4) 97. A person wishes to sell a bond that has a face value of $2 000. The bond has a bond rate of 8% w ith bond premiums paid annually. Four years ago, $1800 was paid fo r the bond. At least a 10% return on Investment Is desired. W hat must be the minimum selling price for the bond in order to make the desired return on investment? (3.9.4) 98. A $5 000,10-year, 8% semiannual bond is purchased at t = 0 for par value. After receiving the twelfth dividend, the bond is sold at a price to yield a 6% annual nominal rate o f return on this original purchase price. (a) What was the selling price? (b) If the purchaser keeps the bond until maturity and redeems It for par value, what approximate annual nom inal rate of return on the bond investment will be earned? (3.9.4) 99. The following labour costs are anticipated over a five-year period: $7 000, $8 000, $10 000, $12 000, and $15 000. It Is estimated that a 6% inflation rate will apply over the time I*1 qu e s ^ o n - The labour costs given above are expressed n i2«-^ a Jr r e r rt d ° " a r s - The time value of money, excluding on, h °in ' *s xe s ^ m a ^ t 0 5%. Determine the present worth equivalent for labour cost. (3.10)

Time Value of Money Operations_______________________________________ __________________________ 135

100. Labour costs over a four-year period have been forecast in then-current dollars as follows: $10 000, $12 000, $15 000, and $17 500. The Inflation rates for the four years are forecast to be 8,9 ,10 , and 10%; the Interest rate (excluding inflation) Is anticipated to be 6 ,6 , 5, and 5% over the four-year period. Determine the present worth equivalent for labour cost. (3.10) 101. Determine the capital recovery cost for an investment of $100 000 over a 10-year period, w ith salvage value of $20 000. Use an interest rate o f 20%. (3.9.5) 102. Determine the capital recovery cost for an investment of $75 000 over a five-year period, with salvage value of -$25 000. Use an Interest rate of 10%. (3.9.5)

Chapter 4

Comparison of Alternatives 4.1 Introduction Chapter 1 recommended that engineers solve problems by formulat­ ing and analyzing the problem, generating a number of feasible solu­ tions (alternatives) to the problem, comparing the alternatives, selecting the preferred solution, and Implementing the solution. The process o f evaluating the alternatives and selecting the preferred solu­ tion Is the subject o f this chapter. Here we apply the time value o f money concept to the comparison of economic investment alternatives. Although multiple objectives are often Involved in performing a comparison of alternatives, for now we concentrate on the comparison of mutually exclusive alternatives on the basis o f monetary considerations alone. The term, mutually exclu­ sive alternatives, means that no more than one alternative can be cho­ sen. The adage of ‘ not being able to have one's cake and eat it to o ' illustrates the notion of mutually exclusive alternatives. A systematic approach that can be used in selection o f investment alternatives Is summarized as follows: 1. Define the set of feasible, mutually exclusive economic investment alternatives to be compared. 2. Define the planning period to be used in the economic study. 3. Develop the cash flow profiles for each alternative. 4. Specify the time value of money to be used. 5. Specify the measured) of merit or effectiveness to be used. 6. Compare the alternatives using the measured) of merit or effectiveness. 7. Perform supplementary analyses. 8. Select the preferred alternative.

__

138

Chapter 4

The procedures for com paring Investm ent a lte rn a tive s outlined In this chapter are intended to aid In m a kin g b e tte r m easurem ents of the quantitative aspects o f Investment alternatives. It cannot be too strongly emphasized that no economic evaluation can replace the sound Judgement o f experienced managers concerned with both the quantitative and nonquantitative aspects o f investment alternatives. T ypical o f th e aspects of alternatives not considered In this chapter are safety, personnel con­ siderations, product quality, customers' satisfaction, environm ental ef­ fects, and engineering and construction capa bility. Such factors are relevant and often control decisions on ca p ita l expenditures. How­ ever, our concern Is w ith the m onetary aspects o f th e alternatives. We Interpret our role to be the d e ve lo p m e n t o f lo g ica l approaches to be used in analyzing investm ent alternatives a n d recommending ac­ tion to be taken, based on econom ic consid e ra tio n s alone. The pro­ cess of deciding which alternative to choose fo r implementation involves the consideration o f m o n e ta ry a n d n o n -m o n e ta ry factors. Approaches that can be used to assim ilate m u ltip le objectives are treated in Chapter 8.

4.2 Defining Mutually Exclusive Alternatives An Individual alternative selected from a set o f m u tu a lly exclusive alter­ natives can be made up o f several investmentproposals. Investment alter­ natives are decision options; Investment proposals are single projects or undertakings that are being considered as investm ent possibilities. Example 4.1 As an illustration of the distinction between investment proposals and investment alternatives, consider a distribution centre that re­ ceives pallet loads of product, stores the product, and ships pallet loads of product to various customer locations. A new distribution centre is to be constructed, and the following proposals have been 1 Method of moving materials from receiving to storage and from storage to shipping a. Conventional lift trucks for operating In 4-m aisles. b. Narrow-aisle lift trucks for operating In 2-m aisles. c. Automatically guided vehicle system. d. Belt conveyor system. e. Pallet conveyor system.

Comparison of Alternatives

2. Method o f placing materials In and removing materials from storage. a. Conventional lift trucks for operating In 4-m aisles. b. Narrow-aisle lift trucks for operating In 2-m aisles. c. Narrow-aisle, operator-driven, rail-guided storage/retrieval vehicle. d. Narrow-aisle, automated, rail-guided storage/retrieval vehicle. 3. Method of storing materials. a. Stacking pallet loads o f material (3 m high, 4-m aisles). b. Conventional pallet rack (6 m high, 4-m aisles). c. How rack (6 m high, 4-m aisles). d. Narrow-aisle, pallet rack (6 m high, 2-m aisles). e. Flow rack (6 m high, 2-m aisles). f. Medium height, pallet rack (12 m high, 2-m aisles). g. High rise, pallet rack (24 m high, 2-m aisles). Given the set o f proposals, alternative designs for the material handling system can be obtained by combining a proposed method o f moving materials from receivlngto storage, a proposed method placing materials In storage, a proposed method o f stor­ age, a proposed method of removing materials from storage, and a proposed method of transporting materials from storage to ship­ ping. Some o f the combinations of proposals w ill be eliminated be­ cause of their incompatibility. For example, lift trucks requiring 4-m aisles cannot be used to place materials in and remove materials from storage when 2-m aisles are used. Other combinations might be eliminated because of budget limitations; a desire to minimize the variation in types o f equipment due to maintainability, avail­ ability, reliability, flexibility, and operability considerations; ceiling height limitations; physical characteristics of the product (crush­ able product might require the use o f storage racks); and a host of other considerations. Characteristically, experience and judge­ ment are used to trim the list of possible combinations to a man­ ageable number.

Example 4.2 lb Illustrate the form ation of mutually exclusive investment alter­ natives from a set o f Investment proposals, consider a situation In­ volving m investment proposals, let x t be defined to be 0 if proposal J Is not Included In an alternative and let x t be defined to be 1 if proposal J is included In an alternative. Using the binary vari­ able x t we can form 2 m mutually exclusive alternatives. Thus, If

139

140

_____________________________________________________________________Chapter*

there are three Investment proposals, w e can fo rm eight mutually exclusive investment alternatives, as depicted In Table 4.1. Table 4.1 Developing Mutally Exclusive Investm ent Alternatives from Investment Proposals X1

x, X,

1

0

0

0

Do nothing (proposals 1 ,2 , and 3 not Included)

2

0

0

1

Accept proposal 3 o n ly

3

0

1

0

Accept proposal 2 o n ly

4

0

1

1

Accept proposals 2 and 3 o n ly

5

1

0

0

Accept proposal 1 o n ly

6

1

0

1

Accept proposals 1 and 3 o nly

7

1

1

0

Accept proposals 1 and 2 o n ly

8

1

1

1

Accept all three proposals

Alternative

Explanation

As pointed out in the discussion o f th e design o f a m a te ria l handling system for the distribution centre, a m o n g th e alterna tives formed some might not be feasible, dep ending o n th e restrictions or con­ straints placed on the problem. To illustrate, th e re m ig h t be a budget limitation that precludes the possibility o f c o m b in in g a ll three propos­ als; thus, Alternative 8 w ould be elim inated. A d d itio n a lly , some of the proposals might be mutually exclusive proposals. For exam ple, Propos­ als 1 and 2 might be alternative com puter system s a n d o n ly one Is to be selected; in this case Alternative 7 w o u ld b e e lim in a te d from con­ sideration. Other proposals m ight be contingent proposals so that one proposal cannot be selected unless a n o th e r p ro p o sa l Is also selected. As an Illustration of a contingent proposal, Proposal 3 m ig h t Involve c o m Pu t e r term inals, w h ic h de p e n d o n the selecZ P™ “ the computer system associated w ith P roposal 2. In such a l nn ♦ho° L ^ i m a t i v e s 2 a r , d 6 w o u | d be Infeasible. Thus, depending aHomJh ^ O n s Pr e s e r r t ' the n um be r o f fe a sib le m u tu a lly exclusive alternatives that result can be considerably less th a n 2 m . te m 2 ^ h ° f^ J

fo rm a liz e d hierarchy for det o funds. Typically, the enasslmmprrf4nh S t h e r a r c h y , n v o , v e s an e n g in e e r w h o Is given an d S K ^ n ^ e .a P r ° b , e m ; t h e Pr o b , e m m a V he o n e requiring the gn of a new product, the Im provem ent o f an e x is tin g manufactur, Z a t , o n s t h e r e is a r a t h e r

u? r g a n , z a t i o n w i n l n v e s t

Comparison of Alternatives_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 141

Ing process, or the development of an improved system for perform­ ing a service. The engineer performs the steps Involved In the problem-solving procedure and recommends the preferred solution to the problem. In arriving at the preferred solution, a number o f alterna­ tive solutions are normally compared; hopefully, the eight-step ap­ proach for comparing investment alternatives was followedl The preferred solution Is usually forwarded to the next level o f the hierarchy fo r approval. In fact, one would expect many preferred solu­ tions to various problems to be forwarded. Each preferred solution be­ comes an Investment proposal, the resulting set of mutually exclusive Investment alternatives are formed, and the process of comparing economic Investment alternatives Is repeated. TTils sequence of opera­ tions Is usually performed in various forms at each level of the hierar­ chy until, ultimately, the preferred solution by the individual analyst or engineer Is accepted or rejected. In this textbook we concentrate on the process of comparing Investment alternatives at the first level o f the hierarchy; however, the need fo r such comparisons at many levels of the organization should be kept In mind.

4.3 The Planning Period In comparing Investment alternatives, It Is important to compare them over a common period o f time. We define that period o f time to be the planning period. In the case o f Investments in, say, equipment to per­ form a required service, the period of time over which the service is re­ quired might be used as the planning period. Likewise, in one-shot Investment alternatives, the period of tim e over which receipts con­ tinue to occur might define the planning period. In a sense, the planning period defines the width of a 'window* that is used to view the cash flows generated by an alternative. In order to make an objective evaluation, the same w indow must be used In view­ ing each alternative. In some cases the planning period Is easily determined; In other cases the duration o f one or more projects Is sufficiently uncertain to cause concern over the time period to use. Some commonly used methods for determining the planning period to use In economic analysis Include: 1. Least common multiple o f Ilves for the set of feasible, mutually exclusive alternatives, denoted f .

Chapter 4

142

2. Shortest project life am ong the alternatives, d e n o te d T s . 3. Longest project life am ong the alternatives, d e n o te d T r 4. Some other period o f time. In the economic analysis literature, th e m o st com m o nly used method of selecting the planning period appears to be th e least com­ mon multiple o f lives. In most cases, such a selection Is m ade Implicitly, not explicitly. When three alternatives are b e in g considered and the individual lives are six years, seven years, and e ig h t years, using such a procedure yields a planning period o f T = 1 6 8 years. If the lives had been either six years, six years, a n d e ig h t years or six years, eight years, and eight years, T = 2 4 years. Clearly, strict reliance on f as the planning period is n o t advisable. If the shortest project life is used to defin e th e p la n n in g period, esti­ mates are required for the values o f the unused p o rtio n s o f the Ilves of the remaining alternatives. Thus, fo r th e s itu a tio n considered above, with Ts = 6 years, the salvage o r residual values a t th e end o f six years' use must be assessed for the other tw o alternatives. If the longest project life, T,, is used in d e te rm in in g th e planning pe­ riod, some difficult decisions must be m ade co n c e rn in g th e period of time between Ts and !,. If the alternative selected is to p rovid e a neces­ sary service, that service must continue th ro u g h o u t th e planning pe­ riod, regardless of the alternative selected. C onsequently, when the shortest life alternative reaches the end o f its p ro je c t life, it must be re­ placed with some other asset capable o f p e rfo rm in g th e required serv­ ice. However, since technological d e ve lop m en ts w ill probably take place during the period o f tim e T s , ne w and im p ro v e d candidates will be available for selection at tim e T s . Thus, th e sp e cification o f the cash flows for the shortest life alternative d u rin g th e p e rio d o f tim e from Ts t ° T, is a difficult undertaking. As a result, T. is se ld o m used as the plan­ ning period. A number of organizations have adopted a sta n d a rd planning pe­ riod for all economic alternatives. Letting T d e n o te th e p la n n in g period specified by the organization, d iffe re n t a p p ro a ch e s are recom­ mended, depending on w h e th e r! < f o r T £ f . If th e p la n n in g period selected is less than the least com m on m u ltip le o f Ilves, th e cash flows for each alternative must be provided fo r a pe rio d o f tim e equal to the p anning period. When the planning period Is g re a te r th a n o r equal to tne least common m ultiple o f Ilves, it is re co m m e n d e d th a t the eco-

Comparison of Alternatives_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 143

nomlc analysis be based on a period of time equal to the least com­ mon multiple of lives. The reason for the latter recommendation Is that at time t a new economic analysis can be performed based on the al­ ternatives available at that time. After f years new alternatives might be available; furthermore, one can more accurately estimate the val­ ues of cash flows occurring after f if one waits until nearer time T to make the estimates. Example 4.3 To illustrate the difficulties associated w ith the selection o f the planning period, consider the tw o cash flow profiles given in Table 4.2. Alternatives A and B have anticipated lives of four years and six years, respectively. Table 4.2 Cash Row Profiles for Two Mutually Exclusive Investment Alternatives Having Unequal Lives Alternative A CF(A)

EOY 0 1-4

Alternative B

- $5 000 -

3 000

EOY 0 1-6

CF(B) -$ 6 0 0 0 -

2 000

Using a least common multiple of lives approach, a planning pe­ riod o f f =12 years would be used. Using a 12-year planning pe­ riod requires answers to the follow ing questions. What cash flows are anticipated for years 5 to 12 If Alternative A Is selected? What will be the cash flows for Alternative B for years 7 to 12? As shown in Table 4.3, the traditional approach is to assume that Alternative A w ill be repeated twice and Alternative B will be repeated once and that Identical cash flows occur during these re­ peating life cycles. Inflation effects, as well as the possibility o f technological improvements, tend to invalidate such assumptions. if the shortest life approach is used, a planning period of four years would be used. In such a case, an estimate of the salvage value of Alternative B should be Indicated at the end of year 4, as denoted In Thble 4.3 by an asterisk. Using the longest life approach yields a slx-year planning pe­ riod. In this Instance a decision must be made concerning the cash flows In years 5 and 6 for Alternative A. If an initial investment must be made to provide the required service for years 5 and 6, It would occur at the end of year 4. The assumption made In Table

Chapter

4.3 Is that Alternative A will be repeated with identical cash flows, and a terminal salvage value of $2500 w ill apply after tw o years of use. Suppose a standard planning period o f 10 years must be used. The same questions that arose in the cases o f T, T s , or T, being the

planning period apply when a 10-year planning period Is used. As depicted In Table 4.3, one might assume that identical life cycles will be repeated until the end of the planning period and provide estimates of terminal salvage values at that time. Table 4.3 Cash Flow Profiles for Various Planning Periods T = TS = 4

7 = 7=12 EQY

—

CHA)

CRB)

EOY

CHA)

7=10 CHBL

E O Y ... C R A ) 0

CRB)

0

-$5000

-$6000

0

-$ 5 0 0 0

-$ 6 0 0 0

-$ 5 0 0 0

-$6000

1

-3 0 0 0

-2 0 0 0

1

-3 0 0 0

-2 0 0 0

1

-3 0 0 0

-2 0 0 0

2

-3 0 0 0

-2 0 0 0

2

-3 0 0 0

-2 0 0 0

2

-3 0 0 0

-2 0 0 0

-3 0 0 0

-2 0 0 0

3

-3 0 0 0

-2 0 0 0

-3 0 0 0

-2 0 0 0

4

-8 0 0 0

-2 0 0 0

+2000*

5

-3 0 0 0

-2 0 0 0

6

-3 0 0 0

-8 0 0 0

7

-3 0 0 0

-2 0 0 0

3

-3 0 0 0

-2 0 0 0

3

4

-8 0 0 0

-2 0 0 0

4

5

-3 0 0 0

-2 0 0 0

6

-3 0 0 0

-8 0 0 0

7

-3 0 0 0

-2 0 0 0

8

-8 0 0 0

9

-3 0 0 0

10

-3 0 0 0

11

-3 0 0 0

12

-3 0 0 0

7 = 7, = 6

-2 0 0 0

EOY

CRA)

C R B)

8

-8 0 0 0

-2 0 0 0

-2 0 0 0

0

-$ 5 0 0 0

-$ 6 0 0 0

9

-3 0 0 0

-2 0 0 0

-2 0 0 0

1

-3 0 0 0

-2 0 0 0

10

-3 0 0 0

-2 0 0 0

-2 0 0 0

2

-3 0 0 0

-2 0 0 0

+2500*

+2000’

-2 0 0 0

3

-3 0 0 0

-2 0 0 0

4

-8 0 0 0

-2 0 0 0

5

-3 0 0 0

-2 0 0 0

6

-3 0 0 0

-2 0 0 0

+2500*

Although the use of a standard planning period has the benefit of a consistent approach In comparing investment alternatives, there are ? dangers that should be recognized. In some cases, the ma­ jor benefits associated with an alternative m ight occur In the later

hS8?,? p r< ^ e c t , l f e - l f t h e planning period is less than the project ra alternatives would seldom be accepted. Just such a practice O n amat e major textile firm to lose Its strong position In the Industry,

ajor modernization of the processing departments had been pro-

Comparison of Alternatives

145

posed, but Its benefits would not be realized until the bugs had been worked out of the new system, all personnel were trained under the new system, and the marketing people had regained the lost custom­ ers. Unfortunately, the planning period specified by the firm was too short in duration, and the modernization plan was not approved. For the example, deciding which planning period to use should de­ pend on the particular situation instead of on the duration of the indi­ vidual alternatives. If, for Instance, Alternatives A and B are two different lift truck designs, and it Is anticipated that the material han­ dlingfunction to be performed by the lift truck will continue for at least 12 years, the least common multiple of Ilves approach might make sense. Likewise, since the lift truck industry is quite dynamic and tech­ nological improvements are quite likely to occur in the future, we might prefer to use the shortest life as the planning period. Example 4.4 As a second illustration of the planning period selection process, consider the tw o cash flow diagrams given in Figure 4.1. The two alternatives are m utually exclusive, one-shot Investment alterna­ tives. We are unable to predict w hat Investment alternatives w ill be available in the future, but we do anticipate that recovered capi­ tal can be reinvested and earn a 10% return. For this type o f situation, a six-year planning period is sug­ gested, with zero cash flows occurring in years 5 and 6 with Alter­ native 1. At the end o f six years, the net future worths fo r the tw o alternatives will be: FW,(10%) = $4 500(F|P10,2) + $3 500(P| A l 0 3 X ^ 1 0 , 6 ) -$ 4 000(FlP10,6) = $13 778.87 FW2 (10%)= $1 000(F| A10,6) + $1 000(A|G10,6XF| A l 0,6) -$ 5 000(F|P l 0,6) = $16 014.01 Thus, we would recommend Alternative 2.

If one did not give careful thought to the situation Involved and blindly assumed a least common multiple of lives planning period, with identical cash flows In repeating life cycles, then Alternative 1 would be recommended. Hence, it Is Important to consider the particu-



t 1

Alternative 2

Aftemattve 1 MARR - 10%

$5000

Fig. 4.1 Cash flow diagram for the example problem.

lar situation Involved and specify the planning period instead of em­ ploying a rale of thumb for establishing a planning period that does not consider the nature of the Investments. It appears that the preferred approach would be to have a 'flexible' standard planning period. All of the 'ro utine ' economic analyses would be based on the standard planning period of, say, five to ten ! S rSJ iT !l r o u t i n e economic analyses would be based on a planning period that was appropriate for the situation.

