This chapter introduces the reader to the field of managerial economics and builds the foundation necessary for decision making under uncertainty. In the next section we examine the definition and scope of managerial economics in order to establish the nature and direction of the discipline you are about to study. This leads to a discussion of the use of models in managerial economics. We will see that models depicting the behavior of consumers and business firms can be of substantial assistance in decision analysis.
In the third section we consider the multiperiod nature of decision making. Decisions made now typically have cost and revenue implications not only for the present period but also for future periods. This necessitates the use of present-value analysis , also known as discounted cash flow analysis, to allow the proper weighting of future profits against present profits for effective decision making.
In the fourth section we introduce expected-value analysis to allow the proper evaluation of decision alternatives when the decision must be made under conditions of risk and uncertainty—the usual situation in the real world.
Which decision is optimal depends on the firm’s objective. In the fifth section of this chapter we consider the appropriate objective function for the firm and the decision criterion indicated under each of four different scenarios. Finally, we provide a brief summary of the chapter.
■ Definition: Managerial economics is concerned with the application of economic principles and methodologies to the decision-making process within the firm or organization. It seeks to establish rules and principles to facilitate the attainment of the desired economic goals of management. These economic goals relate to costs, revenues, and profits and are important within the business and the nonbusiness institution.
Profits are at the heart of the study of managerial economics. Profits are defined as the excess of revenues over costs. If this difference is negative, it is called a loss. For a nonprofit institution, an excess of revenues over costs is called a surplus, while a shortfall of revenues compared to costs is called a deficit. No economic organization, not even the United States government, can continue forever incurring losses or deficits. The objective of a business firm usually requires profit maximization over its time horizon, which may be short term or long term depending on the firm and its circumstances. Nonprofit institutions typically do not seek to make a surplus, but wish to spend their available funds to maximum effect. The decision problems facing business and nonbusiness institutions are therefore essentially similar, involving revenue enhancement (if possible) and cost control, with profit (or the avoidance of a deficit) seen as a measure of managerial effectiveness.
Some people think that profit is a dirty word, associating profit making with “profiteering.” In the normal course of economic activity, seeking profit is a thoroughly honorable activity. Without the promise of profit, there would be no private business firms, and all business activity would have to be organized by the government. History has shown us that this approach is not a superior alternative. Competition for profits among private business firms leads to the ongoing availability of high- quality goods and services at reasonable prices, to increasing productivity, and to continuing economic growth and well-being for the nation’s people. Profit should be seen as the reward for enterprise, the reward for risking capital, and the reward for superior insight and judgment concerning market conditions. Profiteering, or “gouging” the consumer when supply is limited, is typically not a good business strategy (whether or not it is immoral), because consumer ill will is likely to hurt the firm in the longer term.
The scope of managerial economics is quite wide and extends into what some might call accounting, marketing, and finance. There are economic issues of importance to managers in each of these areas, and each decision made will be a better decision if the economics of the problem is clearly understood. Decision making in the areas of revenue enhancement and cost control are the two major concerns of the managerial economist. Chapters 1 and 2 are concerned with effective decision-making methods. Revenue enhancement requires a clear understanding of consumer behavior and of the reactions of consumers to changes in prices, advertising and promotion, and product quality. Estimation and forecasting of demand are of critical importance for the practicing manager. Chapters 3, 4, and 5 are concerned with these issues. Effective cost control requires a clear understanding of cost behavior, the components of cost, the relevant costs for a particular decision, techniques to estimate costs, and cost forecasting. Chapters 6, 7, and 8 are concerned with these issues.
Pricing the firm’s products for optimal revenue enhancement requires a sound knowledge of the theory of markets, pricing under uncertainty, pricing new products, and pricing in markets where price quotes or bids are submitted without knowledge of rivals’ prices. Chapters 9, 10, 11, and 12 address these issues. Advertising and promotion decisions intended to enhance revenue are analyzed in Chapter 13. The quality of the firm’s products and the competitive strategy of the firm (typically involving a dis-
tinct tradeoff between low-cost production and revenue enhancement) is the topic of Chapter 14. Decisions to increase or reduce plant capacity in order to control costs or enhance revenues are covered in Chapter 15.
Thus managerial economics covers a range of topics of critical interest to the practicing manager. Although it is mainly economics, the managerial economics course is to some extent an interdisciplinary and integrative course in the business school curriculum, since it draws upon elements of accounting, finance, marketing, statistics, and quantitative methods for many of its concepts and analytical tools. The economic principles and methodologies of managerial economics are derived largely from microeconomics , the branch of economics that is concerned with the behavior of individual consumers, producers, and suppliers of capital, raw materials, labor, and services. In this book it is assumed that, while you may have taken courses in microeconomics, statistics, accounting, and so on, you are probably pretty rusty on the specifics. Thus each concept is developed from basic principles, presenting little difficulty for those who have no prior experience in these areas and allowing a quick review for those who have.