4.4 Developing Cash H o w Profiles , l y e x d u s i v e alternatives has been specified and m ade' v X S ^ r * 1? 1 h a s cash f l ow profiles can be dem a v e s h a s b e e n D rE J h n .^ K ^ ? emphasized, the cash flow b y SMhS c a r e f u l consideration to future S ffln n c r e y n g completely on past cash flows. The cash flows for an vestment alternative are obtained by aggregating the

Comparison of Alternatives

147

cash flows for all Investment proposals Included In the Investment al­ ternative. Example 4.5 To Illustrate the approach to be taken, suppose a planning period of five years Is used and there are three Investment proposals. Cash flow profiles for the proposals are given in Table 4.4. A budget limitation o f $50 000 Is available for Investment among the proposals. Proposal 2 Is contingent on Proposal 1, and Propos­ als 1 and 3 are mutually exclusive. Based on the restrictions associ­ ated with the combinations of proposals, only four investment alternatives are to be considered. Alternative A is the do-nothing alternative; Alternative B is Proposal 1 alone; Alternative C in­ volves a combination of Proposals 1 and 2; and Alternative D Is Proposal 3 alone. The cash flow profiles for the four alternatives are given in Table 4.5. Table 4.4 Cash Flow Profiles for Three Investment Proposals EOY

CF(1)

CF(2)

CF(3)

0

-$ 2 0 000

- $ 3 0 000

- $50 000

1

-

4 000

+

4000

-

5 000

2

+

2 000

+

6000

+

10 000

3

+

8000

+

8 000

+

25 000

4

+ 14000

+ 10000

+ 40000

5

+ 25 000

+ 20000

+

10 000

Table 4.5 Cash How Profiles for Four Mutually Exclusive Investment Alternatives CF(D)

EOY

CHA)

CHB)

0

0

- $20 000

1

0

-

4000

2

0

+

2 000

+

3

0

+

8 000

+ 16 000

+ 25 000

4

0

+ 14 000

+ 24 000

+ 40 000

5

0

+ 25 000

+ 45 000

+

CH0 - $50 000 0 8000

- $50 000 -

5000

+

10 000

10 000

148

The do-nothing alternative Is the status q u o c o n d itio n and serves as the base against which other alternatives are considered. In some cases, the do-nothing alternative Is n o t feasible (e.g., fa llin g to comply with pollution standards). Moreover, the d o -n o th in g alternative does not necessarily have zero cash flow s associated w ith it. In principle, one should forecast the cash flow s th a t w ill result if the present method Is continued and compare the cash flo w s w ith those associ­ ated with other alternatives. In most, if not all, economic evaluations, it is n o t necessary to de­ velop a detailed forecast o f all item s o f cost, revenue, and Investment associated with an alternative. Costs and revenues th a t w ill be the same regardless o f the alternative selected can be o m itte d . If a cost re­ duction alternative w ill not affect sales revenues, n o forecast of such revenues need be developed. A ttention Is focused o n th e items of cost and revenue that w ill be affected b y th e a lte rn a tiv e selected. Since economic analyses are to be p e rfo rm e d to ju d g e the merits of investment alternatives fo r the future, th e d a ta required differ from those normally provided by the accounting system . The fundamental and traditional purpose o f accounting Is to m a in ta in a consistent his­ torical record o f the financial results o f th e o p e ra tio n s o f the organiza­ tion. Accounting figures are based on d e fin itio n s derived consistent with this objective. Accounting m ethods are n o t designed to deter­ mine the economic w orth o f alternative courses o f action.

4.5 Specifying the Tim e Value o f M o n ey An important step In evaluating Investm ent a lte rn a tive s Involves the specification o f the Interest o r discount rate to be used. Even though a project may be financed entirely fro m in te rn a l sources o f funds, an In­ terest rate is recommended in evaluating in v e s tm e n t alternatives. One reason for doing so is to reflect the cost o f in v e s tin g m o n e y In a par­ ticular project instead o f investing It elsew here a n d e a rn in g a return on the Investment The cost o f fo re going o th e r In ve stm e n t opportunities is referred to as the opportunity cost, as discussed in Chapter 2. Except where other intangible benefits are In vo lve d , th e discount rate should be greater than the cost o f securing a d d itio n a l capital. In­ deed, it should be greater than the cost o f ca p ita l b y an am ount that cover unprofitable Investments th a t a firm m u st m ake for non­ monetary reasons. Examples o f the la tte r w o u ld Include Investments

Comparison of Alternatives_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 149

In antipollution equipment, safety devices, and recreational facilities for employees. The discount rate that Is specified establishes the firm's minimum at­ tractive rate o f return (MARR) In order for an Investment alternative to be Justified. If the present worth fo r an investment alternative were nega­ tive, Indicating that a negative cash flow was equivalent to the Invest­ ment alternative, It would not be recommended for adoption. Some firms establish a standard discount rate or minimum attrac­ tive rate of return to be used In all economic analysis; others maintain a flexible posture. For example, The XYZ Company return on Investment (ratio or earnings to gross Investment) has been, on the average over the past five years, approximately equal to a rate-of-retum o f 15% per year. Accordingly, 15% per year Is being established as a tentative minimum requirement fo r Investment alternatives whose results are primarily measurable in quantitative dol­ lar terms. The principle being applied is that such alterna­ tives should be expected to maintain or to improve overall return-on-investment performance for the XYZ Company. The minimum requirement is based on overall XYZ financial results (as opposed to results for various parts o f the Com­ pany) in order to avoid situations in which Investment alter­ natives o f a given level o f attractiveness are unknowingly undertaken In one part of the Company and rejected in an­ other. The 15% minimum attractive rate-of-retum standard Is In­ tended as a guide, rather than as a hard-and-fast decision rule. Furthermore, it is intended to apply to alternatives hav­ ing risks o f the kind usually associated with Investments which are primarily in plant and facilities. For alternatives In­ volving expenditures with substantially lower risks, such as those solely for Inventories or those In buy-or-lease altemtlves, a lower minimum requirement may apply. Other approaches that are used to establish the MARR include: 1. Add a fixed percentage to the firm's cost of capital. 2. Average rate o f return over the past five years Is used as this years MARR. 3. Use different MARR for different planning periods.

Chapter 4

150

4. Use different MARR fo r different m ag n itu d e s o f Initial Investment. 5. Use different MARR fo r new ventures ra th e r th a n fo r cost improvement projects. 6. Use as a management to o l to stim u la te o r discourage capital Investments, depending on the overall e co n o m ic condition of the firm. 7. Use the average shareholder's return o n In ve stm e n t fo r all companies In the same Industry group. There are a number o f different approaches used b y com panies In es­ tablishing the discount rate to be used In p e rfo rm in g econom ic analy­ ses. The issue is not a simple one. The proper determination o f the discount rate has been the subject of considerable controversy In the e co n o m ic analysis literature for many years. In some ways w e are n o closer to an agreem ent today than we were 40 years ago. FOr this reason, a m o n g others, step 7 (per­ form supplementary analyses) Is Incorporated In th e eight-step proce­ dure for comparing economic alternatives. In m a n y cases, a particular alternative will be preferred over a range o f possible discount rates; In other cases, the alternative preferred w ill be q u ite sensitive to the dis­ count rate used. Hence, depending on th e s itu a tio n under study, it might not be necessary to specify a p a rticu la r v a lu e fo r the discount rate-a range o f possible values m ight suffice. W e e xa m in e this situa­ tion in more depth In Chapter 7. Subsequently, it will be convenient to refer to th e Interest rate or dis­ count rate used as the m inim um attractive rate o f retu rn (/W4/?/?) and to interpret its value using the o p p o rtu n ity cost concept. The argument will be made that money should n o t be Invested In an alternative If it cannot earn a return at least as great as th e MARR, since it Is reasoned that other opportunities fo r investm ent exist th a t w ill yield returns equal to the MARR. In the case o f the public sector, a d iffe re n t In te rp re ta tio n Is required in determining the discount rate to use. Since th is ch ap ter emphasizes economic analysis In the private sector, C hapter 6 discusses establish­ ing the discount rate fo r the public sector.

Comparison of Alternatives_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

151

4.6 The Measures of Merit As noted In Chapter 3 In discussing equivalence, Investment alterna­ tives can be compared In a number of ways. Present worth (PW) and annual worth 0 VV) comparisons are tw o commonly used approaches. Among the several methods of comparing investment alternatives are: 1. Present worth method. 2. Annual worth method. 3. Future worth method. 4. Payback period method. 5. Rate o f return method. 6. Savlngs/investment ratio method. Each of the above measures o f merit or measures of effectiveness has been used numerous times In comparing real-world Investment alter­ natives. They may be described briefly as follows: 1. Present worth method converts all cash flows to a single sum equivalent at time zero. 2. Annual worth method converts all cash flows to an equivalent uniform annual series of cash flows over the planning period. 3. Future worth method converts all cash flows to a single sum equivalent at the end of the planning period. 4. Payback period method determines how long at a zero Interest rate It will take to recover the initial Investment 5. Rate o f return method determines the Interest rate that yields a future worth o f zero. 6. Savlngs/investment ratio method determines the ratio of the present worth o f savings to the present worth o f the Investment. With the exception o f the payback period method, all of the meas­ ures o f merit listed are equivalent methods o f comparing investment alternatives. Hence, applying each o f the measures of merit to the same set of Investment alternatives will yield the same recommenda­ tion (with the possible exception of the payback period method).

Chapter 4

152

Since the present worth, annual w o rth , fu tu re w o rth , rate o f return, and savlngs/lnvestment ratio m ethods are e q u iva le n t, w h y does more than one o f the methods exist? The p rim a ry reason fo r having differ­ ent, but equivalent, measures o f effectiveness fo r eco n o m ic alterna­ t iv e appears to be the differences in preferences a m o n g managers. Some Individuals (and firms) prefer to express th e n e t econom ic worth of an Investment alternative as a single sum a m o u n t; hence, either the present worth method or the future w o rth m e th o d is used. Other indi­ viduals prefer to see the net econom ic w o rth spread o u t uniformly over the planning period so the annual w o rth m e th o d is used by them. Yet another group o f individuals wishes to express th e net economic worth as a rate or percentage; consequently, th e rate o f return method would be preferred. Finally, some individuals p re fe r to see the net eco­ nomic worth expressed as a percentage o f th e In ve stm e n t required; the savlngs/lnvestment ratio is one m eth od o f p ro v id in g such infor­ mation. Since many organizations have established procedures fo r perform­ ing economic analyses, it seems w o rth w h ile to co n sid e r In this chapter the more popular measures o f m erit th a t are used. A m o n g those listed, it appears that the present w orth, rate o f return, a n d payback period methods are currently the m ost popular. H ow ever, a n u m b e r o f gov­ ernmental agencies have recently adopted som e ve rsio n o f the savings/investm ent (or beneftt/cost) ra tio m e th o d f o r purposes of comparing investment alternatives; hence, it is g a in in g in popularity.

4.6.1 Present Worth Method The present worth o f Alternative j can be represented as

'w Ai)=ZM 1+ ')-' w ith ^ /(O

=

present w orth o f A lternative j u sin g MARR o f 1%

n = planning period Ap = cash flow fo r A lternative J a t th e end o f period t I = MARR

Comparison of Alternatives_____________________________________________ _______________________________ 153

The alternative having the greatest present worth Is the alternative recommended using the present worth method. Example 4.6 A pressure vessel Is purchased for $10 000, kept for five years, and sold for $2 000. Annual operating and maintenance costs were $2 500. Using a 10% minimum attractive rate o f return, what was the present worth for the Investment? PW,(10%) = -S I 0 0 0 0 - $2 500(P| A10,5)+$2 000(P| F l 0,5) = -$18 235.20 Thus, a single expenditure o f $18 235.20 at time zero is equivalent to the cash flow profile for the Investment alternative.

4.62 Annual Worth Method The annual worth o f Alternative j can be computed as AW ; ( /) =

W

tn )

L t=o

or A W } (i) = P W j(i)(A \P i, n )

(4.2)

where AVVj(/) denotes the annual worth of Alternative j using I = MARR. Tne alternative having the greatest annual worth Is selected using the annual worth method. Example 4.7 Determine the annual w orth for an anticipated Investment of $10 000 In a computer that w ill last for eight years and have zero salvage value at the time. Operating and maintenance costs are projected to be $1 000 the first four years and $1 500 the last four years. The minimum attractive rate of return Is specified to be 10%. One method o f determining the annual worth Is:

Chapter 4

154

4W,(10%) = -$ W 000) as anticipated. The above example Illustrates that the time value o f money opera­ tions Involved In the IRR method are equivalent to assuming that all monies received are reinvested and earn interest at a rate equal to the internal rate of return. In particular, if the net cash flow In period t is negative, It is denoted by Ct ; If the net cash flow in period t Is positive, It Is denoted Rt . Lettlngr, be the reinvestment rate fo r positive cash flows occurring In period t and /' be the rate o f return for negative cash flows, then the following relationship can be defined:

ZR,(1+r,)'w = ic ,( 1 + n " " t=O

t=O

(4.5)

Chapter 4

150

The future w orth o f reinvested m onies received m u st equal the fu­ ture worth o f Investments. Ifr f equals i', Equation 4.4 becomes O= f ( R , - C , ) ( 1 + t=0

Letting 4, equal R, - C t defines the IRR m e th o d g iv e n by Equation 4.4. Hence, we see that the rate o f return o b ta in e d using the IRR method can be interpreted as the reinvestm ent rate fo r all recovered funds. Since the determination o f the rate o f return Involve s solving Equa­ tion 4.4 for Ij, It Is seen th a t (for a given a lte rn a tive J), It Is necessary to determine the values o f x that satisfy the fo llo w in g n-degree polyno­ mial: 0 = Ao x n + A } x n ~1 + ••• + A ^ x + A n , w h e re x = ( 1 + f) . In general, there can exist n distinct roots (values o f x) fo r an n-degree polynomial; however, most cash flo w profiles encountered In practice w ill have a unique root (rate o f return). The number o f real positive roots o f an n-degree p o lyn o m ia l with real coefficients Is less than o r equal to th e n u m b e r o f changes o f sign in the sequence o f cash flows, A 0 ,A v . . . M n _v A r Since t h e tyP ic a l cash flow pattern begins w ith a negative cash flo w , fo llo w e d by posi­ tive cash flows, a unique root w ill n o rm a lly exist. Example 4.10 As an Illustration of a cash flow profile having multiple roots, con­ sider the data given In Table 4.7. The future w orth o f the cash flow series will be zero using either a 20% or 50% interest rate. Thble 4.7 Cash How Profile EOY

CF

0

-$ 1 000

1

4 200

2 ‘ ........3

-

5 850

_____ 2 700

Comparison of Alternatives______________________________________________ __ ________________________

161

FW(20%) = -$1 000(12)’ + $4 200(12)2 - $5 850(12)+$2 700 = 0 /W(50%) = -$1000(15)’ +$4 200(15)2 -$ 5 850(l5)+$2 700 =0 A plot of the future worth for this example Is given In Figure 4.3. The future worth polynomial Is a third-degree polynomial and there are three changes of sign In the ordered sequence of cash flows (-,+,-,+)•, however, there are only two unique roots, corre­ sponding to I —0.20 and / = 0.50. In this case, there Is a repeated root corresponding to /=0.50, since the future worth polynomial can be written as FW(/%) = $1000(12 - xX15- x) 2 where x = (1 + /). The possibility o f multiple roots occurring In the Internal rate o f re­ turn calculation, coupled w ith the reinvestment assumptions concern­ ing recovered funds, has led to the development o f an alternative rate of return method, called the external rate ofreturn method. The external rate of return (ERR) method consists of the determination of the value of /' that satisfies Equation 4.7. i R , ( 1 + r , ) ”' ' = f c , d

where, as defined above, A t = net cash flow In period t c

*

R

1

=

f A „ l f At < 0 |0, otherwise

=

If A £ 0 [0, otherwise

r t = reinvestment rate for fund*;, recovered In period t I' = external rate o f return Normally, r t equals the minimum attractive rate o f return, since the MARR reflects the opportunity cost for money available for Invest­ ment. The preferred alternative is the alternative with the greatest In­ vestment such that each Increment o f Investment has a return at least equal to the MARR.

Chapter!

Future worth

162

Fig. 4.3 Plot of future worth for the exam ple problem .

Example 4.11 For the data provided In Table 4.6, suppose m oney received from the Intttial Investment Is reinvested and earns 10% Interest At the end of the sixth year the reinvested funds total: $2000(F]P 1 0 ,5 )+ $ 2 500(F|P 1 0 ,4 ) + $ 3 1 31(F]P 1 0 3 ) +$3 510(P|P 1 0 3 )+ $ 4 0 3 0 (F |P 1 0 ,l) + $ 4 4 0 0 = $ 2 4 1 2 7 3 8 Consequently, the external rate o f return Is defined as th e Interest rate such that the future worth of the $10 0 0 0 Investment equals $24127.38. Thus, $ 1 0 0 0 0 (1 + /')’ = $ 2 4 1 2 7 3 8

Comparison of Alternatives

163

Taking the logarithm and solving for /' yields a value of 15.8% as the rate o f return. As another example, If recovered funds are reinvested at 20%, the future worth w ill be $2 000(F|P 20,5)+ $2 500(F|P 2 0 /t) + $ 3 1 31(F|P 2 0 3 ) +$3 510( F|P 20,2) + $4 030 1 then A (B -C )> 0 . That Is, the difference In Incremental benefits and costs may be used In place o f the Incremental benefit-cost ratio.

Example 6.2 Applying this measure to the routes In the previous example re­ sults In the follow ing calculations for comparing Routes A and B. We know that AB8 _X (6) = $2 047 883 per year AC8 _X (6) = $434 846 per year This results In

294

____________________________________________________ Chapter6 A(B—C)b-52000 OOO(f|F /,20)X4|P /3 0> $2 08 000} per year

Searching for file value of / yielding A IV ,. 4 (/) = A B , . / / ) - ACfl . / / ) = 0 we find AW,_4 (30) = $103 8 27 per year AW b . a (3 5 ) = -$ 2 2 6 726 per year

The Interest rate we are seeking Is between 3 0 and 35%, or approximately 32%. In any event / easily exceeds 6%, which Indicates that Route B Is preferable to Route A.

Economic Analysis of Pro|ecte In the Public Sector__________________ ___ ______________ 297

Using the rate-of-retum approach, we can also compare Routes Band C ABC-»(0

= $8 3 2 1 8 5 8 - $7 754 958 = S566 900 per year

AQ-sO

= [$17 300 000+ $1800 OOtXflF /,10)+ $1800 000(P|F /,20)](4|P & ) )

+$317 750 - {[$8 800 000+S1800 000(P|F 1,10)+$1800 000(P|F /,20)](^ P /3 0 > $ 1 76 000} per year

The rate of Interest causing AWc _B (i) = AB c _B (i)-& C c _B (l)= 0 is between 2.5 and 3%, since AWC_B (2 .5 ) = $ 1 9 020 per year

AWc_b (3) = -$8 520 per year Since / is less than 6%, we prefer Route B to Route C

Several different approaches to the same problem have been illus­ trated and shown to be consistent project evaluators. Typically, where government projects are Involved, one o f the benefit-cost criteria ( B /C or B -CT is used. More often than not, the criterion used is the benefit-cost ratio. This Is unfortunate because, just like rate-of-retum analyses In the private sector, the benefit-cost ratio is easy to misuse and misinterpret, and It is very sensitive to the classification of prob­ lem elements as 'benefits' or 'costs.' These problems will be discussed later. The present worth, annual worth, and rate-of-retum methods are seldom used In government analyses. Even though the mechanics un­ derlying these approaches are perfectly suitable, their underlying phi­ losophy is more attuned to return on Investment discounted net profits, and minimum attractive rate of return, Implying measures of return to Investors based on capital Investment by the same Investors. Government Investments, however, do not necessarily result In any monetary incomes to the government and are seldom evaluated solely on the basis of monetary return. Government employees and politicians also prefer benefit-cost analysis, because It allows them to point out specific "benefits' derived by the public as a result o f the gov­ ernment's expenditures of the public's money.

298

6.5 Im portant Considerations In Evaluating Public Projects There are a number of pitfalls that can affect benefit-cost analysis of government projects. In fact, opportunities fo r error pervade benefit­ cost analyses from the very Initial philosophy through to the Interpre­ tation of a B/C ratio. It Is Important to examine the most significant of these, both to help prevent analysts from erring and to help those who may be reviewing a biased evaluation. The major topics to be considered are as follows: 1. point of view (federal, provincial, local, Individual) 2. selection of the Interest rate 3. assessment of benefit-cost factors 4. overcounting 5. unequal lives 6. tolls, fees, and user charges 7. multiple-use projects 8. problems with the ratio &5.1 Point of View The stance taken by the engineer In analyzing a public venture can have an extensive effect on the economic 'facts.' The analyst may take any of several viewpoints, including those of 1. an individual who will benefit or lose 2. a particular governmental organization 3. a local area such as a city or county 4. a regional area such as a province 5. the entire nation The first of these viewpoints is not particularly Interesting from the standpoint of economic analysis. Nonetheless, all too frequently, an Isolated road Is paved, a remote stretch of water or sewer line is ex­ tended under exceptional circumstances, or a seemingly Ideal loca­ tion fo r a public works fa c ility Is suddenly elim inated from consideration. In these cases, the 'benefit-cost analysis,' its review,

Economic Analysis of Protects in the Public Sector

299

and the Implementation decision are usually made from the perspec­ tive o f a small, select group. The other four viewpoints are, however, of considerable interest to those Involved In public works evaluation. Analyzing projects or proj­ ect components from viewpoint 2, that of a particular government agency, Is analogous to making economic comparisons In private en­ terprise. That is, only the gains and losses to the organization Involved are considered. This viewpoint, which seems contrary to benefit-cost optimization to the public as a whole, may be appropriate under cer­ tain circumstances.

Example 6.6 Consider a Department of Public Works construction project for which the water table must be lowered in the immediate area be­ fore w ork can proceed. Any of several water cutoff or dewatering systems may be employed. Water cutoff techniques Include driv­ ing a sheet pile diaphragm or using a bentonite slurry trench to cut off the flow o f water to the construction area. Dewatering methods Include deep well turbines, or wellpoints for lowering the water level. It Is sometimes appropriate to evaluate these different tech­ niques from an 'organization* point of view, since each of the fea­ sible methods provides the same service or outcom e-a dry construction site. Therefore, the most economical decision from the department's standpoint Is also correct from the viewpoint of the public as a whole, since the benefits or contributions to the project are the same regardless of the method chosen.

The third point of view, that of a city or county, Is popular among lo­ cal government employees and elected officials. Unfortunately, seem­ ingly localized projects often have an Impact on a much wider range of the citzenry than those In the Immediate vicinity.

Example 6.7 County officials are to decide whether future refuse service should be county owned and operated or whether a private contractor should be employed. The Job requires front-end loader compac­ tion trucks, as well as roll on-off container capability. These trucks would be travelling primarily along rural roads, and from 1 con­ tainer (e.g., a roadside picnic area) to 50 containers (e.g., a large rurally located Industrial plant) must be collected at each stop. Front-end loader containers range from 2 to 8m 3 while roll on-off containers are sized from 15 to 45m 3 . Several trucks and drivers

300

will be required, along with a base fo r operations and mainte­ nance. The cost In dollars per tonne of refuse collected, removed, and disposed of. Is given below: Personnel services

$ 6.08 per tonne

Materials, supplies, utilities

5.13

Maintenance and repair

5.62

Overhead

3.41

Depreciation

3.61

Five percent Interest on the half financed by bonds

1.80

Tbtal county cost

$25.65 per tonne

Federal taxes foregone

$ 1.87 per tonne

Provincial taxes foregone

0.62

Property taxes foregone

1.54_________

Not necessarily paid by county

$ 4.03 per tonne

County cost to provide refuse service w ill be $ 2 5 .6 5 /t However, the county Is not required to pay the additional $4.03/t for federal and provincial taxes and local property taxes, as would a private Ann. 1()01) )] 2

+2[E(X,X 2 ) - E ( X 1)E(X i )] +

2 [E (X ,X 3 ) - E ( X ,) E ( X 3 ) ]

+2[E(X j X 3 )-E (X 2 )E(X3 )J Since e( x , 2 ) - [ E ( x , ) ] 2 defines the variance of the random variable X, and E(X„ X , ) - E (X , ) E ( X ,) defines the covariance of the random variables X p and X *, we see that

358

Chapter 7

Var(V) = V a r(X j) + V ar(X 2 ) + V ar(X 3 )+2C ov(X 1X 2 ) +2Cov ( X 1X 3 )+2C ov (X 2 X 3 )

(77)

where Cov(X p X k ) denotes the covariance of X p and X k . Generalizing to the sum o f (n + 1) random variables, If

y = fx

;

J=o

(7.8)

then the expected value and variance o f Y are given by

E(V) = f

e( x ; )

J=o

Var(Y) = i ; v a r ( x J ) + 2 S J=O

£

C o v ( x ,X ,)

j=o k=j+l

(7.9)

Additionally, If o, and a2 are constants and X , , X 2 , and Y are random variables related by

y = a1X 1 + d 2 X 2

(7.10)

Var(Yj = o12V a r(X 1 )+ a 2 2 V ar(X 2 )+ 2 a 1o2C ov(X 1X 2 )

(711)

recall that

Combining the relationships given In Equations 7.9 and 7.11, the variance of present worth Is found to be

Break-Even, Sensitivity, and Risk Analyses

359

Var(PW ) = X V a r ( c y )(1 + i ) ' 2 } n -1

+2X

J=O

n

Z c o v ^ C . X z + l) - 1'-*’ k=J+1

(7.12)

Hillier2 argues that It Is probably unrealistic to expect Investment analysts to develop accurate estimates for covariances. Hence, the net cash flow In any year should be divided into those components of cash flow that are reasonably independent from year to year and those that are correlated over time. Specifically, it Is assumed that (7.13)

where the X. values are mutually independent over j but, for a given value of h, Yo h , y, „ ,..., Yn h are perfectly correlated. When two random variables X and Y are perfectly correlated, one can be expressed as a linear function of the other. Hence, If Y = a+bX

(7.14)

where a and b are constants, then the covariance o f Xand Vis given by C o v (X y ) = E(XY) - E(X)E(Y)

(7.15)

However, from Equation 7.14, we note that E(Y) = a + bE(X) and

2 Hillier, F. S., *The Derivation of Probabilistic Information for the Evaluation o f Risky Investments,' Management Science, 9 (3), April 1963.

(7.16)

360

Chapter?

V ar(V ) = b 2 V a r(X )

(7.17)

Substituting Equations 7.14 and 7.16 Into Equation 7.15 gives Cov(XY) = E (aX + b X 2 ) - E (X )[a + b f ( X ) ] = aE(X) + b E (X 2 ) - aE(X) - b [E (X )] 2 = b V a r(X )

(7.18)

Recalling that the standard deviation o f a random variable is defined as the square root of the variance of the random variable, we see that the Equation 7.18 can be expressed as C ov(X Y ) = bSD (X)SD (X)

(7.19)

where SD(X) denotes the standard deviation o f X. However, from Equa­ tion 7.17 we see that SD(y) equals bSD(X). Hence, COV(XY) = SD(X)SD(Y)

(7.20)

where X and Y are perfectly correlated. The model suggested by Hillier yields the follow ing expressions for the expected present worth and variance o f present worth:

(7.21) and

Bt»ak-Ev>n, Sensitivity, and Risk Analyses

361

Var(PW) = £ V a r(x ,)(1 + /)’ 2 ' + £ j=o

£

sd( y, )(1 + 1)-'

n=i L y=o

(7.22) where SD(Y) h ) denotes the standard deviation of the random variable A very Important theorem from probability theory, the central limit theorem, Is usually Invoked at this point. The central limit theorem es­ tablishes under very general conditions that the sum of Independently distributed, random variables tends to be distributed normally as the number of terms in the summation approaches infinity. Hence, it is ar­ gued that present worth, as defined by Equation 7.1, Is normally distrib­ uted with mean and variance, as given by Equations 7.21 and 7.22. Of course, this assumes that the Cy are statistically independent How­ ever, the view Is usually taken that the normal distribution Is a reason­ able approximation to the distribution of present worth. Given that present worth is assumed to be normally distributed, one can compute for each Investment alternative the probability of achiev­ ing a given desired level. For example, one is usually Interested in knowing the probability that present worth Is less than zero. Some analysts Interpret this to be a measure o f the risk associated with an In­ vestment alternative. In performing an analysis of the risk associated with an investment alternative, we have treated the simplest situation; only the cash flows were considered to be random variables and the measure of effective­ ness employed was present worth, not rate of return. When either the discount rate or the planning period are treated as random variables and when the probability distribution for rate of return is desired, simu­ lation approaches are usually employed.

Example 7.13 As an Illustration of the analytic approach In developing the prob­ ability distribution for PW, consider the following. A flight school operator Is considering the alternatives of pur­ chasing a utility category training aircraft versus purchasing an ac­ robatic version of the same aircraft. Having been In the flight

362

Chapter 7

training business for a number of years, the operator has reason to believe that Income and expense from a utility category aircraft will be nearly Independent from year to year. The pertinent data are summarized In Thble 7.7. Table 7.7 Estimated Net Cash Rows for a Utility Model Aircraft Year

Expected Value

Source

0

Purchase

1

Range

Standard Deviation

—$11 000

$9 5 0 0 -1 2 500

$500

Income-expense

2 200

2 0 5 0 - 2 350

50

2

Income-expense

2 200

1 9 0 0 -2 5 0 0

100

3

Income-expense

2 200

1 9 0 0 - 2 500

100

4

Income-expense

2 000

1 7 0 0 - 2 300

100

5

Income-expense

1 000

7 0 0 -1 300

100

5

Salvage

6000

4 8 0 0 -7 2 0 0

400

Investment in the acrobatic aircraft is a more risky but promis­ ing Investment The acrobatic aircraft Is also expected to have a life of five years. The operator feels that maintenance costs will be nearly independent year to year with this aircraft. However, since the flight school has never offered an acrobatic course before, there is some uncertainty regarding the demand for time In such an aircraft It Is felt that the net cash flow from the sale of flight time for each o f the five years will be perfectly correlated. The pertinent data are summarized in Tbble 7.8. An Interest rate o f 10% is to be used in the analysis. From Equations 7.21 and 7.22, we obtain the following: E(PW,) = $190 VarfPW,) = 334 741

E(PW2 ) = $355 Var(PW 2 ) = 11631930

Assuming normally distributed PW, the probability of an equiva­ lent present worth less than zero Is found to be Pr(pW, < 0 |/= 1 0 % )= 0 3 7 for the utility model aircraft and Pr(PW2 < 0 |/= 1 0 % )= 0 .4 6

Break-Even, Sensitivity, and RW ( Analyses

383

for the acrobatic model aircraft Here we are faced with a choice between the alternative having the greatest expected value and the alternative having the smallest variance. The choice will de­ pend on the owner's attitudes toward risk. If the decision were yours to make, which model aircraft would you choose? Why? Table 7.8 Estimated Net Cash Flows for an Acrobatic Model Aircraft Year

Source

Expected Value

Range

Standard Deviation

0

Purchase

-$ 1 4 0 0 0

$13100-149 00

$ 300

1

Expense

- 10000

9 1 0 0 -1 0 900

300

2

Expense

- 10000

9 1 0 0 -1 0 9 0 0

300

3

Expense

- 11 000

10100-11 900

300

4

Expense

- 12 000

1 0 8 0 0 -1 3 200

400

5

Expense

- 12 000

10 8 0 0 -1 3 200

400

1

Income

12 500

9 5 0 0 -1 5 500

1000

2

Income

13 500

10 5 0 0 -1 6 500

1000

3

Income

13 500

10 8 0 0 -1 6 2 0 0

900

4

Income

14 500

1 2 1 0 0 -1 6 9 0 0

800

5

Income

13 500

11 4 0 0 -1 5 600

700

5

Salvage

7 500

6000 -

900 0

500

7.6 Computer Simulation Even th o u g h a n a ly tic approaches can be used in some situations to perform the risk agg re gation step In risk analysis, sim ulation ap­ proaches are ty p ic a lly required to cope w ith th e com plexities o f a realworld situation. S im ulation, in th e general sense, m ay be th o u g h t o f as perform ing expe rim e nts o n a m odel. Basically, sim ulation is an 'if..., then.. .* device (l.e., If a certain Input Is specified, then the output can be determined). Some o f th e m a jo r reasons fo r using sim ulation In risk analysis are: 1. A nalytic solutions are im possible to obtain w ith o u t great difficulty.

364

Chapter 7

2. Sim ulation is useful in selling a system m o d ific a tio n to m anagem ent. 3. Sim ulation can be used as a ve rifica tio n o f analytical solutions. 4. Sim ulation is ve ry versatile. 5. Less background in m ath e m a tica l analysis and proba bility th e o ry is generally required. Some o f th e m a jo r disadvantages o f sim u la tio n are: 1. Sim ulations can be quite expensive. 2. Sim ulations introduce a source o f random ness n o t present In analytic solutions. 3. Sim ulations d o n o t reproduce th e in p u t d istrib u tio n s exactly (especially th e ta ils o f th e distribution). 4. V a lidation is easily o ve rlo o ke d in using sim ulation. 5. Sim ulation is so easily applied it is o fte n used w h e n analytic solutions can be easily o b ta in e d a t considerably less cost

Example 7.14 lb illustrate the simulation approach in risk analysis, consider the following situation. An individual is planning on purchasing a used computer for $1 000 and performing certain billing and account­ ing functions for several retail businesses In the neighbourhood. It Is anticipated that the business will last only four years because of the growth of competition. If income In the third year exceeds ex­ penses by more than $400, operations w ill continue the fourth year. However, If income is less than or equal to $400 more than expenses In the third year, the computer and software will proba­ bly be sold at the end of the third year. For simplicity, a probability of 0.70 is assigned to the possibility o f selling out at the end of the third year, given that income is not greater than $400 above ex­ penses In the third year. Let Ej = expense for year J l t = Income for year j n = life of the Investment Sn = salvage value based on an n-year life.

Rraak-Even, Sensitivity, and Risk Analyses

365

Assuming a zero discount rate for simplicity of calculations, the present worth of the Investment Is given as P W = - $ 1 0 0 0 + £ ( / y - eJ+S, J- 1

The probability distributions assumed to hold for this example are provided In "fable 7.9. In practice, a computer would be used to per­ form the simulation. However, to Illustrate the technique, we win manually perform 10 simulations of the investment A table of two-digit random numbers Is gtven In "fable 7.10. Using the work­ sheet given In "fable 7.11,10 simulations of the investment yielded an average present w orth o f $805 for the Investment Of course, 10 simulations is not an adequate number o f trials to draw strong conclusions concerning the Investment However, the example does Illustrate the simulation approach. To Illustrate the approach taken, the first random number se­ lected will provide the simulated value for expenses in the first year. A random number o f 90 was obtained from row 1, column 1 in "fable 7.10. Consulting Table 7.9, it is seen that a random number of 90 represents an expense of $400; hence, $400 is entered ap­ propriately on the worksheet given in "fable 7.11. The second ran­ dom number is selected from row 2, column 1 o f "fable 7.10 to generate the income for year 1. A random number of 78 ts ob­ tained and, from Table 7.9, a simulated Income o f $600 is obtained for the first year. Continuing through the third year, it Is found that income exceeds expenses by $500; hence the business will con­ tinue through the fourth year. A random number of 97 is drawn to generate the salvage value of $400 for the Investment Note that In the second trial, It was decided that the business should be discontinued after three years. Furthermore, income was never less than expenses In any years; this illustrates that in­ sufficient observations (trials) were obtained since, In year 2, ex­ pense can exceed income w ith probability Pr(E 2 = 600 and l 2 = 500 )= 0 .1 0 (0 .2 0 )= 0 0 2 and in year 4 expense can exceed Income with probability P f a = 7 0 0 and/ 4 = 6 0 0 )+ Pr(E4 =800 and /« = 600) = 0.125 As an a lte rn a tive to using th e discrete p roba bility distributions given In "fable 7.9, w e m ig h t e m p lo y continuous distributions such as th e nor­ mal, gam m a, o r beta distrib u tio n s to represent Income and expenses.

366

Chapter?

lhble 7.9 Data for the Risk Analysis Example Problem p fe )

RN

A

p (O

RN

$200

0.25

0 0 -2 4

$ 400

0.50

00-49

300

0.50

25-74

600

0.50

50-99

400

025

7 5 -9 9

p (f2 )

RN

>2

p (O

RN

$300

0.10

0 0 -0 9

$ 500

0.20

00-19

400

0.40

10-49

750

0.40

20-59

500

0.40

5 0 -8 9

1000

0.40

60-99

600

0.10

9 0 -9 9

p(e3 )

RN

/3

X /3)

RN

0.20

00-19

$ 800

0.30

00-29

500

0.30

2 0 -4 9

1000

0.50

30-79

600

0.30

5 0-79

1200

0.20

80-99

700

0.20

8 0 -9 9

p t e .)

RN

A

pO J

RN

$500

025

0 0 -2 4

$ 600

0.25

00-24

600

0.25

2 5 -4 9

800

0.25

25-49

700

0.25

50-74

1000

0.25

50-74

800

025

7 5-99

1200

0.25

75-99

f

3

$400

N

p0V|/3 - E 3 £$4OO)

RN

3

0.70

0 0 -6 9

4

0.30

7 0-99

Break-Even, Sensitivity, and Rtok Analyses

387

Ibble 7.10 TVvo-DIgIt Random Numbers 90

43

78

83

82

99

54

58

98

02

78

31

68

09

87

51

81

42

80

35

21

42

03

97

15

62

93

95

07

56

60

39

27

37

12

63

31

35

66

93

79

39

44

22

83

96

51

00

89

61

73

29

43

84

91

34

29

38

30

84

90

18

00

10

97

64

33

29

17

48

26

04

07

64

15

02

44

32

92

99

82

13

50

83

35

39

50

51

59

83

21

30

86

90

16

09

04

46

19

63

60

53

33

97

96

54

91

43

44

40

09

02

31

27

71

18

03

65

53

62

03

45

70

42

22

16

67

13

08

35

45

92

79

97

46

02

37

60

80

55

05

35

75

57

90

43

63

17

56

21

69

09

22

07

69

85

38

74

02

58

05

33

79

00

69

29

67

08

48

97

91

14

53

00

03

42

94

68

64

58

97

32

27

80

15

39

85

87

82

38

52

16

09

37

81

73

37

01

66

84

JvO

Chapfef7

■fable 7.11 Simulation of the Sample Investment Problem 90 51 27 89

$400 500 500 800

RAK/) 78 93 79 29

/ $ 600 1000 1000 800

1 2 3 4

07 59 96 *“

200 500 700

82 04 31

600 500 1000 -

400 0 300

1 2 3 4

37 22 48 15

300 400 500 500

90 05 94 09

600 500 1200 600

300 100 700 100

—

1 2 3 4

31 97 39 38

300 600 500 600

81 37 61 64

600 750 1000 1000

300 150 500 400

—

1 2 3 4

13 46 27 35

200 400 500 600

83 54 45 60

600 750 1000 1000

400 350 500 400

—

1 2 3 4

07 97 39 78

200 600 500 800

33 68 37 58

400 1000 1000 1000

200 400 500 200

—

1 2 3 4

15 44 30 15

200 400 500 500

12 73 33 50

400 1000 1000 1000

200 600 500 500

—

1 2 3 4

19 71 45 63

200 500 500 700

91 70 80 69

600 1000 1200 1000

400 500 700 300

—

1 2 3 4

91 85 83 35

400 500 700 600

64 81 98 95

600 1000 1200 1200

200 500 500 600

—

1 2 3 4

22 84 02 30

200 500 400 600

29 29 83 63

400 750 1200 1000

200 250 800 400

—

TOal Meat 1 1 2 3 4 2

3

4

5

6

7

8

9

10

RN B ) = P (A ) P (B )

(8.7)

An extension o f Equation 8.6 for three dependent events Is given by

418

Chapter 8

P ( A n B n C ) = P (A ) P(B\A) P(C\A,B)

(8 .8)

which was the relationship used in the previous um example. For the purpose o f the development to follow, Equation 8.6 Is now written In Its complete form as P(A r^B ) = P (A ) P(BlA) = P(A)B) P(B)

(8.9)

from which It follows that

(8.10)

or

(8.11)

Example 8.7 Either Equation 8.10 or 8.11 Is a form of Bayes's theorem, which may provide a logical guide to sequential decision making under risk. To illustrate this claim, let us consider another um example where one um, Uv contains tw o whtte and eight black balls; the other u m ,l/ 2 , contains five white and five black balls. Suppose the um contents are known to you, but the urns themselves are hid­ den from your view. Now, another person selects a single ball from one o f the ums. W ithout know ing the colour of the bail drawn, which um would you guess was selected? Then, If the person re­ ported a black ball had been drawn, which um would you guess had been selected? W ithout benefit o f the sample Information concerning the col­ our o f the ball drawn, most persons would feel that it was equally lik e ly th a t Uy o r U2 h a d been c h o s e n . T h a t is, since p (l/ 1 ) = p(u 2 ) = % guess either um. However, after the sample in­ formation of a black ball Is received, intuition suggests that U, was the um selected because the probability o f a black ball In U} Is greater than the probability of a black ball In U2 . We now wish to formalize this example and show that Bayes's theorem can be used to support the intuitive guess of U, after receiving the report Before receiving the report o f a black ball, the prior belief that I/, or U2 was selected is subjectively stated as

DecWon Modeto

419

P(U^) = the prior probability of selecting U, =y2 and P(U 2 ) = the prior probability of selecting U2 -y

2

Note that W = the event of a white ball being drawn and B = the event of a black ball being drawn Also, before receiving the sample information, the likelihood of a particular report (W or B), given ttiatU, orU 2 was selected, can be determined, based on knowledge of the contents o f each urn. That Is, f ( k v m ) = the conditional probability of a white ball being drawn given that U, was selected for the draw 2_ 10 Similarly,

In the literature on sequential decision making, these conditional probabilities, either objectively known as above or subjectively as­ signed, are termed likelihood statements. Ultimately, It is desired to assess the probability of whether U, or U2 was chosen based on the sample Information received (the re-

420

Chapters

port of a blade ball In this case). Thus, the posterior probabilities of p( u2\b\ P(U,\w), and P(t?2 1w) need to be determined.

In the calculation o f these posterior probabilities, It Is conven­ ient to calculate the probability o f event W and event B before the ball Is drawn. It is reasoned that event W could occur If either U1 or U2 were chosen. The same Is true for the event B. Thus, B could oc­ cur if (1) Uy were selected and B occurred, or (2) U2 were selected and B occurred. That is, AB) = P(Uy and B or U 2 and B)

= p(u, c\b\ju2 c\b) = P(Uy ob)+ p(u2 db) Then, using Equation 8.9, A B )= P(Uy )P(B[U ,)+P(U 2 )p(B\U2 ) and, from the previous data,

Similarly, A W ) = p(Uy )p (w \U ,)+ P (U 2 )P(W\U 2 )

With these results, the posterior probabilities may be calculated from Bayes's theorem as

a

W lW )

P(w\Uy)p(Uy) W IV )

-

(2 /IO X V 2 ) 7 /2 0

=

2 7

Decision Models

421

,,.A 1 7

K w | u 2 )p(u 2 ) P(W)

(5/10X1/2) 5 7/20 "7

8 It Is noted from the results of P(u, | B) = — and p(u 2 1b ) =

that

the event of B should therefore increase the belief that Ut was se1 lected. That Is, the prior probability of I f was - and the posterior g probability, given the event B, of U, is — if, after receiving the in­ formation of B, the guess has been Uu then the above calculations would support the Intultltlve guess. If a second sample of a ball is taken from an um, then the posterior probabilities p ( u ,|B ) and

p(U2IB) become the prior probabilities second trial.

and p(u 2 ) for the

8.63 The Value of Perfect Information In order to use the logic of Bayes's theorem in a more practical way to aid decision making, it is of Interest to evaluate the value of the sample information In a monetary sense. Recalling the previous example, sup­ pose the person who drew the ball from the um had offered to pay a $10 prize If the correct um were guessed. However, the person will charge for the sample Information of whether a white or black ball is drawn. The question then Is how much should be paid for the sample information. Before answering this particular question, a rationale for determining the value o f perfect information is considered (I.e., the knowledge o f exactly which um was selected for the draw). The prior probabilities of P (L f) = P(U2 ) = 1 are now recalled and written wtth the more suggestive notation of PP(U3) = PR(U2 ) = | . Without any sample Information, one should be indifferent as to which um Is guessed, and the prior expectedprofit {PEP} would be $5, as determined from the matrix o f Table 8.8.