A review of basic calculus and analytical geometry (graphs), with extension to various optimization techniques, is provided in Appendix A at the end of this book. Your instructor may wish to cover this material at this point, or you may simply be asked to refer to it as needed throughout the course in order to better understand the material being covered.
In simple microeconomics, economists use models of consumer and firm behavior that assume full information, or certainty, about the shape and location of demand curves, cost curves, and so on. That is, they presume to know exactly how much will be demanded at each price, and exactly what the cost of production will be at that output level. In the real world, however, business firms typically operate under conditions of incomplete information, or uncertainty , and must estimate or forecast the quantity demanded at any price and the costs, based on the limited information they have at hand or can obtain by conducting information-search activity.
Managerial economics has evolved out of microeconomics to provide guidance for business decision making in an environment of uncertainty. In managerial economics we nevertheless examine several microeconomic models that assume full information, as a prelude to the incorporation of uncertainty into the decision problem. In many cases we can effectively estimate the demand and cost data necessary to apply the microeconomic model, but in other cases the information search cost is prohibitive and we must look for an alternate solution procedure.
In economics, the distinction is made between those areas that are “positive” and those areas that are “normative.” Positive economics is descriptive: it describes how
economic agents or economic systems do operate within the economy or society. Normative economics, on the other hand, is prescriptive: it prescribes how economic agents or systems should operate in order to attain desired objectives.
Managerial economics is primarily normative, since it seeks to establish rules and principles to be applied in decision making to attain the desired objectives. But managerial economists must always be mindful of the actual practices in the business or institutional environment. For example, if firms choose their price level by applying a markup to their direct costs rather than by equating marginal revenue and marginal costs as implied by microeconomic principles, managerial economics should be concerned with determining the optimal level of the markup rather than attempting to persuade decision makers to use the marginalist principles. The approach taken in this textbook is to integrate business practice with economic principles. Thus, business practices, even if divergent from the strictures of normative microeconomics, are discussed in terms of the microeconomic principles. By reference to the microeconomic principles, the business practice can be evaluated in terms of its efficiency in attaining the desired objectives.
■ Definition: A model is defined as a simplified representation of reality. Models abstract from reality by ignoring the finer details which are not essential to the purpose at hand. They therefore concentrate on the major features and interrelationships existing among those features without obscuring the picture with less important details. 1
Symbolic models use words and other symbols to represent reality, and they include descriptive speech, diagrams, and mathematical expressions. Descriptive speech usually simplifies or abstracts from reality, such as in the statement “A horse is a four-legged animal with a long brushy tail.” Such a statement is a symbolic model of the members of the equine family, giving the general idea of what a horse is without mentioning the complexities involved in the physical or psychological makeup of a horse or the differences between a Thoroughbred and a Percheron. Diagrams similarly represent a situation of reality by means of lines, shading, and other features, and they abstract from the finer details. Mathematical models are usually further removed from reality and are used because reality is very difficult to depict visually or verbally or because it involves interrelationships that are extremely complex.
In this book we shall use mostly verbal and diagrammatic models to explain concepts and to analyze decision problems, although some mathematical models will be used, mainly in footnotes where these are appropriate. Diagrams (and mathematical expressions) are a very efficient means of representing reality; they are quicker to make and comprehend (once you understand them), and there is less ambiguity in
'This discussion relies on the excellent discussion in I. M. Grossack and D. D. Martin, Managerial Economics (Boston: Little, Brown & Company, 1973), pp. 5-10.
such models, as compared to descriptive speech. A diagram (or mathematical expression) states precisely and concisely the relationships assumed to exist among facts, observations, or variables. If these assumed relationships are inaccurate, the inaccuracies will be exposed by later testing the model against reality; and our model may be modified to reflect the feedback received from this testing.
The Purposes of Models. There are three main purposes of models. First, models are used for pedagogical purposes: They are a useful device for teaching individuals about the operation of a complex system, since they allow the complexity of reality to be abstracted to a framework of manageable proportions. By excluding from consideration the minor details that do not affect the basic features or relationships that are at issue, the model allows us to deal with the central issues of the problem without the added complexity of relatively minor influences.
Second, models are used for explanatory purposes. They allow us to relate observation of objects and subsequent events in a logical fashion. The assumed link between the objects and the events may be tested for its authenticity by reference to real-world situations. If an observation that contradicts the link is made, then this link is refuted, while the link is supported (although not proved) by the repeated observation of supporting evidence. Models may aid the researcher in discovering relationships that exist between and among variables. If a high correlation is found to exist between two variables, the researcher may then attempt to discover the reasons for this relationship, construct a model incorporating the assumed relationship, and test for this relationship with future observations.