422

Chaptwg

■fable 8.8 Matrix for the Um Example

Actual Um Selected

if

u2

if

$10

$ o

u2

$ o

$10

Um Guessed

The prior expected profit is $5, as determined from either of the two calculations below. E( profit] guess U ,) = f l ^ $ 1 0 ) + ( 1 j($ 0 ) = $5

E(profit|guessl/ ; ) = ^ 1 ^ $ 0 ) + ^ 1 ^ $ 1 0 ) = $ 5 If these expected profits were different, the alternative with the larger expected value would be chosen, and the prior expected profit is the larger value. Now, given that the person drawing the ball states that U} was se­ lected and that the person is perfectly reliable, L f would obviously be guessed and a $10 payoff received. The same Is true, of course, for the report of U2 . The expected profit given perfect irrformatlon (EP\P1) Is thus $10 In this example. It is now reasoned that the maximum amount one should pay for perfect Information is $10 - $5 (expected profit without any information)=$5. We can formalize the expected value of perfect in­ formation (EVP!) as the difference between the expected profit given per­ fect Information (EP\PI) and the prior expected profit (PEP), or EVPI = EP\P1 - PEP

(812)

8 K 4 The Value of Imperfect Information In most real-world decision situations, Information Is Incomplete and not perfectly reliable, as in the urn example Just discussed. The report that a black ball had been drawn gave additional Information to the decision maker, but still did not provide the absolute answer as to

Decision Models

423

which urn was selected, in order to define the expected value of sample Information (EVSI), the posterior probability (with new notation) results are repeated below. FOr the event B, the posterior probabilities were

po(u,|b) . A and PO(U2 | B ) = A 8 5 Thus, since — > — , the best guess is that I/, was selected. The actual um chosen could have been guess of U-t is

or U2 , and the expected profit given a

E(profit]guess U y ) = (expected profrt|guess U, and Is U ,)P O (y,) + (expected profit | guess

=(*o)(£>W(^ =

and is U2 )PO(y 2 )

$80 13

For the event W, the posterior probabilities were P 0(U ,|W ) = |

P O K IN ') = y Thus, If the report had been W, and since y > y , the best guess would be U2 with

424

F( profit | guess U2 ) = (expected profit |guess U2 and Is U y )PO(Uy ) + ( expected profit | guess U2 and Is U2 )PO(u2 )

=

$50 7

However, before the sample Is taken and the report Is given, both events B and IV are possible with P(B)=

and P(IV)=

The expected

profit given sample Information (EP\SI) Is EP\SI = (expected profit |B )P (B )+ ( expected profit | IV)P( IV) $80" - - 0 , 5 , the best guess given B would be U r 13 b b i

The appropriate calculations at D3 are r / E(guesst/ + (-$ 1 .5 0 ) = $6.5

_____ . . .

7

Since -j 3? '5

>

the best guess given IV would be U2 .

Replacing D2 and O3 with these best expected values yields the re duced decision tree o f Figure 8.11. From Figure 8.11, the appropriate calculations a tD 1 are E(buy sample inform ation) = $60.5 JY ll 3l ) T = $5

($39.5 7

DecWon Models

427

Fig. 8.11 Reduced decision tree for um example.

E(no sample information and guess U2 ) = E(no sample Information and guess U } ) = $5 Thus, the three alternatives atD , are equally desirable given that the charge for sample Information Is $1.50. If the charge Is less than this amount, the decision m aker should choose the 'buy sample Informa­ tion' alternative In order to maximize expected profits. For a similar sequential decision problem Involving only costs, the basic relationships to determine expected value of Information be­ come EVP/ = prior expected cost - expected cost given perfect Information = PEC-EC \PI

(8.14)

428

EVSI = p rio r expected cost -e x p e c te d cost given sam ple Info rm ation = PEC -E Q S /

(

Example 8.85 The editor of the ABC Book Company Is considering a manuscript for an historical novel. The editor estimates the book can be m ar keted for $5, with the company receiving $4 and the author $1. Since other publishers are Interested in the manuscript, the author wants a commitment from the ABC Book Company In the very near future. ABC has already invested $300 In a preliminary re­ view o f the manuscript and a quick market survey. The editor con­ siders this manuscript competitive w ith other recently published historical novels and the author, although relatively new, has other published works that have received good reviews. The editor Identifies the immediate alternatives: (1) reject the manuscript, (2) accept the manuscript, or (3) obtain an additional review o f the manuscript by an expert. An additional review will cost $400, and the expert's opinion will not perfectly predict the historical novel's success in the m arket The editor reasons that at best the expert can only rate the manuscript good, fair, or poor. If the manuscript Is accepted fo r publication, the editor specu­ lates on the possible market outcomes; there could be (1) low mar­ ket demand, or (2) the novel could be a best-seller. Additionally, there Is the question o f movie rights. There may be no Interest In film ing the novel, but, on the other hand, a best-seller could yield $30 000 for the movie rights. Even If there is low market demand, a film company might pay $4 000 for the rights to make a lowbudget, grade B film. The ABC Book Company has available historical data on 60 re­ cent manuscripts of similar books, given in Table 8.9.

SThls example was presented at the Sixth Triennial Symposium, Engineering Economy Division, ASEE, June 19-20,1971, in a paper entitled 'Introduction to Decision Theory" by Barnard E. Smith. The article appears In the publication, Decision and Risk Analysis: Powerful New Toolsfo r Management, The Engineering Economist, Stevens Institute of Technology, Hoboken, NJ, and Is presented here by the permission o f the publisher.

Decigton Models___________________________ ____________________________________ 429

"fable 8.9 Sbdy Manuscript Dedsions/Outcomes Outside Reviewer's Evaluation Dedslon/Outcome When Marketed

No Outside Good Expert Review

Fair

Poor

Total

Not published

5

0

14

12

31

Low market demand

2

0

4

3

9

Best-seller

1

2

4

0

7

Best-seller + movie

1

4

3

0

8

Low market + B movie

1

0

3

1

5

10

6

28

16

60

Total

From available sales figures, the editor can estimate reasonably well the revenues given a particular outcome for a published book. The editor considers the present worth of after-tax revenues over the effective life of a book using a 10% expected rate of re­ turn. For a best-seller, a present worth of $70 000 is estimated, and a low-demand publication yields an estimated $5000 present worth. Offers for movie rights, If any, are usually made one year af­ ter the manuscript Is under contract, and these after-tax revenues should be added to the book sales (assume that $4 0 00for a grade B film and $30 000 for a 'best-seller film* are appropriate figures.) It costs $15 000 to publish a book. If it Is assumed the editor has a linear utility function for dollars and uses a 'm axim ize present worth’ principle for choice among alternatives, what decision should be made at the present time? The editor faces these Immediate alternatives: A,—reject the manuscript without any additional review by an expert A2-accept the manuscript without obtaining an expert's review A3 -o b ta in an expert's review The value of alternative A, Is the $300 cost, which has already been spent on a preliminary review (l.e., v (A 1)= $ 3 0 0 ]. Indeed, the $300 for a preliminary review Is a sunk cost and applies whether alternative Ap A2 ,o r A3 Is chosen at the present time. This $300 cost can therefore be dropped from consideration and v( A,) = 0 with this adjustment

430

Chapters

For alternative A2 there could be four possible market out­ comes, defined as e ,-to w market demand 0 2 - th e best-seller

9 3 -best-seller plus movie rights e4 - lo w market demand plus grade B movie Thus, the value associated w ith each o f these outcomes Is deter­ mined as v(e, | A2 ) = revenues - publication costs = $5 0 0 0 -$ 1 5 0 0 0 = -$ 1 0 000 V(0 2 | A2 ) = $ 7 0 0 0 0 -$ 1 5 000 = $55 000 v (e 3 1A2 ) = $70 000+$30 0 0 0 - $15 000 = $85 000 V(0 4 |A 2 ) = $5 000+$4 0 0 0 -$ 1 5 000 = -$ 6 0 0 0 For alternative a 3, the expert's evaluations could be either Z ,—Good Z 2 -F alr

Z j-P o o r For each o f these evaluations, the editor must make one of two possible decisions, defined as X j-re je ct the manuscript X 2 -accept the manuscript Given the decision o f X, for each o f the three possible evaluations ( z ; ), the cost to the ABC Book Company would be the same - the additional cost of an expert's review. That is, v ( x , | A3 ,Z j) = -$400,

for j = 1, 2, 3

Then, for the decision o f X 2 associated w ith each of the reviewer's evaluations, there are the four possible market outcomes, which were defined previously. The value for any given market outcome and decision X 2 Is the same for all Z t evaluations. This value Is equal to the values for alternative A2 minus $400 for the expert's review. That Is,

Dectoton Models

431

V 0 )+ ( $ 5 4 .6 ) ^ + ( $ 8 4 £ ) ^ j+ ( - $ 6 4 X 0 ) = $74.6 In thousands = $74 600 and the best decision at D2 Is therefore X 2 (accept the manuscript). The calculations fo r D 3 are E (X ,) = -$ 4 0 0 £ (X 1 ) = ( - $ 1 0 4 ) ^ p $ 5 4 . 6 ) ^ j + ( $ 8 4 4 ) ^ ] + ( - $ 6 4 ^ ^ = $ 2 9 3 8 6 In thousands = $29 386 and the best decision at O3 ls therefore X 2 (accept the manuscript). The calculations fo r D4 are E (X ,) = -$ 4 0 0 E (X 2 ) = (- $ 1 0.4)( | ]+ ($ 5 4 .6 > 0 )+ ($84.6X 0)+(-$6.4 = -$ 9 .4 In th o u sa n d s= -$ 9 400 and the best decision at

Is therefore X, (reject the manuscript).

The reduced decision tree Is given by Figure 8.13, and the appro­ priate calculations fo r D| are

434

Fig. 8. 13 Reduced decision free for ABC Book Com pany example (value In thousands of dollars). E(A 3 ) = ($ 7 4 6 0 0 ^ + ( $ 2 9 3 8 6 ) | ^ + ( - $ 4 0 0 ) | j | ) = $25 280 E(A 2 ) = (-$ 1 0 0 0 0 ) ( |} + ( $ 5 5 0 0 0 ) ^ + ( $ 8 5 0 0 0 ) ^ + (-$ 6 0 0 0 ^ j = $22 800 £ (A )= $ 0

and the best decision at D, In order to maximize the expected pres­ ent worth Is alternative A3 (obtain an expert's opinion). Then, If the evaluation Is either good or fair, accept the manuscript for publica­ tion; but if the evaluation Is poor, reject the manuscript The ex­ pected value of sample Information In this example is the difference between e( a3 ) and e( a2 ) plus the $400 paid to the ex­ pert, or $2 880.

EVSI = EV\SI-PEP+SAOO = $25 2 8 0 - $ 2 2 8 0 0 + $ 4 0 0 = $2 8 80

DecWon Models_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 435

dearly the expert's opinion Is well worth the $400 fee In this ex­ ample.

& 65 Sequential Dedslons-Summary Comments The decision-tree representation of a sequential-decision situation is an effective device fo r visualization. By developing the tree, serious at­ tention Is given to explicitly Identifying and describing alternatives explicitly, to assessing the outcomes that might occur as a result of the alternative choice, and to making any necessary future decisions. Even if the probabilities for outcomes cannot be adequately assessed or values for the 'branches' accurately estimated, considerable Insight Into the decision situation may be achieved and decision making aided. On the other hand, whether the expectation principle of choice among alternatives Is an appropriate criterion or not is a matter of per­ sonal preference, and any real-world sequential-dedsion situation may readily become too large for effective description or convenient solution. As the number of alternatives, the number of outcomes per alternative, and the number of decision points increase, so do the un­ certainty of data and the burden o f calculation. Furthermore, the ex­ pense In time and money to obtain the Information required may become prohibitive when compared with the benefits expected from the analysis. An evaluation o f such a trade-off is largely, If not primar­ ily, a matter o f Judgement

8.7 Multiple Objectives When managers are unable to describe goals In terms of one objec­ tive, they often seek multiple objectives, and desire proposals to achieve these in some combination. Establishing the relevant objec­ tives is an important problem for a given decision maker, and the problem Is further complicated by conflicting objectives and the diffi­ culty (if not Impossibility) of obtaining a measurement of these objec­ tives. The literature on the subject of multiple objectives and the theory of measurement is very extensive, encompassing the general areas of psychology, statistics, mathematics, and management. Most of the Is­ sues discussed In the literature are beyond the scope of this book. The discussion that follows here Is necessarily very limited. For the subse­ quent discussion, It will be assumed that objectives can be precisely stated and are relevant to the decision.

436

Chapters

8.7.1 Classifying Objectives According to Importance Assessing the relative Importance of multiple objectives, measuring the outcomes of alternatives In terms o f these objectives, obtaining a common measure for multiple outcomes having different measures, and combining all these Into a single measure o f merit form the heart of the multiple-objective problem and the source of much difficulty and controversy. We cannot offer a solution to the problem, but can only discuss approaches to it, citing cautions at appropriate points. When objectives can be sorted into classes such that each objective In the class is equivalent, then this Is nothing more than a classification system or nominal system of measurement. However, when objectives can be sorted into classes that can then be ordered, this Is termed an ordinal system o f measurement. To illustrate a nominal classification of objectives, consider a case where objectives are first categorized on a 'm u st' or 'desirable' basis. For example, a university library must serve the faculty and student population, but perhaps It Is only desirable to serve residents of the lo­ cal community. Once objectives are classed on a 'm ust' versus 'desir­ able' basis, the desirable objectives m ay be considered only subjectively thereafter, whereas the must objectives are deserving of greater attention and, if possible, quantification. Some of the ap­ proaches taken to quantifying objectives will now be treated; in the discussion, the term goal Is used synonymously with objective, and the symbolism adopted to denote the kth goal or objective of interest Is C*.

8.7.2 Ranking If a decision maker is capable o f stating preferences among objec­ tives, objectives may be ranked in terms o f relative importance. A technique to assist the decision m aker In m aking consistent prefer­ ence statements is the method o f paired comparisons. In order to Il­ lustrate the method, assume that fo u r goals (objectives) are relevant: G v G2 , G ,, and G 4 . The m ethod o f paired comparisons enu­ merates all possible pairs

2

, makes preference state­

ments, and deduces a ranking fo r each goal. In this example, the possible pairs are:

Decision Models

G, versus C2

G, versus C3

437

C2 versus G3

C2 versus G4

c 3 versus C4

G, versus G4 If the symbol > represents 'preferred to ' and the symbol < repre­ sents 'not preferred to,' suppose the results of the decision about the pairs were: G ,>G 2

g,< g3

G, >G 4

C2 < C 3

g2 > g4

G3 >G 4

Rewriting so that all comparisons are of the 'preferred to ' variety yields: C3 >G,

g3 > g2

C

1

>

C

2

C

1

> C

4

g2 > g4

g3 > g4

It can then be concluded that C3 Is preferred to all others, G, is pre­ ferred to two others, G2 is preferred to one other, and G4 Is preferred to zero others, and the following ranks should be assigned: G3 = i, G, = II, G2 =111, andG 4 = IV. If, In the above procedure, the preference statements had been G} > G2 , G2 > G 4 , and G, < G4 , the decision maker should have re­ considered the Inconsistent judgements. These results are 'intransi­ tive' In a mathematical sense. The ranking method o f classifying objectives according to relative Importance Is certainly useful; indeed, ranking was Implicitly used in the comparison of alternative Investments under conditions of as­ sumed certainty In the earlier chapters of this textbook. There are, however, obvious deficiencies In the ranking method. For Instance, the ranks do not indicate the extent to which one objective is preferred over another. Furthermore, because rankings are an ordinal scale of measurement, the mathematical operations of addition, subtraction, multiplication, division, and so forth, cannot be performed. Therefore, a higher-order scale of measurement (Interval or ratio scale) is gener­ ally required to determine choice among alternatives under condl-

*38____________ ____________________________________________ ________Chapter 8

tlons o f risk and uncertainty. Unfortunately, accepted methods for measuring objectives on a ratio scale do not currently exist.

8.73 Weighting Objectives Before proceeding with a discussion o f certain methods to assign rela­ tive weights to multiple objectives, a distinction between objectives and outcomes should be made. This is perhaps most easily done by means of an example where a machine-replacement situation Is of In­ terest, and the following tw o objectives are defined: G,: minimize the equivalent annual cost over a five-year planning period G2 : minimize the number of production personnel transferred to other jobs within the firm Let Q)l( - the outcome resulting from taking the /th alternative course o f action In terms o f the kth objective or goal )

=

value

the outcome resulting from selecting the

/th alternative In terms o f the kth objective Suppose, in this example, that - the alternative o f keeping the present machining centre ^2 = the alternative of replacing the present centre with a new machining centre The outcomes for each o f these alternatives in terms of the two ob­ jectives, G, and C2 are: - an annual cost of $80 000 0,2 = no persons transferred 02i - an annual cost of $60 000 022 = two persons transferred In order to compare the tw o alternatives In a quantitative fashion, val­ ues must be assigned to each outcome; the values must be weighted In some way to reflect the relative Importance o f the two objectives, and some type o f aggregate number must be determined for each al­ ternative to enable a comparison and final choice. For example, the end result (without regard for validity at this point) might appear as

Decbion Models

439

v '(^ )= * v Iv (e „)+ w 2v(e 12)

v(A 2 )= w ,v(e 2,)+w 2v(o 22) where V (^ 2 ) > V ( / l 1 ).

The w* (or w ,, w2 ) values are weighting factors identifying the rela­ tive Importance of the objectives Gk (or G v G2 ) and hence are applied to the appropriate outcomes (e A ) of the/th alternative. Determining the weighting factors Is thus one Issue on quantifying multiple objec­ tives. The values V (e ) may be In different dimensions and, If so, may have to be transformed to another common dimension. Determining the transformation function is another issue In quantifying multiple objectives. Finally, the values for each alternative, V (4 y ), were deter­ mined by a linear function. Determining the mathematical functional form of the model Is yet another Issue in quantifying multiple objec­ tives. Again, all these issues will not be resolved In this chapter, but some approaches that have been taken will be presented. One method for assigning relative weights to multiple objectives in order to indicate the extent by which one objective Is preferred to an­ other Is simply to assign weights by judgement. That is, in the case of four objectives (Gv G2 ,G 3 , G 4 ), the decision maker may reason thatG, Is twice as Important as G v three times as Important as G2 , and five times as Important as G4 . Assigning a weighting value of 1.00 to G3 , it can be reasoned that the set of weighting values is ForG 3 :

w 3 =1.000

ForGp

w, =0.500

For G2 :

w 2 =0.333

ForG 4 :

w 4 = 0.200

These weights may be transformed to a scale from 0 to 1.0, or 'nor­ malized,* by dividing each weight by the sum of all weights; that Is, 2.033

= 0 .4 9 1 8

° - 5 92 = 0 .2459 2.033

0 .3 3 3 2 .0 3 3 0 .2 0 0

< =2 .0 3 3

= 0 .1 6 3 9

= 0 .0 9 8 4

A n o th e r m e th o d 6 o f assigning w e ig h ts to m u ltip le objectives by Judgem ent depends upo n th e v a lid ity o f th e fo llo w in g tw o assump­ tions. For m ultiple objectives G

Gm :

1. It m ust be possible fo r th e decision m a k e r to th in k about and Judge the value o f an y c o m b in a tio n o f objectives. That Is, It m ust be possible to consider n o t o n ly th e im portance or w e ig h t value fo r, say, G v b u t also th e w e ig h tin g value for sums o f objectives. 2. Values are assumed to be a d d itive. G iven th e individual w e ig h tin g values fo r, say, G, and G 2 , it is assum ed th a t the w e ig h tin g va lu e fo r b o th objectives is th e sum o f th e ir in d ivid u a l w e ig h tin g values. The general procedure fo r th e m e th o d n o w fo llo w s. 1. Rank th e objectives in o rd e r o f Im po rtance, w h e re G, indicates th e m ost im p o rta n t, G 2 th e n e xt m ost im portant, and so fo rth , and G m is th e least Im p o rta n t. 2. Assign th e w e ig h tin g v a lu e o f 1.00 to G , (i.e., w , = 1 .0 0 ) and w e ig h tin g values to th e o th e r objectives to reflect th e ir im portance relative to G r These tw o steps actually complete th e ju d g e m e n t process, and th e fo llo w in g steps serve to refine the in itia l ju dgem ents and aid consistency. 3. Com pare G, to th e lin e a r c o m b in a tio n o f all o th e r objectives. That is, com pare G, versus (G 2 + G 3 +■■■ + G m ). (a) If G, > ( c 2 +G 3 + ■ •+G m ), adjust (if necessary) the value of w, such that w, > ( w 2 + w 3 +- - + w m ). Attempt, In all adjustments

6 Adapted from C, West Churchman et al., Introduction to Operations Research (New York: John Wiley, 1957), with commentary from William T. Morris, TheAnalysisq Management Deaslons, rev. ed. (Toronto: Irwin, 1964).

Ded»ion Mod«l»

441

of the procedural steps, to keep the relative values within the objective group Invariant Proceed to Step 4. (b) If G, = ( g 2 +G 3 4---+G m ), adjust so that w, - ( h' z + w 3 + -+w m ) and proceed to Step 4.

W If Gi < ( f i 2 +G 3 +---+G m ), adjust (tf necessary) the value of w, s o th a tw 1 < ( w 2 + w 3 + ••• + w m ).

Now compare G, versus ( g 2 +G 3 +•• +G m . 1 ), [I] If G, > ( g 2 +G 3 + - + G m _1 ), adjust (If necessary) the values so that w, > ( w 2 + w 3 + •+w m _,)and proceed to Step 4. Pl] If G, = (G 2 +G 3 + •••+Gm _1 ), adjust Of necessary) the values such thatw , = ( w 2 + w 3 +•• + *% _,)and proceed to Step 4. [Ill] IfG, < (G 2 +G 3 + - + G m_1 ), adjust (If necessary) the values such that w, < (w 2 + w 3 + •••+i r , . , ). Then,

compare G,versus ( g 2 +G 3 + •• +G m_2 } and so forth,

either until G, is preferred or equal to the rest, then proceed to Step 4, or until the comparison of G, versus ( g 2 +G 3 ) Is completed, then proceed to Step 4. 4. Com pare C 2 versus (G 3 + G 4 +••• + G m ) and proceed as In Step 3. 5. C ontinue u n til th e com parison o f G m _2 versus (G m _, + G r o ) Is com pleted. 6. If desired, co n ve rt each w* Into a norm alized value, d ividing b y th e sum ^ w K .

Exam ple 8 .9 Assume that It is desired to determine weighting values for four ob­ jectives where these have been ranked In order of Importance such that G, Is most im portant and G4 Is the least Important Step 1 of the procedure is thus accomplished. Step 2. The objectives are tentatively assigned the weights w, =1.00, w 2 =0.80, w 3 =0.50, and w 4 = 0.20. Step 3. Assume that is preferred to the linear combination of the other objectives. That Is, G, > (G2 +G 3 +G4 )■ Since

442

Chapter 8

w, < (w 2 + w 3 +w 4 ), it is necessary to adjust w, to be greater than the sum of 1.50, e.g., =1.75. Then proceed to Step 4. Step 4. Comparing C 2 versus ( g 3 +G 4 ), assume that G2 < ( g 3 +G4 ), and the Step 3 procedure is repeated. In this case, the weighting Values do not agree w ith the assumption [w 2 = 0.80 > (0 .5 0+ 0.2 0)], therefore, w 2 Is adjusted to be 0.65. [The adjustment to decrease w 2 does not violate the previous preference of G, > ( g 2 +G 3 +G4 ) with the new value of w, =1.75 to reflect this preference.) Since, In this example of only four objectives, the comparison of C m _2 versus (G m _, +G m ) or G2 versus ( g 3 +C 4 ) has been completed at this point, Step 5 o f the procedure has been done. If it Is desired to normalize the results, new weighting values are calculated as 1.75 "1.7 5 + 0 .6 5 + 0 .5 0 + 0 .2 0

= 0.5645

OSO "^ 3 .1 0

= 0.2097

' 0.50 ’ = 3.10

= 0.1613

0.20 3.10

= 0.0645

W

* *

=

Total

1.0000

For a small number o f objectives, this method is relatively easy to use, but becomes cumbersome as the number o f objectives In­ creases. However, it is believed that many real-world problems re­ duce to a few primary objectives. It should also be emphasized again that the method still depends on the decision maker's Judge­ ment in assigning the relative weights.

8.74 Determining the Value of Multiple Objectives In the previous section on weighting objectives, an example of a ma­ chine replacement situation was hypothesized to Introduce the discus­ sion. This example Is now recalled, where the objectives were: G,: minimize the equivalent annual cost over a five-year planning period

Decision Models

443

G2 : minimize the number of production personnel transferred to other Jobs wtthln the firm For the tw o alternatives of A } (keep present machining centre) and A^ (replace with new machining centre), the outcomes were 4,: $80 000 annual cost ( 0 n ); no persons transferred (e 1 2 ) Az: $60 000 annual cost (e 21 ); two persons transferred (e ^ ) Let It be assumed that, by some method, the objectives have been weighted asw 1 = 0 .6 0 and w2 =0.40. One method to assign values to the outcomes of the two alterna­ tives Is to make a second series of Judgements as to the degree to which each outcome succeeds in meeting the objective. Suppose these Judgements are made In the form of numbers on an arbitrary scale from zero to one where the higher scale value, the closer to meeting the desired objective. Assume the results of such Judgements are as given below. Outcome

Value 0.50

e>2

1.00

®21

0.85

§22

0.60

Then, assuming that a linear model Is appropriate, the values for each alternative are calculated as v ( A 1) = w y ( e

11 ) +

w2 v ( 9 1 2 )

= (0 .6 0 ) ( 0 .5 0 ) + (0 .4 0 )(1.00) =0.70 v ( A 2 ) = w y ( e 21 )+ w 2 v ( e 2 2 ) = ( 0 .6 0 )(0 .85) + ( 0 .4 0 )(0 .60) = 0.75 Thus, since V ( A j) > V ( A ,) alternative Aj would be chosen, and the present machining centre would be replaced. Clearly, these results are sensitive to the weighting values assigned by Judgement. Furthermore, the magnitude of the difference In values between V (A ,) and W A j) Is a function of the arbitrary scale (from zero to one) chosen. The linear model assumed Is also questionable.

All these points are criticisms o f this method of quantifying multiple objectives. In the literature on the theory o f utility functions, It Is ar­ gued that If a decision maker's utility function can be established for each objective, then values of outcomes (e.g., $80 000 annual cost) can be converted to a utility value (e.g., $80 000 annual cost equals 0.75) from the decision maker's utility curve. Such a utility value could be determined for each objective, and our model for the alternatives above would become

u (x ! ) =

w

,u (e ! 1 ) + w 2 M(e 2 2 )

Then, the decision maker should choose to maximize the weighted utility value. Whether such utility functions fo r each objective can be readily obtained for a single decision maker Is still questionable In our opinion, and even If this could be accomplished, there Is yet another question of whether Interaction between multiple objectives would be totally missed. As stated in Chapter 2, the study o f value measurement, and utility theory in particular, Is Interesting, but questions remain on the practical application and Implementation o f utility theory to busi­ ness and engineering problems. In any event, a treatment of utility theory is outside the scope of this textbook.

8.8 Summary In this chapter, some prescriptive or normative models for decisions under risk and uncertainty have been presented. For decisions under risk and uncertainty where the matrix model Is appropriate, criteria that are commonly cited In the literature fo r choosing among mutually exclusive alternatives were discussed. None o f these criteria were cited as the best or optimal basis fo r selecting an alternative; they merely represent observations on how decision makers seem to decide. The decision-tree model fo r a sequence of decisions where the out­ comes of alternatives are chance events was also presented, and the expectation-variance criterion was suggested as a guideline for choice among alternatives. Some fundamental laws of probability were re­ viewed, and Bayes's theorem of conditional probability was Illustrated as a guideline for revising one's opinion In sequential decisions under risk when only Imperfect Information Is available to the decision maker.

DecWon Modeto

445

The last section o f the chapter was an Introductory presentation of some approaches to quantifying multiple objectives. The discussion offered and examples used were restricted to decisions under as­ sumed certainty. However, the methods Illustrated are easily extended to decisions under risk and uncertainty and the matrix model used for analysis.

446

Chapter 8

Problems 1 Assume the following decision matrix where the cell values are units o f gain.

$4

$ 2

$0

$1

-4

10

3

7

4

8

2

3

2

4

9

5

If a person's Index of optimism Is estimated as a =0.25, predict the person's choice o f alternative by applying the various principles o f choice for a decision under uncertainty. (8.4.1,8.5) 2. Analyze the cost matrix below by the use o f the Hurwicz principle. (8.5.4) 5,

S3

$5

$5

$5

9

8

0

4

6

3

3. Apply the various dedslon-under-certalnty principles of choice to the matrix below. The values In the matrix are cost units. (8.4.1, 8.5) S3

■$«

$13

$13

$5

$9

8

8

8

8

0

21

5

5

9

17

5

5

5

7

7

5

4. A particular production department supervisor Is known to be very conservative In business matters. For the decision model below, where values are equivalent annual profits (In thousands of dollars) for a MARR - 15% and a five-year

Petition Models

447

planning period, predict the supervisor's choice o f an alternative by each o f th e principles fo r a decision under certainty. (8.4.1,8.5)

Si

S2

,

S3

s*t

$16

$16

$ 8

$12

9

20

10

6

15

10

8

9

3

14

6

11

8

16

9

12

5. Assume th e fo llo w in g decision m atrix. Pi

Si

P2

.

Si

Pi

.

sa

$ 20

$100

$1 200

190

190

190

500

120

100

(a) TTeat th e decision as one under uncertainty and the values In the m a trix as costs. (1) If a = an Index of optimism, for what value of a is one indifferent between A 2 and A3 ? (8.5.4) (2) Which alternative would be chosen by the Savage principle? (8.5.5) (b) TTeat th e decision as one under risk w here p , = 0 .2 0 , p 2 = 0 .7 0 , and p 3 = 0.10, and the values in the m atrix are profits. W h ich a lte rn a tive w o u ld be chosen by the expectation-variance principle? (8.4.2)

6. Given th e decision m a trix show n below (with cost elements), determ ine th e preferred alternative using the fo llo w in g principles o f choice. (8.5) (a) m in im a x (b) m in im in (c) m in im a x regret

448

Chapters

(d) expected cost, if each future state Is expected to occur with equal probability (e) Hurwlcz (with a = 0.40)

s3 $120

$50

$ 10

60

60

60

70

50

60

20

20

150

7. Given the decision m atrix shown below, determine the recommended alternative under the follow ing principles of choice: expectation, most probable future, maximax, maximin, and m inim ax regret Entries In the m atrix are profits. (8.4,8.5)

Pi =0.1

p2 = 0 3

,$ 2

-$ 5 0

$100

p 3 = 0 .4

p 4 =0.2

. 5a, $200

$400

r

- 20

50

500

100

100

-1 0 0

50

100

200

- 50

50

200

8. Shown below Is a m atrix o f costs for three Investment alternatives under three future states. Determ ine the preferred alternative using the following decision rules. (8.5.2,8.5.3, 8.5.5) (a) m inim ax (b) minimln (c) m inim ax regret

,

s3

$300

$200

$100

150

180

200

210

110

175

Deptolon Models

449

9. An analysis yields a decision under risk given In the matrix below, where the matrix values are annual profits In thousands of dollars.

-S3

a $6

° $9

1

5

20

7

3

7

9

" 5

5

0

12

15

2

8

-2 0 0

-1 0

22

9

0

4

12

$12

$5

-$8

7

0

3

(a) Which alternative should be chosen In order to minimize the probability o f a loss? (8.4.4) (b) Which alternative should be chosen In order to maximize the probability o f an annual profit of at least $9 000? (8.4.4) (c) Which alternative should be chosen in order to maximize the probability that annual profits will be between $3 000 and $10 000? (8.4.4) (d) Would the most probable future principle be a reasonable one to follow In selecting an alternative? (8.4.3) 10. An electronics firm has recently received a government contract to produce a certain quantity of expensive electronic guidance systems. The contracted number of units can be produced In tw o years, and the contract terminates then. The firm won the contract with cost estimates based on using present manufacturing equipment If the firm had more specialized equipment (with almost zero salvage value immediately after purchase), the unit cost of production would be reduced, and profit per unit would thus be Increased. However, this would be true only If the contract were for at least four years Instead of two years. The firm feels there Is a 50% chance fo r an additional two-year contract, a 30% chance for an additional four-year contract, and a 20% chance of no contract renewal. Using only present equipment, total profit on this job for a two-year, four-year, and slx-year contract is estimated as

450

Chaptar 8

$40 000, $80 000, and $120 000, respectively. If the specialized equipment is purchased, total profit on this Job for two-year, four-year, and six-year contracts Is estimated as -$100 000, $90 000, and $180 000. If an expectation principle is used, should the specialized equipment be purchased? Does the most probable future criterion seem reasonable in this decision? (8.4.1, 8.4.2,8.4.3) 11. A highway construction firm Is considering the purchase of a used mobile crane. TVvo such cranes are available, A, and if either Is purchased, the firm w ill use a six-year depreciation schedule. The cranes differ slightly In capacity, age, and mechanical condition, but both are presently in operating condition and have the capacity to do the Job expected. The firm expects that a major overhaul of each will eventually be required, but when this will be necessary is uncertain. Estimates o f the operating expenses (excluding labour) for each crane are given below:

A

^2

Year

Expense Schedule 1

Expense Schedule 2

Expense Schedule 1

Expense Schedule 2

1

$ 5 000

$ 5000

$14000

$ 6 00 0

2

5 000

11 000

6 000

6000

3

5 000

5 0 00

6 0 00

14000

4

11 000

5000

6000

6000

5

5000

5000

6000

6000

6

5000

5 0 00

6 000

6000

Other data for the tw o used cranes are:

First cost Salvage value at the end of six years

A

$30 000

$20000

12 000

10000

If the firm uses a before-tax MARR = 20% and a present worth method of comparing alternatives, which crane would be purchased if the Laplace principle were used? (8.4.1,8.5.1)

DecWon Modeto

451

12. Suppose a coin Is biased such that, when the coin Is tossed, the probability o f a head (H) showing is 0.60. Now assume that the coin Is tossed three consecutive times. (a) Sketch a probability tree for the three tosses and determine the probability values for each of the eight possible outcomes: HHH, HHT, and so on. For the three tosses, the following states are now defined: S, = the event o f three heads (3H) S2 => the event o f two heads, one tail-occurring In any order (2H,1) S, = the event o f two'tails, one head-occurring In any order (2T,H) * S4 = the event of three tails (3T) (8.6.1,8.6.2)

(b) Create a decision matrix where four alternatives are to guess the events defined above. Now assume that before tossing the coin three times, a person offers a $4 prize if the (3H) event is guessed correctly, a $3 prize If the (2H, T) event is guessed correctly, a $4 if the (2T, H) event Is guessed correctly, and a $5 prize If the (3T) event Is guessed correctly. However, there is a $1 charge to play the game. If the game Is played, would the expectation principle and the most probable future principle select the same alternative? (8.2,8.4.2, 8.4.3) 13.7 A vehicle manufacturer plans to develop and operate a public bus system for a community. The manufacturer's purpose Is to demonstrate the profitability of the bus system and then sell the system to the community. The manufacturer's most serious concern Is whether the Public Utilities Commission will authorize a competing bus system. It Is reasoned that there Is a small chance that such an event will happen; therefore, the manufacturer subjectively assigns a probability of 0.2 to the event of competition. A detailed analysis by the manufacturer results In an estimate of 50 vehicles needed In the event of no competition, and 25 vehicles needed If competition results. Furthermore, It Is reasoned that operating either a 50-vehide or a 25-vehlde 7 Problem 13 taken from deNeufvIlle and Stafford, Systems Analysisfo r Engineers and Managers, New York: McGraw-Hill, 1958, by permission of the publisher.

452

Chapter g

system will not affect the probability o f the Public Utilities Commission authorizing or not authorizing competition. If the manufacturer initially develops a 50-vehlde system and there Is no competition, it Is estimated the system will yield a $250 000 profit. In the event o f competition, a $120 000 loss Is projected. If a 25-vehtde system Is Initially developed and competition occurs, the manufadurer plans to sell the system as quickly as possible and estimates a $25 000 profit in this instance. On the other hand, if there is no competition, the 25-vehlde system will be inadequate. Poor service w ould therefore result, and the value of the demonstration would be reduced, thereby adversely affecting the sales price. Thus, In the event of no competition, the m anufadurer reasons that another decision must be faced: whether to expand from 25 vehldes to 50 vehicles or not to expand. If there Is no expansion, the manufadurer would sell soon to avoid the bad publicity for expeded poor service, estimating a $35 000 profit in this case. If, on the other hand, the system Is expanded and the total system is sold as soon as possible, tw o outcomes are predicted. One outcome is that poor service could result during the expansion and a net $10 000 loss would result (a probability of 0.10 is subjectively assigned to this event). The second outcome is that poor service would not result, and a net $140 000 profit Is expeded In this instance. Create a dedsion-tree model o f this situation and solve by use of the expedation principle. (8.6.1,8.6.2,8.4.2) 14. Company Able purchases tw o models, Model R and Model G, of Produd Tau from three suppliers, A, B, and C. These suppliers have produced, and are expeded to continue to produce, according to the follow ing table:

Supplier

Proportion of Total Produd Tau

Proportion o f Produd Tau by Model R

G

A

0.60

0.20

0.80

B

0.10

0.55

0.45

C

0.30

0.60

0.40

Decision Models

453

The Quality Control Department of Company Able has recently determined that many type R models have been defective and Intends to send a representative to each of the suppliers to observe their manufacturing procedures. Using the logic o f Bayes's theorem, which supplier is the most likely, next most likely, and least likely defect producer? (8.6.1 8.6.2) 15. Let It be supposed that, out of your sight, an experimenter rolls a green, red, and white die. The experimenter covers two of these. Data concerning the dice are that (1) the green die has five surfaces marked H, one surface marked T, (2) the red die has two surfaces marked H, four surfaces marked T, and (3) the white die has three surfaces marked H, three surfaces marked T. Also A = the event that the green die Is uncovered B = the event that the red die is uncovered

C= the event that the white die is uncovered h = the event that a surface marked H is showing on the

uppermost surface of the uncovered die

t = the event that a surface marked T Is showing

Now, the experimenter reports that the letter H is showing on the uppermost surface of the uncovered die. (a) Using the logic o f Bayes's theorem, what is the best guess concerning the colour of the uncovered die? (8.6.1,8.6.2) (b) Assume that the experimenter, after the report but before the guess, offers to pay $6 If the green die is guessed correctly, $15 If the red die Is guessed correctly, and $15 if the white die Is guessed correctly. However, there Is a $3 charge to play the game. Given that the game is played, model this situation as a declsion-under-rlsk matrix and determine the best guess by the expectation principle. (8.2,8.4.2) 16.®After receiving somewhat unreliable information concerning enemy troop movement along a supply route, a military commander of an artillery battalion faces the following combat 8 Problem 16 taken from Agee, Marvin et al., Quantitative A naly^ fo r Management Decision, Englewood a iffs NJ, Prentice-Hall, 1976, by permission of the publisher.

^54__________________________________________________ Chapter 8 decision (assume the matrix values are loss units, arbitrarily chosen). Pi =0.7 S1 = Ttoop Movement

p2 =03 S2 = No TYoop Movement

4 = bombard supply route

0

4

A i = n o t bombard supply route

12

0

The commander can choose between A y and 4 2 without additional information, or he can send out a reconnaissance plane for additional information. If the plane is sent out, he reasons the mutually exclusive and collectively exhaustive outcomes o f the flight are: Outcome Oy :

the plane is shot down, with a loss o f 3 units.

Outcome O2 :

the plane returns safely w ith negative information about troop movement, with a loss of 0.2 units.

Outcome O3 :

the plane returns safely w ith a report of suspicious activity, w ith a loss o f 0.2 units.

The commander assigns the follow ing subjective conditional probabilities to these outcomes: P(0,|S 1 ) = 04

P (0,|S 2 ) = 0.2

p (o 2 | S i) = a i

p( o 2 |s2 ) = ob

P (0 3 |S ,) = 0.5

P(o 3 |S2 ) = OjD

(a) Determine the posterior probabilities. (8.6.2) (b) Create a decision tree for this problem. (8.6.1) W Using the mlnlmtzlng-expected-loss principle o f choice, what action should the commander take? (8.4.2) (d) What Is the expected value of sample Information In this problem? (8.6.3, 8.6.4)

Decision Models

455

17. A manufacturer Is considering the possibility of Introducing a new product and the advisability of a test marketing prior to making the final decision. The alternatives are A,: = market the product A2 : = do not market the product

For simplicity, only three possible futures are considered; they are shown below together with the prior probabilities associated with them. Profit Sy:

the product captures 10% of the market

$10000000

0.70

1 000000

0.10

- 500 0000

0.20

S2 : the product captures 3% of the market S3 : the product captures less than 1% o f the market

Prior Probability

If the test marketing is made, three possible results are considered. Z y : test sales o f more than 10% of the market Z 2 : test sales of 5 to 10% of the market

Z 3 : test sales of less than 5% of the market

The conditional probabilities of the test results are

z.

z.

P (2 |S ,)

0.6

03

0.1

P (Z|5 2 )

0.3

0.6

0.1

pCZ|53 )

0.1

0.1

0.8

(a) Determine the prior expected profit (8.6.1,8.6.2,8.63) (b) Determine EVP/. (8.63) (c) Determine EVS/. (8.6.4) (d) If the test marketing costs $250 000, what would be the expected net gain from the sample information? (8.6.4)

456

Chapters

18. A textile firm In a rural tow n owns tw o trucks that transport raw materials from and finished goods to a nearby city. One of the trucks (tractor and trailer) is eight years old, and annual maintenance expenses are increasing significantly. Management is considering the purchase o f a new truck but, because of a mild national economic recession, textile sales have declined during the past tw o years, and management Is reluctant to make a large capital expenditure at present. Although in the midst of a recession, inflation Is also occurring; if the decision to purchase is delayed fo r a year, the cost of a new truck will probably increase 10% or more. Adopting a two-year planning period, the following estimates and Judgements are made. Present Truck The present market value Is approximately $9 000. If kept one more year, operating (excluding labour and depreciation) and maintenance expenses are estimated as either $6 000, $10 000, or $12 000 with subjective probabilities of 0.25,0.50, and 0.25, respectively. The salvage value at the end of the year is about $8 000. If kept one more year, a new decision to keep an additional year or buy a new truck w ill be made. If the truck is kept tw o additional years, operating and maintenance expenses (excluding depredation) for the second year are estimated to be either $7 000, $10 000, or $15 000 with subjective probabilities o f 0.10, 0.70, and 0.20, respectively. The salvage value at the end of the second year Is about $7 500. New Truck A new truck can be purchased now for $30 000. Average annual operating, maintenance, and depreciation expenses for each o f the first tw o years are estimated to be $15 000 with reasonable certainty. If the purchase of the truck Is delayed for one year, an Increase in purchase price Is virtually certain. It is Judged there Is a 50-50 chance that the purchase price w ill be $32 000 or $34 000. If it Is $32 000, total operating, maintenance, and

DecWon Models

457

depreciation expenses w ill be $16 000; If It Is $34 000, total expenses w ill be $17 500. For simplicity, assume MARR = 0% and (a) create a decision-tree model fo r this situation, and then (b) determine the best present decision by the expectation principle. (8.6.1, 8.4.2) 19. The Jax Tool and Engineering Company Is experiencing considerable problems with in-process material handling and storage o f finished goods. A new layout of machinery is possible, but this alternative Is subjectively judged by management to be prohibitively expensive, and the alternative Is discarded. They wish, therefore, to consider a new material-handling system and Identify the following objectives: G,: minimize annual costs over a 10-year planning period

C2 : minimize the disruption of production during the installation of the new system

G3 : Install material-handling equipment that has flexibility to meet different handling requirements

G4 : the material-handling equipment should have a high index of repairablllty G5 : material-handling personnel should require little training in order to operate the equipment

Management ranks these goals in the following order of Importance: G n = I, C 3 = II, G2 - III, Gs = IV, and G4 = V. Using the Churchman, Ackoff, and Am off method of weighting objectives (see p. 440, assign weights to these five objectives. (8.7.4) 20. Suppose a decision may be modelled as p, =0.6 S, = Contract Renewed

p 2 =0.4 S2 = Contract Not Renewed

A, =use present equipment

v(e„)

v(e12)

A2 = purchase new equipment

v(e21)

Assume that analysis produces the following descriptions of the outcomes:

e „ « no new capital required, 20 people required for at least four years, average annual profit units are 85 e,2 - no new capital required, 20 people required for two years and then laid off, average annual profit units are 67 e21 = tw o units of new capital required, 8 people required for at least four years, average annual profit units are 150 Ojj - two units o f new capital required, 8 people required for two

years and then laid off, average annual profit units are 400

Presume the relevant values for objectives are V (profit units)

= + 0.003 (profit units) + 1.0

V (capital units required)

= e~u , where u = the number of capital units required

V (work-years required)

= -0.0145 (work-years required) +1.0

Furthermore, the goals o f 'capital units required' and 'annual profit units' are considered equally important, but the labour consideration is only half as im portant as either of the other two. On the basis o f this Information, which alternative would you recommend? (8.2,8.7, 8.7.2)

Chapter 9

Accounting Principles 9.1 Introduction As already mentioned, the engineer should have some understanding of basic accounting practice and cost accounting techniques in order to obtain data from the firm's accounting system. If accounting is clas­ sified Into general accounting and cost accounting, then cost account­ ing is Judged the most important to the engineer as a source of data for making cost estimates pertinent to engineering projects. Cost account­ ing will therefore receive the greater emphasis in this text; in either case, the treatment o f accounting is cursory and directed toward fun­ damental accounting concepts instead of comprehensive accounting detail. In virtually all businesses, general accounting information Is sum­ marized in at least tw o basic financial reports: 1. A balance sheet, or statement of financial conditions provides a summary listing of the assets, liabilities, and owners' equity (shareholder's equity) accounts of the firm as of a particular date. 2. An income statement (profit and loss statement or statement of earning^ shows the revenue and expenses Incurred by the firm during a stated period of tim e-a month, quarter, or year.

9.2 Balance Sheet The records of financial transactions and the variety of Internal reports that are put Into the accounting system provide Information on sales and other revenues and the expenses Incurred In obtaining the reve­ nues. Revenues and expenses for a specified period are then summa­ rized on the profit and loss statement. The net profit or loss resulting for this period Is then transferred to the owners' equity section of the balance sheet Thus, although the two basic financial reports provide different financial pictures of the firm, they are directly related. Before Illustrating this fact, discussion of certain terminology is necessary.

460

The Items listed on a balance sheet are usua lly classified Into three m ain groups: assets, liabilities, and o w n e rs' e q u ity Items. Subgroups m ay also be Identified, such as cu rre n t and fix e d assets, and current and long-term liabilities. Assets are prope rties o w n e d b y the Ann, and li­ abilities are debts ow ed b y th e firm against these assets. The difference betw een assets and liabilities is th e o w n e rs' equity, o r shareholders' equity, w h ich is th e Investm ent m ade b y th e ow ners o r shareholder o f th e business plus a n y accum ulated p ro fits le ft In the business by the ow ners o r shareholders. A fu n d a m e n ta l a cco u n tin g equation is thus defined: assets - lia b ilitie s - o w n e rs' e q u ity Rewriting, w e have assets = lia bilities + o w n e rs' e q u ity and th e usual fo rm a t o f a balance sheet fo llo w s th e equation in this form . Current assets include cash and o th e r assets th a t can be readily con­ verted in to cash; an a rb itra ry perio d o f o n e ye a r is usually assumed as a criterion fo r conversion. Sim ilarly, current liabilities are the debts that are due and payable w ith in o n e ye a r fro m th e date o f the balance sheet In question. Fixed assets, then, are th e properties owned by the firm th a t are n o t readily co nve rted in to cash w ith in a one-year period, and long-term liabilities are debts due and paya ble after one year from the date o f th e balance sheet. Typical current-asset Items are cash, ac­ counts receivable, notes receivable, ra w m a te ria l Inventory, work In progress, finished goods in ve n to ry, and p repa id expenses. Fixed-asset Items are land, buildings, e qu ipm ent, fu rn itu re , and fixtures. Items that are typ ica lly listed und er cu rre n t lia b ilitie s are accounts payable, notes payable, Interest payable, taxes payable, prepaid income, and divi­ dends payable. Long-term lia b ilitie s m a y also be notes and bonds pay­ able, m o rtg a g e s p a y a b le , a n d so fo rth . O w n e rs' equity Items appearing o n a balance sheet are less standard and, to a degree, de­ pend on w h e th e r th e business Is a sole proprie torship, a partnership, o r a corporation. The size o f th e co rp o ra tio n Is also an Influencing fac­ to r o n Item designation. H ow ever, item s such as capital stock, retained earnings, capita l surplus o r earned surplus alw ays appear under own­ ers' equity. A n exam ple o f a balance sheet fo r a hypothetical firm Is ex­ hibited In Table 9.1.

Accounting Principles

461

fa b le 9.1 Sample Balance Sheet Jax Tool and Engineering Company, In c Balance Sheet December 31 ,2 0 0 0 Assets

Current assets Cash Accounts receivable

$ 25000 11500 0

Raw materials

8 5 00

Work In progress

7000

Finished goods Inventory

15 5 00

Total current assets

$171 0 00

Fixed assets Land

30000

$200000

Building Less: Accumulated depreciation Equipment

50000

150000

750000

Less Accumulated depredation

150000

600000

Office equipment

10000

Total fixed assets

790000 $961 0 0 0

Total assets Liabilities a n d Ow ners' Equity

Current liabilities Accounts payable faxes payable

$ 32000 ________15QQQ. $ 47000

Total current liabilities Long-term liabilities Mortgage loan payable

130000

Equipment loan payable

350000 480000

Total long-term liabilities

527000

Total liabilities Owners' Equity Common stock Retained earnings fatal equity fatal liabilities and equity

325000 109 0 0 0 434000 $961000

462

Chapter 9

The sample balance sheet In Table 9.1 Is balanced (i.e., assets = liabilities + owners' equity), and It gives a statement of the financial condition of the organization as o f a specific date-the close of an ac­ counting period. Although depredation Is covered In Chapter 5, note the Inclusion of accumulated depredation In the fixed-assets portion of the balance sheet. For example, the building originally cost $200 000, and depredation expenses have been charged annually so that the total depredation charges as o f the date of the balance sheet have been $50 000, which is the amount entered as the accumulated depredation for the building. The first cost of the depreciable asset (the building, In this case) minus the accumulated depreciation equals the book value. A similar explanation applies for the fixed asset of equip­ m ent In this balance sheet, the equipment account Is an aggregate for all equipment owned by the company instead of an individual listing of the equipment, which could be the case, depending on accounting practice.

9.3 Income Statement The second basic flnandal report compiled by the accounting system is the Income statement {profit and loss statement or statement of earning^. For the current accounting period, the Income statement provides management with (1) a summary o f the revenues received, (2) a sum­ mary o f the expenses Incurred to obtain the revenues, and (3) the profit or loss resulting from business operations. The format of the in­ come statement varies, and the revenue and expense Items depend on the type o f business involved. Thus, the Income statement given In Table 9.2 is Illustrative only and still concerns our hypothetical Jax Tool and Engineering Company. Let us further assume that the period o f time covered Is one year, which has ended as o f the date of the bal­ ance sheet given in Table 9.1.

9.4 Interpretation of Financial Statements Recall that one purpose of the accounting system Is to interpret the fi­ nancial data o f an organization. Among the most commonly used techniques for Interpretation Is the calculation of accounting ratios and other accounting quantities derived either from balance sheet or income statement Information. From balance sheet data working capi­ tal, current ratio, the add-test ratio, the equity ratio, the debt to equity ratio, and other less frequently used accounting ratios can be calculated.

Accounting Principles

~

■* t t p f

’. . f - ?;

463

Table 9.2 Sample income Statement Jax Tool and Engineering Company, Inc. Income Statement Year Ended December 31,2000 Sales Less cost of goods sold Direct labour Direct materials Indirect labour Depreciation Repairs and maintenance Utilities Gross profit Less other expenses Administration Marketing Z/ Interest payments Net profit before tax Less income taxes Net profit

$1200000 $420000 302000 112000 98000 41 500 11 500

985000 $ 215 000

$ 76000 49000 35 000_______ 160000 $ 55000 _______ 26000 $

Working capital, or the capital used to meet the dally commitments of the organization, Is the excess of current assets over current liabili­ ties, so that working capital = current assets - current liabilities The working capital Is an Important reflection of the financial posi­ tion of the company. From Table 9.1, the working capital of theJaxTbol and Engineering Company can be determined: $171 000 - $47 000 = $124 000 The company's working-capital condition can also be expressed as a ratio. This ratio, called the current ratio, Is defined as current ratio =

current assets current liabilities

29000

464

Chapter 9

From "fable 9.1, the current ratio fo r the Jax Company Is $171000 $47 000 which Implies that cunent assets would cover short-term debts 3.638 times. The current ratio assumes that the current assets o f Inventories are convertible to cash w ithin one year. A more conservative ratio of the li­ quidity o f the company is the add-test or quick asset ratio, which Is de­ fined as add test ratio

current assets - inventories - prepaid expenses current liabilities

"The add-test ratio for the Jax Company can be calculated as follows: (In this case, inventories include the cost o f raw materials, work in progress and finished goods, and there are no prepaid expenses.) $ 1 7 1 0 0 0 -$ 3 1 0 0 0 $47 000

= 2 g79

A ratio that measures the finandal strength o f the firm is the equity ratio: equity ratio =

' total assets

o w n e rs

From "fable 9.1, the equity ratio is $434 000

=0.451

$961000 which means that 45.1 % of the Jax Tool and Engineering Company is owned by the shareholders. (The $109 000 retained earnings are also owned by the shareholders.) The debt to equity ratio expresses the relative magnitude of debt to equity capital. It Is defined as debt to equity ratio =

lo n

g~t e n n " a b l ^

owners' equity

Accounting Principles

465

From Table 9.1, th e d e b t to equ ity ratio is ^

9

=1.106

0 0

$434 0 0 0 Therefore, fo r every d o lla r o f equ ity the shareholders have in th e ir company, th e creditors have p ut in $1,106. In addition to these fo u r ratios th a t can be computed using on ly bal­ ance sheet data, a n u m b e r o f com m on accounting ratios are calcu­ lated using data fro m b o th th e balance sheet and the incom e statement. The m ost Im p o rta n t o f these are the operating ratio, the In­ come ratio, th e 'g ro ss'rate o f return on investment, the inventorytum over ratio, and th e collection period. The operating ratio Is defined as follow s: o p e ra tin g ra tio

tota l revenues

=

to ta l expenses Using Table 9.2, th e o p e ra tin g ratio is t1 2 0 0 0 0

° =1.048

$11 4500 0 where the to ta l expense fig u re excludes the Income taxes paid. (In­ come taxes m a y o r m a y n o t be included In calculating the ratio.) An operating ra tio g re a te r th a n 1.0 indicates th a t a net p rofit (before in­ come taxes in th e e xa m p le used) Is being m ade and is, therefore, desir­ able. Such o p e ra tin g ratios are perhaps m ore m eaningful w hen they are com puted fo r d iffe re n t product lines o r different plants w ith in a m ultiplant corp o ra tio n . They can then be used by m anagem ent to as­ sess the effectiveness o f in d ivid u a l product lines or plants. The Income ratio Is defined as Incom e ra tio =

netProffl

to ta l revenue

x l0 0 %

and, using th e a fte r-ta x data fro m Table 9.2, Is calculated to be $2

9 0 0 0

.. x l 00% = 2 .4 1 7 %

$1200 000

46 6

Chapters

This figure fo r th e Jax C o m pany Is ra th e r lo w fo r a m anufacturing firm and Is m o re characteristic o f large re ta il organizations. The ratio Indi­ cates th e net after-tax p ro fit m a rg in o n gross revenues; management w o u ld be pa rticu la rly interested In tre n d s (rising, decreasing or static) o f th e ratio. The net after-tax p ro fit fo r a g iv e n ye a r can also be used to calculate a ‘gross' rate o f return on Investment T h a t Is, gross rate o f retu rn = — — Pr o ^ ------x l 00% to ta l in ve stm e n t D ifferent versions o f this ra tio arise p rim a rily because o f widely vary­ in g de fin itio n s o f to ta l investm ent. T he to ta l investm ent o f the Jax C om pany Is defined as to ta l assets, o r th e sum o f ail fixed assets at th e ir o rig in a l cost m inus accum ulated d e p re cia tio n plus all current as­ sets (see Table 9.1). Thus, th e gross rate o f re tu rn fo r th e accounting pe­ riod covered b y th e in com e sta te m e n t is '

=

$29 000

$961000

x l 0Q % = 3 Q 2%

This Is a v e ry lo w rate o f retu rn o n in ve stm e n t and n o t typical of suc­ cessful m a n u fa ctu rin g firm s. A lth o u g h this ra tio is c o m m o n ly used In th e business world, none of Its m any versions gives a co m p le te ly accurate fin a n cia l picture, since th e ratio does n o t ta k e In to account th e tim e v a lu e o f money. This ratio also has little bea ring o n th e discussion In th is text, m ost o f our study being based o n th e tim e v a lu e o f m oney. The Inventory turnover ratio show s h o w m a n y tim es the organization has sold its com plete in v e n to ry d u rin g th e past accounting period, thus m easuring th e rate a t w h ic h in v e n to ry Item s are moving. It Is de­ fined as . . . cost o f goods sold In ve n to ry tu rn o v e r ra tio = ----------- - --------------in v e n to ry w h ere th e in v e n to ry includes th e cost o f all fin ish e d goods the com­ pany has produced d u rin g th e a cco u n tin g period. Similarly, Inventory tu rn o ve r ratios can be calculated fo r in ve n to rie s o f raw material and

Accounting Principle*

4«7

work In progress. For the Jax Tool and Engineering Company the fin­ ished goods Inventory turnover ratio Is $985 000 $15 500

=63.55 per year.

A small Inventory turnover ratio Indicates an accumulation of slow moving goods. The collection period Indicates the promptness with which receiv­ ables are collected. It is defined as collection period

=

. ^ n t s receivable average dally sales

For the Jax Tool and Engineering Company, the collection period is $115 000 ($1200 000/365 days)

= 34.98 days

Indicating that accounts are paid in 34.98 days on the average. All the data necessary for calculating these accounting ratios can be found easily In the consolidated financial statements of a public com­ pany. The following example illustrates the method of determining ra­ tios from this source. Example 9.1 Determine the accounting ratios fo r Excor Industries and Subsidi­ aries on the basis o f Its Second Quarterly Report In 2000. The Con­ solidated Financial Statements o f the company are reproduced In Table 9.3. The tota l current assets o f the company are $1 942 805 000, and the tota l current liabilities are $924 437 000; therefore, the w orking capital Is w o rkin g ca pita l= current assets-current liabilities = $1942 805 0 0 0 -$ 9 2 4 437 000 = $1018 3 6 8 0 0 0

The current ratio Is

468

Chapter 9

Thble 9.3 Consolidated Financial Statements - June 3 0 ,2 0 0 0 (In Thousands o f Dollars) o f Excor Industries and Subsidiaries Statement o f Earnings Second Q uarter 2 00 0 Net sales........................................................................................................

$720953

Other in c o m e ..............................................................................................

7259 $728212

Cost o f sales and operating expenses....................................................

$487 547

Selling, general and adm inistrative expenses.......................................

72 349

Research and d e v e lo p m e n t.....................................................................

12 945

E x p lo ra tio n .................................................................................................

6 079

Interest, net o f amounts c a p ita lize d ........................................................

41 506

Currency translation a d ju s tm e n ts ...........................................................

8882 $629308

Earnings before income and m in in g t a x e s ...........................................

$ 98 904

Income and m ining t a x e s ........................................................................

52 782

Net e a r n in g s ..............................................................................................

$ 46122

Dividends on preferred shares.................................................................

6 592

Net earnings apltcable to com m on s h a r e s ...........................................

$ 39530

Net earnings per com m on s h a r e ........................................................

$0.53

Common shares outstanding at end o f p eriod .................................

75 526 290

Balance Sheet as at June 3 0 ,2 0 0 0 71 824

Notes p a y a b le ................

$374986

Accounts receivable. . .

558 232

Accounts payable . . . .

436811

Inventories.......................

1 2 87 310

Current taxes payable. .

112658

25 4 3 9

Total current lia b ilitie s .

$ 924437

$1 942 805

Long-term d e b t .............

1 015160

Cash and securities . . .

Prepaid expenses . . . . Total current assets . .

$

Deferred ta x e s ................ Property, plant and equ ip m e nt-ne t . . . . Cost In excess o f net assets acquired . . . . Other a ssets...................

$2 517 057 2 8145 94 332 $4 582 339

O ther liabilities................ Preferred sh a re s.............

453000 70231 346 948

Com m on shares.............

116016

Retained earnings and capital su rp lus.............

1 656 547 $4 582339

Accountinfl Principles

48®

current ratio =

current assets current liabilities $1982 805 000 $924437 000

= 210

The Inventories are $1 287 310 000, the prepaid expenses $25 439 000; therefore, add test ratio

current assets - inventories - prepaid expenses current liabilities $1942 805 0 0 0 -$ 1 287 3 1 0 0 00-$25 439000 $924437 000 = 0.682

The total assets are $4 582 339 000, and the shareholders' equity (the sum o f preferred shares, common shares, and retained earn­ ings and capital surplus) is $2 119 511 000; therefore, equity ratio =

owners' equity . — -r— total assets

$2119 511000 ~ $4 582 339 0 0 0 “

0 463

The long-term liabilities are $1 015 160 000; thus, debt to equity ratio =

longterm liabilities owners' equity $1015160000 $2119511000

From the Statem ent o f Earnings, the tota l revenues are $728 212 000 for the quarter (the sum of net sales and other In­ come) and the total expenses are $629 308 000 (Including cost of sales and operating expenses; selling general, and administrative expenses; research and development; exploration; and interest and currency translation adjustments); therefore,

470

Chapters

operating ratio =

total revenues to ta |e x p e n s e s

$728 212 000 “ $629 308 0 0 0 “

1157

The net profit for the quarter Is $46122 000; accordingly, Income ratio =

net profit total revenue

x !0 0 %

$46122 000 $728 212 000

x100% =633%

The total Investment Is $4 582 339 000; therefore, gross rate o f return

net profit total Investment $46122 000 $4 582 3 3 9 0 0 0

x100% x 1 00%=1.007%

The Inventory is $1 287 310 000, the cost o f goods sold Is $487 547 000+$72 349 000 = $559 896 000 (cost o f sales and operating expenses plus selling, general and ad­ ministrative expenses), and the Inventory turnover Is calculated as: , , cost o f sgoods sold Inventory3 turnover = :— -?---------Inventory $559 896 000 " $1 287 310 000 = 0.435 per quarter The accounts receivable are $558 232 000, and the average dally sales are $720 953 000 91 days

= $7 922 560/day

Accounting Principles

471

Therefore, u n ts r e c e tv a t,|e collection period . ^ average dally sales

$558 232000 “ $7 922 560/day = 70.46 days

9.5 Cost Accounting The balance sheet and th e Incom e statement in Section 9.4 are consid­ erably rem oved b o th In tim e and In detail from decisions at the usual engineering project level. M ore im portant to the engineer as a source of cost In fo rm a tio n Is th e cost accounting system w ith in a particular firm. The firm m a y be in vo lve d In m anufacturing o r providing services, and If it Is Involve d In m anufacturing, production m ay be on a jo b ­ shop o r process basis. There are som e fundam ental differences In cost accounting procedures fo r determ ining m anufacturing costs versus determ ining th e cost o f p ro v id in g a service; also, there are differences In accounting procedures if m anufacturing is on a Job-shop or process basis. In o rd e r to concentrate on basic principles instead o f details, the cost accounting system assumed w ill be th a t o f a Job-shop m anufac­ turing firm . Thus, th e em phasis w ill be on determ ining the per-order costs fo r a jo b order. The to ta l cost o f p ro d u cin g a ny Job order consists o f direct material, direct labour, and overhead costs. An additional item o f cost could be special to o lin g o r e q u ip m e n t purchases strictly fo r the Job order In question. In o rd e r to sim p lify th e presentation, this definition o f tota l cost does n o t break d o w n th e overhead cost Into factory overhead, general overhead, and m a rke tin g expenses. Materials fo r a given Job order m ay include purchased parts and In-house fabricated parts, and the cost fo r direct m aterials is determ ined prim arily from purchase In­ voices. Q uestions o f scrap allow ances and averaging m aterial costs, which fluctuate over tim e, present problem s In obtaining accurate di­ rect m aterial costs, b u t d e te rm in in g such costs is reasonably straight­ forward. D irect la b o u r tim e spent on a jo b order Is norm ally recorded by operators o n la b o u r tim e cards, and direct labour cost is deter­ mined by a p p lyin g th e appropriate labour cost rates. The labour rates, as determ ined by th e accounting system, w ill norm ally include the cost of em ployee frin g e benefits in addition to the basic hourly rate. Al-

472 ______________ _________________________________________

Chapters

though accurately determining the direct labour cost for a given job Is a major accounting problem, It Is more readily determined than the overhead cost. Overhead costs cannot be allocated as direct charges to any single Job order and must, therefore, be prorated among all the job orders on some arbitrary basis. Common methods fo r distribution are: 1. The rate per direct labour hour. 2. A percentage o f direct labour cost. 3. A percentage o f prime cost (direct material plus direct labour cost). If a single overhead rate fo r the entire manufacturing firm Is to be used, this would, of course, be an average overhead rate for the entire factory, assuming the expense o f providing and using an hour of fac­ tory facilities Is about the same throughout the factory. To Illustrate the three methods listed, it Is assumed that a company experienced the following costs In the previous year. Total direct labour hours

$ 48 000

Total direct labour cost

$480 000

Total direct material cost

$600 000

Total overhead cost

$360 000

Then, the overhead rate per direct labour hour would be overhead rate =

costs direct labour hours o v e rh e a d

_ $360 000 48 000 h = $7.50 per direct labour hour If a particular job order requires 40 h o f direct labour with an average rate of $12.50 per hour and $850 o f direct materials, then the total cost of the job would be computed as

Accounting Prlndpleg

473

Direct material cost

=

$ 850

Direct labour cost

- 40 h x $12.50/h «

500

Overhead cost

= 40 h x $7.50/h =

300

Total cost $1650

Determining the overhead rate as a percentage of direct labour cost for this company will yield overhead rate %

overhead costs x100% direct labour cost $360 000 x100% $480 000 = 75% of direct labour cost

For the same Job above, the total cost would be computed as Direct material cost

=

$ 850

Direct labour cost

=

500

Overhead cost

= 75% x $500 =

375

Total cost $1725

Determining the overhead rate as a percentage of prime cost for this company will yield overhead rate % = ---------------- overhead costs x 100% direct labour cost + direct material cost = $360000 $1080 000

x10Q %

= 33X% For the same Job above, the total cost would be computed as

474

Chapter fl

Direct material cost

=

$ 850

Direct labour cost

=

500

Overhead cost

= 3 3 }^% x$ 1 3 5 0 =

450

Total cost $1800

Determining the overhead cost fo r a job order by the 'rate per direct labour h o u r method will yield the same result as the 'percentage of di­ rect labour cost' method, provided that the rate per direct labour hour used on the job Is equal to the factory average rate per labour hour. The 'percentage of prime cost' method w ill normally yield a different assignment of overhead to a Job order than the other two methods. The choice among these three methods Is arbitrary; Indeed, other methods are used by cost accountants In distributing overhead costs to a given job order. The rate per direct labour hour method Is perhaps most commonly used. Whatever method is chosen from the above for distributing over­ head costs to job orders In a current year, the rates or percentages are based on the previous year's cost figures. Thus, overhead rates may change from year to year w ithin a particular firm. Since an average overhead rate fo r the entire factory may very well be too gross an estimate when actual overhead costs differ among de­ partments within the factory, cost accounting may determine Individ­ ual overhead rates for departments or cost centres. Furthermore, the hourly rates for direct labour may vary am ong these cost centres. A further refinement can be made by determining overhead rates for in­ dividual machines within cost centres. Then, as particular job orders progress through departments and/or machines, the direct labour time (or machine time) spent on the jo b order In the various cost cen­ tres is recorded, the appropriate labour or machine rates and over­ head rates are applied, and the total cost fo r the job is calculated. The following example Illustrates one of the variety of methods used to calculate the overhead rate for a given cost centre and the to­ tal cost of a job.

Example 9.2 The following information has been accumulated for the Deetco Company's two departments during the past year. (See table be­ low.)

Accountinfl Principles

475

The Deetco Company distributes depredation overhead based on (1) the first cost of equipment In each department, (2) zero sal­ vage value o f the equipment In 10 years, and (3) a constant annual {or straight-line) rate of depredation. All overhead other than de­ predation Is first distributed to each department according to the number of employees In each department, and an overhead rate per direct labour hour Is computed. What selling price should the company quote on Job Order D if raw material costs are estimated as $900, estimated direct labour hours required In Departments A and B are 30 hours and 100 hours, respectively, and profit is to be calculated as 25% of selling price? The summary o f costs Incurred by the two departments of Deetco Company during the last flnandal year Is given below. Department A

Department B

Direct material cost

$720000

$240000

$960000

Direct labour cost

$260 000

$140 000

$400000

25 200

16200

41 400

Number of employees

14

9

23

First cost of equipment

$250 000

$200000

$450000

Annual depreciation

$ 25 000

$ 20000

$ 45000

Direct labour hours

Total

Other factory overhead

$150000

General overhead

$350000

For Department A, the total overhead allocated is determined as Annual depreciation

-

J 25 000

Other factory overhead

= (14/23X$150 000) =

91 304

General overhead

= (14/23XS350 000) =

213043

Tbtal overhead costs

$329 347

Thus, the overhead rate for Department A per direct labour hour Is

S^oS/h = $13 07

d,reCt ,ab°Ur h°Ur

476

Chapter}

For Department B, the total overhead allocated Is determined as Annual depreciation

=

$ 20000

Other factory overhead

- (9/2 3)($ 15 0 000) -

58 696

General overhead

= (9/23)($350 000) =

136957

Total overhead costs

$215 653

Thus, the overhead rate for Department B per direct labour hour Is $215 653 -g u ~= $1331 per direct labour hour 1o zoo n Then the estimated total cost fo r Job Order D Is computed as Direct material cost

-

Direct labour cost for Dept. A

$260000 _ "2 5 2 0 0 /h x 3 0

h

Overhead cost for D ept A

= ($13O 7/hX 30 h)

Direct labour cost for D ept B

$140000 ~ 1 6 2 0 0 /h

Overhead cost for D ept B

= ($ 1 3 3 1 /h X l 00 h)

_ _ ° h

X 10

Total cost If x - the selling price o f Job Order D, then x = total cost+profit = $3 7 9 6 3 2 +Q25X and x= I

$3 7 9 6 3 2 0.75

= $5 062.43

$ 900.00

=

309.52

=

392.10

=

864.20

=

1331.00

ss

$3796.82

Accounting Principles

477

Problems 1. K.Z. Moley purchased a small engine repair shop and opened on July 1,2000. At the date of opening, he had invested $8 000 of equity funds with the following breakdown: $4 000 In equipment, $3 000 In Inventory Items, and $1 000 In operating cash. (a) Prepare a balance sheet for the business as of July 1,2000. (9.2) The data below summarize the gross sales and expenses for the business during the first three-month period. Gross sales

$13 500

Purchases

7 000

Salaries

30 0 0

Advertising expense

250

Rent expense

600

Expense for utilities

400

For the purchases, $6 000 was paid with cash and $1 000 Is still owed. At the close of the three-month period, the end-of-perlod Inventory Is worth $2 600. (b) Prepare an Income statement for the three-month period covered. (9.3) (c) If, at the end of the three-month period on October 1,2000, the owners' equity account Is $9 850, determine the amount of the cash account in order to balance the accounting equation as of October 1,2000. (9.2) 2. A successful building contractor purchased a farm to operate on a part-time basis and raise beef cattle. The purchase price of the farm was $75 000. The contractor paid $25 000 cash and financed the remainder over 10 years at a 9% Interest rate with a mortgage, payable to a local bank. Soon after the purchase of the farm, the contractor purchased cattle for $8 000, paid $3 000 In cash, and gave a promissory note to the seller for the balance. The note carried an 8% annual Interest rate and was to be paid o ff within five years. During the first full year, the farm operation resulted In the following revenues and expenses.

478

Chapter 9

$6 500

Calves sold Hay sold

200

Labour expenses

500

Expenses for machinery rented

2 000

Veterinarian fees

175

Fertilizer purchased

1 200

Property taxes

450

Expenses for repairs

375 4 900

Internal expenses Expenses for miscellaneous supplies

125

Prepare an income statement for the farm ing operation and determine the net profit (loss) before income taxes. (9.3) 3. An Income statement and balance sheet for the WAC Company covering a calendar year ending December 31 are as below: Income Statement $247000

Cross income from sales

138 800

Less: cost of goods sold

$108200

Net Income from sales Operating exenses Rent Salaries

$ 9 700 30 200

Depreciation

5 800

Advertising

4 300

Insurance

1 500

$51 500

Net profit before Income taxes

56700

Less: Income taxes

23 973

Net profit after Income taxes

$32 727

Accounting Principles

479

Balance Sheet Assets

Liabilities

$ 94227

Cash Accounts receivable

8000

Notes payable

$ 25000

Accounts payable

Raw materials Inventory

10000

Dedared dividends

Work-in-progress Inventory

15 000

Total liabilities

Finished goods Inventory

18 500

Land

30000

Building

$80 0 00

Less: accumulated depredation

$ 8QQQ

Equipment

$40 000

Less: accumulated depreciation

$ -.4 0 0 0

Total assets

72 000

20000 $ 51000

Shareholders' Equity

Capital stock

$ 20 0 00

Retained earnings

- 3 2 727

Total net worth 36000

6000

$232 727

Total liabilities and shareholders' equity

$238 72

If land, building, and equipment are fixed assets, and notes payable Is a long-term liability, compute (a) the current ratio, (b) the add-test ratio, (c) the equity ratio, (d) the operating ratio, (e) the Income ratio fo r net profit after taxes, (f) the debt to equity ratio, (g) the gross rate of return, (h) the Inventory turnover ratio, and (I) the collection period for the WAC Company. (9.4) 4. An order for 500 units of Part D-142 Is received by the J.T. Kling Engineering Company, a small machine shop. The finished dimensions of the rectangular part are 19 mm x 17 mm x 45 mm (neglecting tolerances). The raw material for this part is 20 mm x 20 mm x 1 000 mm steel rectangular bar stock, w ith each unit costing $12 and yielding 20 part blanks. The basic manufacturing sequence, with standard machining times per part, and machine overhead rates (per machining hour) Is given below.

$283 727

480

_ _ _ _ _ _ _ Chapters

S tandard T im e p e r Part

Machine Overhead Rate

C u to ff on p o w e r saw

1 m ln

$0.30/h

M ill tw o sides

4 m ln

$0.50/h

D rill th re e 4 -m m -d la m e te r holes

2 m ln

$0.40/h

Surface g rin d o n e side

2 m ln

$0.35/h

0 .2 5 m ln

-

O perating

Package

H ie direct labour time per part is the same as the machine (or operation) time per p a rt The tooling cost fo r this job order is estimated to be $500. Excluding tooling costs and machine overhead, other factory overhead costs (for indirect labour, utilities, indirect materials, etc.) are $9 per direct labour hour. If the average direct labour hour rate (including fringe benefits) is $10.75 per hour, determine (a) the total estimated costs for the job order of 5 000 units, and (b) the unit selling price if profit is to be 30% of the total cost (9.5) 5. The welding department of a m ining equipment manufacturing plant consists o f four cost centres: manual arc welding (A), semiautomatic welding (B), furnace brazing and heat treating (C), and finishing (D). Some oxyacetylene cutting is also done in Centre B. Assume that it is possible to allocate departmental overhead expenses directly to each cost centre and that the following data for the welding department were compiled last year by the accounting system. Cost D ep a rtm e n ta l Centre Expenses

D irect Labou r H ours

Cost

Direct Material

A

$10 5 0 0

10000

$52 0 0 0

$8 000

B

6800

4000

16 0 0 0

8 000

C

4 600

1 500

4 500

3 000

D

2 400

2 800

6 000

20 0 0

Compute the overhead rate (or rates) applicable by the following methods (9.5) (a) b la n k e t (departm enta l) percentag e o f d ire ct labour cost (b) b la n k e t percentage o f p rim e cost

Accounting Principles_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

(0 blanket hourly rate per direct labour hour (d) percentage of direct labour cost for each cost centre (e) percentage of prime cost for each cost centre (f) rate per direct labour hour for each cost centre

6. Consider the welding department and four cost centres given In Problem 5. The direct labour hours and cost for each cost centre remain the same as In Problem 5, but new and additional data are given below (assume cost data are for the previous year). Cost Centre

Square metres occupied

Cost of Machinery

Number of Direct Labour Employees

A

900

$ 5000

5

B

400

6000

2

C

600

90 0 0

1

D

500

... 3J0Q

1

2 400

$23000

9

Total

Expenses other than fo r direct labour and materials chargeable to the welding department last year were: Maintenance Gas and electricity Supervision and other Indirect labour Miscellaneous supplies Equipment depreciation Building depreciation

$4 000 10 000 24 000 5 000 2 300 4 000

Determine an overhead rate per direct labour hour for each cost centre If the welding department expenses above are first allocated to each cost centre as follows: (9.5) (a) Maintenance expenses and equipment depreciation expenses are allocated according to the value of equipment (percent of total) In each cost centre. (b) Building depreciation expenses are allocated according to the floor space occupied by each cost centre.

482

(d Supervision and other Indirect labour expenses are allocated according to the number of direct labour employees of each cost centre. (d) Supplies and gas and electrldty expenses are allocated according to the number of direct labour hours for each cost centre.

Chapter 10

Fundamental Economic Concepts 10.1 Introduction The main goal o f engineering economic analysis is to develop a logical methodology for choosing an engineering project from among sev­ eral feasible, competing alternatives where the criterion for selection is a measure o f economic effectiveness. Although the subject of engi­ neering economic analysis can be studied on Its own, some knowl­ edge of basic economic principles and some experience in engineering problem solving are particularly helpful in understanding the objectives and the methods of the subject In this chapter, two ba­ sic economic concepts closely related to the subject of engineering economic analysis are outlined; the supply and defnand Interaction and the theory o f production. It is important that readers without any previous exposure to econom ic theory study the material presented In this chapter. However, readers who are already familiar with these fundamental economic concepts may omit i t

10.2 Supply and Demand In a free-market economy, the laws of supply and demand deter­ mine the price of goods and the quantities of these goods that are sold. It Is Important for the practising engineer to understand the funda­ mental concepts of supply and demand, their interaction, the way this Interaction determines prices and quantities of goods sold, and the method of predicting the change In price and quantity sold due to change In demand and supply.

10.2.1 Demand The quantity of a commodity that would be purchased by all consum­ ers in a market, termed the quantity demanded, is a specific quantity of goods desired per time period (e.g., 23 000 snowmobiles per year, 7 000 tractors per week, or 15 000 kg of chromium per month). Quan­ tity demanded Is also a function of a number of factors: the price of the

484

Chapter 10

com m odity, the size o f th e p o p u la tio n served b y the market, the In­ com e o f th e p o p ula tion, prices o f related com m odities, tastes and pref­ erences o f th e p o p u la tio n , advertising, e tc T he q u a ntity demanded n o rm a lly changes If a n y o n e o f these factors changes. For example, the num be r o f sn ow m o biles d em an ded In th e Canadian market will increase If th e price o f th e s n o w m o b ile drops, if th e Canadian popula­ tio n increases, o r tf th e average Incom e o f Canadians rises. Even an In­ crease in air fares to Florida w o u ld te n d to increase the number of snow m obiles dem anded In th e C anadian m arket. O f th e factors a ffe cting q u a n tity dem anded, price Is the most signifi­ c a n t Q u antity dem anded Is th e q u a n tity desired a t a given price, and it Is n o t necessarily th e sam e as th e q u a n tity a ctu a lly purchased. For In­ stance, let us assume th a t a t a price o f $3 0 0 0 pe r unit, 23 000 snow­ m obiles are dem anded in a year, b u t o n ly 18 0 0 0 units are available fo r sale at this price in th e m a rk e t Thus, regardless o f the fact that no m ore th a n 18 0 0 0 snow m o b ile s can be a ctu a lly purchased in this par­ ticu la r year fo r th e price o f $3 0 0 0 per u n it, th e q u a n tity demanded is still 23 0 0 0 snow m obiles p e r year. The relationship betw een q u a n tity d em an ded and price can be ex­ pressed in general as =

f(P > X i , x 2 ,... x „ )

w here Qd = th e q u a n tity dem anded, P = th e price, and x , to x „ = other factors. The m ost Im p o rta n t va ria b le in th is e q u a tio n is th e price o f the com­ m odity. If all factors except th e price are h e ld constant, the equation becomes a univariab le re la tio n sh ip b e tw e en q u a n tity demanded and price

This relatio nship Is called th e dem a n d fu n c tio n , o r sim ply the demand. The term s quantity demanded a n d dem and m ust be clearly distin­ guished. The q u a n tity d em an ded refers to a specific quantity desired per tim e period a t a g ive n price, such as 23 0 0 0 snow m obiles per year at $3 000, w hereas th e dem a n d Is th e fu n c tio n a l relationship between price and q u a n tity dem anded. The d e m a n d fu n c tio n has the following Im p o rta n t properties: as th e price o f th e c o m m o d ity falls, the quantity o f th e co m m o d ity dem an ded b y consum ers Increases and as the price rises, the q u a n tity dem an ded drops. T he dem a n d function of a com-

Fundamental Economic Concepte

485

fable 10.1 Demand Schedule Price ($ per unit)

Quantity Demanded (units per year)

1 000

55 000

2000

35 000

3000

23000

4000

15000

5000

10000

6000

9000

m odify Is usually specified in a table called a demand schedule, or by a graph representing a dem and curve, o r occasionally by a m athem atical function. A h yp o th e tica l dem and schedule fo r snowmobiles in the Ca­ nadian m a rke t is g ive n in "fable 10.1, and the corresponding dem and curve Is sh o w n in Figure 10.1. The slope o f th is ty p ic a l dem and curve is negative at every point, re­ flecting th e fu n d a m e n ta l relationship th a t exists between price and quantity dem anded: as th e price drops, all other things being equal, the q u a n tity dem an ded increases; and as the price rises, all other things b e ing equal, th e q u a n tity dem anded falls. Each point on the de­ mand curve represents a specific price-quantity com bination. At p o int A the price Is $2 0 0 0 per unit, and the corresponding quantity de­ m anded is 35 0 0 0 units per year. A t point B, the price is $3000 per unit, and 23 0 0 0 units per ye a r are dem anded at this price. As the price rises from $2 0 0 0 to $3 0 0 0 per unit, the quantity demanded Is reduced from 35 0 0 0 to 2 3 0 0 0 units per year. This change is represented by the a rro w o n th e Figure p o in tin g fro m A to B, and is called movement along the dem and curve.

486

Chapter 10

Fig. 10.1 Demand curve.

10.Z2 Shift In the Demand Curve *S

graphical representation of the unlvarlable price and quantity demanded of a commodity, r e m a , n constant. If factors other than the price of th** F n r ^ ? I ? 1? d ^ i ? 1 a n g e ' t h e d e m a n d curve responds to that change, m m a J? E T ' ” t h e C a n a d , a n population increases or if the average InZ™ * S e population Increases, the quantity o f snowmobiles deat any given price w ill increase, and the demand curve of

Price, $ p er unit

Fundamental Economic Concepts

Fig. 10.2 Shift in the d e m a n d curve caused by changes In factors other than the price

snowmobiles will shift to the right as shown In Figure 10.2. Similarly, a change In factors resulting In a reduction In the quantity demanded at any given price shifts the demand curve to the left

488

Chapter 10

10.23 Supply The quantity of a commodity supplied by all producers In a market, called the quantity supplied, Is expressed as a rate or flow of goods us­ ing dimensions of 'units per time,’ such as 10 000 tractors per week or 17 000 kg of chromium per month. Quantity supplied also depends on a number of factors: the price o f the commodity, the cost of produc­ tion, the state o f technology, etc The quantity supplied Is a function of these factors and, In general, it Is expressed as Qs = ^

y

v

y 2 ,...y m )

where Qs = the quantity supplied P = the price y , to y m = other factors The most Important factor In this equation is the price of the com­ modity. If all other factors are constant, the quantity supplied becomes a univariable function o f the price. This functional relationship, Qs = 's called the supply. A clear distinction must be made be­ tween quantity supplied and supply. The quantity supplied Is a specific quantity supplied per time period at a given price, such as 57 500 snowmobiles per year at $2 000, whereas the supply refers to the functional relationship between price and quantity supplied. The sup­ ply function o f a commodity is normally specified In a table, called the supply schedule, or by a graph, called the supply curve. Table 10.2 shows a hypothetical supply schedule for snowmobiles in the Canadian market and the corresponding supply curve Is shown in Figure 10.3. Table 10.2 Supply Schedule Price ($ per unit) 1 000 2 000 3 000 4 000 5 000 6 000

Quantity Supplied (units per vear) 25 000 38 500 48 000 57 500 65 000 72 000

489

Price, $ per unit

Fundamental Economic Concepts

The slope o f a ty p ica l supply curve Is positive at each point o f the curve, Indica ting th a t th e a m o u n t o f com m odity available fo r sale (the quantity supplied) Increases as the price Increases If all other factors re­ main unchanged. O n th e o th e r hand, the quantity supplied decreases If the price o f th e c o m m o d ity falls, all other things being equal. A point on the curve represents a specific price-quantity com bination. For ex­ ample, at p o in t C th e price o f a snow m obile Is $2 000, and the corre­ sponding q u a n tity supplied Is 38 500 units per year. As the price rises

490

Chapter 10

from $2 OOO to $3 000 per unit the quantity supplied Increases to 48 000 units per year. This change Is movement along the supply curve, and It Is Indicated by the arrow pointing from point C to point D In Fig­ ure 10.3.

10.2.4 Shift in the Supply Curve The supply curve Is the graphical representation o f the unlvarlable re­ lationship existing between the price o f a commodity and quantity of this commodity supplied, all other factors remaining constant. If a fac­ tor other than the price changes, the quantity supplied responds to the change. For example, if the cost o f production increases, the quantity supplied decreases, because a number o f manufacturers or producers will no longer be able to manufacture or produce the commodity at a profit under these conditions. The reduced quantity supplied by the manufacturers and producers results in a shift In the supply curve to the left. However, other changes m ight tend to increase the quantity supplied at any given price, which would result in a shift of the supply curve to the right (see Figure 10.4). For example, an extended strike at Sudbury In the nickel mines would have a considerable effect on the nickel supply in Canada, resulting In a shift to the left In the supply curve of nickel. On the other hand, the rapid advances that have oc­ curred in microprocessor technology have resulted In a shift to the right In the supply curve of computers.

1 0 2 5 Price For the purpose of determining the price o f the commodity at the mar­ ketplace, the demand and supply curves shown in Figures 10.1 and 10.3 are combined in figure 10.5. It can be observed that at a price of, say $4 000 per unit, only 17 000 snowmobiles per year would be de­ manded, but 53 000 units would be supplied. This results In a large ex­ cess supply or surplus of 36 000 unsaleable snowmobiles. Sellers normally cut prices when there is a surplus, In order to be able to sell their products. The large surplus at the $4 000 price would result In a considerable downward pressure on the price, and the $4 000 price would fall rapidly. On the other hand, at a price o f only $1 300 per unit, 52 000 units per year would be demanded, but since only 30 000 units per year would be supplied, there would be an excess demand or shortage of 22 000 units. Buyers norm ally bld prices up If there Is a shortage of goods. The shortage at the $1 300 price would result in a

4P1

Price, $ per unit

Fundamental Economic Concept*

Fig. 10.4 Shift In the supply curve caused by changes In factors other than the price

considerable upward pressure on the price, and the $1 300 price would rise quickly. W h at Is the stable (equilibrium) price of this com­ modity and how can this price be determined? At the equilibrium price dearly neither shortage nor surplus can ex­ ist Therefore, the quantity demanded at the equilibrium price must be exactly equal to the quantity supplied at that price. There is only one price that satisfies this condition, the price corresponding to the point of Intersection o f the supply and demand curves (point E In Rgure 10.5).

Price, $ per unit

492

Fig. 10.5 The equilibrium point. D =dem and curve S = supply curve E = equilibrium point Pe = equilibrium price Q e - equilibrium quantity

Fundamental Economic Concepts_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

453

The point where the demand and supply curves Intersect Is called the equilibrium point, and the corresponding price and quantity are termed equilibrium price and equilibrium quantity. In Figure 10.5 the equilibrium price Is $1 900 per unit and the equilibrium quantity Is 37 000 units per year. At prices above the equilibrium price, there is a surplus and a downward pressure on the price. At prices below the equilibrium price, there Is a shortage and an upward pressure on the price. The price moves toward the equilibrium price in a stable market free from governmental or monopolistic interference. Once the equi­ librium price is reached, It remains stable unless disturbed by changes in market conditions (shifts In the demand or supply curves). The equilibrium point moves if there is a shift In either demand or supply. If there Is a shift to the right In the demand curve, a shortage develops at the original equilibrium price, creating an upward pres­ sure on the price, and the equilibrium point moves upward along the supply curve (see Figure 10.6). At the new equilibrium point, E’ , the equilibrium price is higher and the equilibrium quantity is larger than at the original equilibrium point, E. On the other hand, a shift to the left in the demand curve causes both equilibrium price and equilibrium quantity to fall. A shift to the right in the supply curve (see Figure 10.7) results in a surplus at the original equilibrium price and, therefore, the equilibrium point moves downward along the demand curve from point E to point E*. The new equilibrium price, P ' c , Is lower, and the equilibrium quantity, Q * £ , Is higher than the original values of P£ and Qe . A shift to the left in the supply curve causes the equilibrium price to rise and the equilibrium quantity to fall.

494

Price, $ pe r unit

Chapter 10

Fig. 10.6 The effect of rising dem and on the equilibrium price. As the demand rises (the demand curve shifts to the right from position D to position D’l, a shortage develops at the original equilibrium price PE , and the equilibrium point moves from point E to point E*, resulting In a higher equilibrium price, P*e , and a higher equilibrium quantity, Q*t .

496

Price, $ per unit

Fundamental Economic Concepts

Fig. 10.7 The effect of rising supply on lhe equSbrium price. As the supply rises (the supply curve shifts to the right from position S to position S*), a surplus develops at the original equilibrium price P£ , and the equilibrium point moves from point E to point E’ , resulting In a lower equllbrium price, P*E , and a higher equilibrium quantity, Q’ £ .

Price, $ per unit

Fig. 10.8(a) The effect of a shift In supply: Steep dem and curve As the supply curve shifts from S to S‘, the equilibrium point moves from E to E'.

10.Z6 Elasticity of Demand As a result o f a shift In supply, the point o f equilibrium moves and the equilibrium price changes. For example, In Figure 10.8(a), a shift In sup­ ply causes the equilibrium price to drop sharply. However, In Figure 10.8(b), the same shift In supply results In a small drop In equilibrium price. Ih e change In equilibrium price and equilibrium quantity resulting from a shift In supply is clearly dependent on the magnitude of the shift In supply, and also on the shape o f the demand curve. For exam-

Price, $ per unit

Fundamental Economic Concepts

Fig. 10.8(b) The e ffe c t o f a shift In supply:Flat d e m a n d curve As the supply curve shifts from S to S*, the equilibrium point moves from E to E*.

pie, In Figures 10.8(a) and 10.8(b), the supply curves are shifted by ^entical amounts; however, the resulting changes i n - P and equilibrium quantities are different because of ^ e r e n ^ _ o „ | (u shape of the demand curves. Mathematicians and engineers generally characterize the shape o f a curve by Its slope and mists characterize the shape o f the demand curve by Its el city. The elasticity o f demand (or price elasticity), which Is shape of the demand curve, measures the responseof the quantttydemanded to the change In price. It is defined as the ratio of the per

498

Chapter 10

age change in quantity demanded to the corresponding percentage change In price. E '

» Q, /

AQ

- - ( A Q » )P P (AP)Q a

where Ea = the elasticity of dem and P = the average price Q, = the average quantity dem anded AP = the change in price AQd = the change in quantity dem anded The purpose o f the negative sign in the equation Is to make the de­ mand elasticity positive (since the slope of the demand curve is nega­ tive). The elasticity o f demand is norm ally different at every point of the demand curve. Demand elasticity may vary In magnitude from zero to Infinity. The demand is said to be elastic if Ea >1, and Inelastic If Ed < 1.

Example 10.1 Determine the elasticity of demand for snowmobiles at prices of $1 500 and $3 500. The demand schedule for snowmobiles Is listed below: Price $ Quantity Demanded (units per year) 1000

55 000

2000

35 000

3000

23 000

4000

15 000

5000

10 000

Fundamental Economic Concepts

499

At a price o f $1 500 AP= $2 0 0 0 -$ 1 0 0 0 = $1000 Q, = (5 5 000+35 000)/2 = 45 000 AQ, = 35 0 0 0 - 55 0 0 0 = -2 0 000 and therefore ‘

(-20000)1 500 1000(45 000) = ° ' 6 7

At a price of $3 500 AP= $ 4 0 0 0 -$ 3 0 0 0 = $1000 Qd = (23 000+15 000)/2 =19 000 AQd = 1 5 0 0 0 - 23 0 0 0 = -8 0 0 0 and (-8 0 0 0 )3 500 = - - ---- ----- -------r=1 .47 1000(19000) Therefore, the demand curve Is Inelastic In the vicinity of a price of $1 500, but elastic In the neighbourhood of a price of $3 500. Most demand curves are elastic at the high price end, Inelastic at the low price end, and have an elasticity of one at a point between.

102.7 Total Expenditure The total amount spent on a commodity by all consumers In the mar­ ket is called the toto/ expenditure, and It Is expressed as T=Q P where T = the total expenditure Q = the quantity sold and P = the price For obvious reasons the total expenditure of consumers, which is the total revenue o f producers, is quite Important The total expenditure

500________________________________________________________________ Chapter 10

generally changes as th e e q u ilib riu m p o in t m oves because o f a shift In e ith e r supply o r dem and. Let us n o w exam in e th e change In to ta l e xp e n d itu re resultlngfrom a m ovem ent o f th e e q u ilib riu m p o in t a lo n g th e dem an d curve because o f a sh ift in supply. As th e price rises a lo n g th e dem an d curve from PA to PB (see Figure 10.1), th e change in to ta l e xp e nditure Is

AT = Qb Pb -Q a Pa

= (Qb - ^ ) / ,b -Q .(^-P 8 ) = &QPb +APQa

w h e re AQ = Q 8 - Q 4 Is th e c h a n g e In q u a n tity dem anded and AP = PB - P A is th e change in price. In term s o f average values AT = A Q (P + A P /2 ) + A P (Q + A Q /2 ) w h ere th e average price is P = (P , + P 8 ) /2 , a n d th e average quantity Is Q = (Q A + Q B y/2, and since AP/2 < P a n a AQ /2 < Q, w e have the follow­ in g a p p ro xim ate re la tio n : AT = A Q P + A P Q ( A Q P "] = APQ 1+ APQ or

/

D u rin g a m o ve m e n t a lo n g th e d e m a n d curve fro m po in t A to point B (see Figure 10.1), th e change In price (i.e., th e q u a n tity AP) Is positive; therefore, th e change In to ta l e x p e n d itu re Is also positive If Ed < 1; zero If Ed = 1; and neg ative If Ed > 1. D u rin g a m o v e m e n t a long the demand curve in th e o p p osite d ire ctio n , th e price drops and AP Is negative; therefore, th e change In to ta l e x p e n d itu re Is neg ative If Ed < 1; zero If Ed =1; and positive if Ed >1 (see T^ble 10.3).

Findanwntal Economic Concepts

501

Table 10.3 The change In total expentiture during a movement along the demand curve Price

Quantity

Rises

Decreases

Falls

Increases

Elasticity of Demand Total Expenditure Elastic, Ea >1

Decreases

Unitary, Ea =1

Unchanged

Inelastic, Ed 1

Increases

Unitary, Ed =1

Unchanged

Inelastic, Ea 1, and Inelastic If E, < 1.

Fundamental Economic Concepts

503

1 0 2 3 Price Control The gove rn m en t m a y in te rfe r w ith the operation o f the free m arket system by setting th e m in im u m o r m axim um price o f goods and serv­ ices sold. The effect o f this g overnm ent action on production and con­ sumption w ill n o w be exam ined by studying the supply-demand interaction. Price Celling The m a xim u m price o f goods o r services m ay from tim e to tim e be set by the g o v e rn m e n t This action usually takes the form o f rent con­ trol, price control, etc. The price ceiling (maximum price) has little effect on the m arke t if it is set abo ve the equilibrium price, since the com­ m odity w ill co n tin u e to be traded at its equilibrium price (below the celling). If, on th e o th e r hand, the price ceiling Is set below the equilib­ rium price (see Figure 10.9), th e quantity supplied at the ceiling price (quantity Q s ) w ill be less th a n the quantity demanded at the price, quantity Q a , and a sh ortage develops. In a free m arket w ith out price control, th e sh ortage q u a n tity, Qd - Q s (see Figure 10.9) w ould disap­ pear after fo rc in g th e price to rise to its equilibrium level. However, as the price Is n o t a llo w e d to rise above the ceiling, the shortage win re­ main perm anent. The q u a n tity supplied at the m axim um price would, In a free m arket, be n o rm a lly sold at a price o f PB (see Figure 10.9) above the ceiling. A black m arket usually develops under these cir­ cumstances fo r th e c o m m o d ity to be sold above the legal m axim um price. The black m a rk e t price is often substantially higher than the equilibrium price th a t w o u ld exist in the absence o f price controls. M inim um Price The m in im u m price o f goods and services m ay also occasionally be set by the gove rn m e n t. G overnm ent-controlled m inim um wage falls Into this category. The m in im u m price has little effect on the m arket If It Is below th e e q u ilib riu m price o f the com m odity, since the com m od­ ity w ill contin ue to be traded at Its equilibrium price above the m ink mum price set. If, o n th e o th e r hand, the m inim um price Is set above the equ ilibrium price (see Figure 10.10), a surplus develops. The quan­ tity o f the c o m m o d ity supplied at the m inim um price quantity Q s is larger than th e q u a n tity dem anded at this price. In the absence o f price control, th e surplus, q u a n tity Q s - Q d , on Figure 10.10, w ould dis­ appear as th e price fe ll to Its equilibrium level. However, as the price is not allow ed to fa ll b e lo w its controlled m inim um value, the surplus

504

Price, $ per unit

Chapter 10

Fig. 10.9 Price ceding. Pe = equilibrium price Pc = price celling (maximum price) Q£ = equilibrium quantity Q d -Q s = shortage at the price celling

Fig. 10.10 Minimum price. Pf = equilibrium price Pc - controlled minimum price O f = equilibrium quantity Qs - Q

tf

= surplus at the minimum price

506

would be permanent. A minimum wage law, fo r example, tends to raise the wages of those who can keep their Jobs, but It reduces the number o f people employed, and creates a surplus of labour.

10X10 Price Support The periodic (or seasonal) fluctuation in the supply of certain com­ modities, mainly agricultural commodities, results In widely fluctuat­ ing prices In the free market. Since wide fluctuation In the price of basic commodities is generally undesirable from the point of view of both producers and consumers, the government often intervenes In the op­ eration of the free market in order to stabilize the price of basic com­ modities. Price fluctuation can be reduced (or eliminated) by Introducing a price support scheme. Suppose that the government is w illing to support the price of a commodity at its equilibrium price level. The production of this com­ modity might fluctuate. In one particular period, production might rise above the equilibrium quantity to quantity shown in Figure 10.11, and a surplus o f quantity Q, - Q e is created at the equilibrium price level. This surplus would normally depress the price to P, below equi­ librium price level. However, if the government intervened by purchas­ ing the surplus from producers at the equilibrium price, the producers would be able to sell all o f their product at the equilibrium price. The equilibrium quantity would be purchased by the consumers and the surplus quantity (Q, - Q f ) by the government. In some other period, production might fall below the equilibrium quantity, to quantity Qr The price would then normally rise to P2 . However, if the government intervened by selling the quantity Qf -Q 2 from Its inventory at the equilibrium price, P£ , it would prevent the price o f the commodity from rising. The equilibrium price, therefore, would always be maintained and the demand of consumers satisfied. Price support schemes generally w ork well, provided that the price Is supported at the equilibrium level. If, however, the price Is supported above the equilibrium level the government subsidises producers, pro­ duction increases, consumption drops, and the inventory held by the government grows. If, on the other hand, the price Is supported below the equilibrium price level the government subsidises consumers, pro­ duction drops, and consumption increases. For example, supporting oil prices in the home market below the International equilibrium price of the commodity for an extended period may cost billions of dollars annually to the government.

Fundament* Economic Conotpto

Price, $ per unit

507

Fig. 10.11 Price support. S - supply curve D - demand curve

SOS

Chapter 10

10.3 Production The work of engineers Is dosely related to the production of a wide va­ riety of goods and services. Engineers are frequently Involved In Im­ portant production dedslons to determine the best way to use available plant and machinery, w hat new plant or facility to build and what machinery to acquire, what new technology to Introduce, etc To be effective in the area o f production dedsion making, It Is necessary to be fam iliar with the basic prlndples o f the theory o f production.

103.1 The Production Function One of the primary objedlves o f an engineering company Is to use the most economically effident method o f produdlon. Using the produc­ tionfunction, one can determine which factors need to be modified In order to achieve produdlon at optimum level. Rrms that are engaged in produdlon use manpower, machines, equipment, raw materials, etc., to produce goods and services. Man­ power, machine, equipment, raw material, and other elements needed for production are called inputs Into the produdlon process. An Input Is either fixed or variable. A variable input is one that can be changed (In­ creased or reduced) on short notice In response to a change In market conditions. Fixed inputs, on the other hand, cannot be changed easily. Fadory buildings, power plants, oil refineries, and large computer Instal­ lations are fixed inputs. Labour, machinery, tools, and raw materials are variable inputs. From the long-term point o f view, all Input Is variable; everything can be changed given suffldent time. The size of the labour force or the quantity of raw material supply may be altered In a few weeks In response to changes In demand, but a change In the capacity of a nudear power plant might take several years to Implement. Conse­ quently, some Inputs are considered fixed for short-run decisions, even though all Inputs are variable for long-run decisions. The firm uses the inputs (fixed and variable) In Its produdlon pro­ cess to produce goods and services called outputs. The relationship be­ tween inputs and outputs Is the production function. The produdlon function Is an empirical relationship normally specified In the form of a table, a graph, or, In exceptional cases, a mathematical fundion. The produdlon fundion is generally a multivariable fun dion having at least tw o inputs (labour and capital). With the help o f the production function, one Is able to determine the best combination of Inputs to produce a spedfled output.

Fundamental Economic Concepts

509

1 0 3 2 Th e U nivariable Production Function A typical unlvarlable production function Is listed in the first two col­ umns of Table 10.4. This function may represent the activity of an en­ gineering company specializing In the production of automatic packaging machines. The variable Input Is labour, measured In thou­ sands of labour hours per year, the fixed input Is capital In the form of buildings, equipment, and machinery. The output, or total product, is the number of machines produced by the company yearly. The total product divided by the variable Input (l.e., the number of machines produced per thousand labour hours in the present case) Is the average product, or productivity. It can be expressed as AP = — L where AP = the average product TP = the total product (output) L - the labour (Input) table 10.4 The Production Function (1)

(2)

(3)

80

67

0.84

(4)

Values of the average product were computed and listed In Column 3 of table 10.4. The table shows that the average product Increases Ini-

51®_ _ _ _ _ _ _ _ _ _ _ _ ._ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ Chapter 10

tially, reaches a maximum value, and then begins to decrease. This Is the general behaviour of a typical production function. Economists ex­ plain this behaviour using the 'law ’ o f diminishing returns which states that if Increasing quantities of a variable factor are added to a given quantity of a fixed factor, a point w ill be reached after which each addi­ tional unit of variable factor will result In a smaller increase In output than did the previous unit (l.e., the average product begins to de­ crease). Figure 10.12 shows a typical production function. Coordinates of a point on the curve are corresponding input and output values. The av­ erage product at a point on the curve is equal to the slope of the ray drawn from the origin to the point. For example, the average product TP at point C Is - A which Is the slope o f line OC. The average product Increases until Its maximum value is reached at point B. This point is called the polntofdlmlnlshlng average productivity. The average product drops if the input is increased beyond this point. The marginal product, or Incremental product, is the change in total product resulting from one unit increase in the variable input (i.e., the derivative of the total product), and may be expressed as A£ where MP = the marginal product ATP = the change In total product (output) A£ = the change In labour (variable Input) Values of the marginal product were computed and listed In Column 4 of Table 10.4 for the production function listed In the table. In Figure 10.12 the marginal product is equal to the slope of the tangent to the production function curve. It can be observed that the marginal prod­ uct is positive and increasing (with increasing Input) if 0 £ L . At point C the marginal product reaches Its maximum value. In the range of Lc L La , the marginal product Is positive but decreasing in value (with Increasing Input). At point A the total product Is maximum and the marginal product Is zero. It should also be observed that at point B the average product is maximum and the average and marginal prod-

ciindjmental Economic Concepts

O u t p u t , u n its p e r

511

Input, 1O’ labour hours per year IP = total product curve AP = av erag e product curve MP = marginal product curve

Fig. 10.12 The production function.

512

uct values are equal. Furthermore, In the region L > LA , the marginal product Is negative and the total product decreases with Increasing In­ p u t Point C, where the marginal product Is maximum, is the point of di­ minishing marginal productivity. Three stages of production are normally distinguished on a typical production curve such as the one shown In Figure 10.12. In the first stage o f production, productivity increases w ith increasing variable in­ p u t indicating that the fixed Inputs are not fully utilized. It is advanta­ geous to Increase production In this stage at least until the limit of the region, point B, is reached. In stage three both the total product and the productivity decrease as the variable input Is increased. This stage o f production is rather inefficient. Production should not be sustained in this stage. Production is normally performed in the second stage of the pro­ duction function, if the desired output is not available in this stage, the fixed inputs should be changed to alter the shape of the production function, and to make the desired output available in the second stage o f production. The characteristics o f the three production stages are listed in Table 10.5.

Table 10.5 The Stages of Production Input

Total Product (Output)

Average Product (Productivity)

Marginal Product

First stage

0 < l< lB

Increases

Increases

Positive

Second Stage

L8 S £ < L 4

Increases

Decreases

Positive

Decreases

Decreases

Negative

Third Stage

A typical property of the production function is that beyond a cer­ tain point both the marginal and the average products diminish as the variable inputs are Increased, due to the “law" o f diminishing returns. One of the main objectives o f the firm Is to set up the best method of production. The best method o f production Is either the one that uses the smallest quantity o f inputs, or the one that costs the least to obtain the required ou tpu t The method which uses the fewest inputs to pro­ duce a specified quantity of output Is the technologically most efficient

Fundamental Economic Concepts

513

method o f p ro d u ctio n . The m ethod th a t costs the least to produce a specified o u tp u t Is th e economlcaltymosteffidentmethod o f production.

10.33 M ultivariable Production Function Production g e n e ra lly requires a num ber o f different Inputs. For exam­ ple, the firm engaged In producing autom atic packaging machines uses raw m ate rials (steel, plastic, rubber), finished goods (microproces­ sors, sensors, switches), m achinery (lathes, m illing machines, grinding machines), la b o u r (engineers, technologists, machine operators), etc, In the p ro du ction process. A ll o f these items are inputs to the produc­ tion. However, fo r th e purpose o f explanation, we w ill consider the simple case o f o n ly tw o inputs: labour and capital. The production function fo r tw o inputs Is expressed as: 7P = / ( / . , K ) where TP = the total product (output) L = th e labour (input) K = th e c a p ita l (input) This fu n c tio n is represented by a curved surface, and the contour lines o f this surface can be used fo r a two-dim ensional graphical repre­ sentation o f th e fu n c tio n . A contour line is defined by the specific rela­ tionship th a t exists betw een K and L fo r a given output quantity. This relationship, co rre sp o n d in g to a constant output and computed from the pro d u ctio n fu n c tio n , can be represented by a two-dim ensional curve (see A g ure 10.13). This curve is called the isoquant, or constant quantity curve, and th e o u tp u t corresponding to any point on this curve Is c o n s ta n t An iso q u a n t show s all alternative combinations o f inputs for produ cing a g ive n outp u t. For example, at point A, the required la­ bour Input Is La and th e capita l in p u t is K A . At point B, the labour input Is La and th e capita l In p u t K B . As w e m ove from point A to point B on the Isoquant curve, capita l Is being substituted for labour w ith the out­ put re m a ining unchanged. In the firs t tw o stages o f production, the m arginal product is posi­ tive, in d ica tin g a reductio n In o u tp u t as the Input Is reduced. Therefore, If there are tw o Inputs, as o n e Input Is reduced, the other must be in­ creased to keep th e o u tp u t constant. Thus, the slope o f the isoquant

Input, K

51*_________________________________________________________________Chapter io

Fig. 10.13 The isoquant curve.

Fundamental Economic Concepts

515

curve is negative. T he slope o f the isoquant curve Is called the marqinal rate o f substitution, and It Is expressed as M g 5

_ A K _ A7P/AL AL

\T P fb K

MPk

where MRS = th e m arginal rate o f substitution AK = th e c h a n g e in Input K AL = th e c h a n g e in Input L ATP = th e c h a n g e in output (total product) MP l = th e m arginal product In terms of input L

MPK = th e m arginal product in terms of input K A group o f iso quan t curves obtained from a production function and representing d iffe re n t p ro d u ctio n outputs form an isoquant map, see Figure 10.14. The iso q u a n t m ap is the contour map o f the three-dimen­ sional surface representing th e production function. Isoquants corre­ sponding to h ig h e r levels o f o u tp u t are further aw ay from the origin on the iso quan t m ap.

10.34 Cost o f Production The cost o f p ro d u ctio n is th e sum o f the costs o f all inputs. In the case of tw o Inputs (labo ur and capital) the total cost can generally be ex­ pressed as T C = p LL + p K K+FC where TC = th e tota l cost p L = th e unit cost of Input L (labour) p K - th e unit cost o f Input K (capital) FC = th e co st o f all fixed Inputs This e q u a tio n Is v a lid o n ly If the to ta l cost Is linearly dependent on the variable inputs, w h ich Is n o t always the case. However, fo r the sake o f s im p licity w e w ill consider only linear cost functions In this

Input, K TP = total product TP,