Third, models are used for predictive purposes. The predictive value of a model is usually based upon the ability of that model to explain the past behavior of a system, and it uses this past relationship to predict the future behavior. Suppose we have found that two events are causally related, such as low winter temperatures and high consumption of heating oil; we may predict increased consumption of heating oil the next time we observe temperatures falling. The model should specify the extent to which heating oil consumption is related to the temperature so that we may predict with some degree of accuracy the consumption of heating oil during a particular cold spell. This model will remain a good predictor as long as the relationships it represents stay constant. For example, if many households were to change to electric heating, the model would no longer represent reality as well as it did, and its predictions would be less accurate than before. A new or revised model would then be called for.
Friedman has argued that it is not necessary that the assumptions underlying a predictive model be valid and testable. 2 For example, a spurious model of the behavior of a sunflower, which twists around to follow the sun each day, is that the sunflower swivels at the base in response to a heat sensor located in the petals. This model,
although simplistic, would nevertheless allow us to use the direction which the sunflower is facing to predict the location of the sun on a partly cloudy day.
For predictive purposes such a model could be quite useful, although for pedagogical and explanatory purposes such a model has no virtue. Even for predictive purposes, however, we may be somewhat cautious and skeptical about the model’s ability to predict accurately on its next test. If the relationship postulated is obviously false and the real determinants are unknown, the latter could change and our prediction would turn out to be wrong. For example, the installation of a bright light nearby might cause the sunflower to look at the light all day and to wrongly predict the location of the sun. Thus we need to be more cautious in using such a model to predict, since it may fail as a predictor at any time if the underlying (true) relationship changes without our knowledge or observation.
How Do We Evaluate Models? It should be clear from the preceding discussion that a model must be evaluated with an eye to its purpose. If a model is intended for pedagogic purposes, then it must be evaluated on this basis. For example, simple models of oligopoly (an oligopoly is a small group of competing firms) do not perfectly explain the actual behavior of real firms in the real world, yet they do introduce students to the problems of mutual interdependence in business situations. Criticisms of such models as being too simplistic or unrealistic are thus unwarranted to the extent that such models best introduce these concepts to the student.
Models designed for explanatory purposes must be evaluated on the basis of how well they explain reality. If an observation is generated that is at variance with the model, then that model is refuted as an explanatory device. If there are two or more separate models that purpor/t to explain a sequence of events, the best model is the one that most accurately depicts the important variables and relationships between and among these variables.
Models designed for predictive purposes must be judged by the accuracy of their predictions in subsequent tests. A predictive model is superior to an alternate predictive model if its predictions are more accurate more of the time. As the situation being depicted changes or evolves, we would expect existing models to become less accurate, since they must be modified to represent the evolving or changed situation.
In all cases when evaluating models, it is important to keep in mind that the model is a simplified representation of reality and is thus intended to represent the general features of reality rather than the specific features of a particular instance in reality. The model is an abstraction that loses some of the finer points and details; consequently, the predictive or explanatory power of a model will generally not be exact for any particular instance, because variables and relationships have been excluded from the model for the sake of simplicity or expedience. For example, a model may predict that firms will increase the size of their inventories of raw materials when the price of those raw-material components is seasonally reduced. If such a model abstracts from the cash-flow situation of the firm, it may imperfectly predict the behavior of certain firms that are facing a liquidity problem at that particular time.
In the following chapters you will notice the considerable use of graphical models to explain and analyze economic phenomena. Understanding graphs requires a basic knowledge of analytical geometry and calculus, and specifically, an understanding of such concepts as slopes, intercepts, linear functions, curvilinear functions, convexity, concavity, maximums, and minimums, among others. If your analytical geometry and basic calculus is rusty, I’d suggest you take half an hour to read about functions, graphs, and derivatives in Appendix A. Doing so may save many hours later, since graphs are an efficient tool for both understanding and expressing managerial economics. Someone once said “a picture is worth a thousand words.” A graph is worth at least as much.
Jargon: The Use of Verbal Models for Efficient Communication. The study of managerial economics involves a lot of specialized words, or jargon. Since the start of this book, you have been bombarded with “new” words, in the sense that these words convey a special meaning or connotation. Even words like costs, revenues, profits, labor, capital, and risk have special, more precise, meanings in the jargon of economics. Each of these jargon words and many others are formally defined at appropriate points in this book. We use a lot of jargon in managerial economics (as do all other disciplines of inquiry) to facilitate our discussion of complex phenomena.
Jargon words are really only verbal models of things or phenomena. They allow us to communicate with each other more efficiently, since they offer a concise and precise means of conveying the information we wish to convey. For example, “utility” is substantially quicker to say than is “the psychic satisfaction which a consumer expects to derive from the consumption of a product or service.” And since jargon words are definitional terms, there should be no ambiguity about the precise meaning of these words. Thus communication between economists, between managers, or between you and me through the medium of these printed pages is facilitated and greatly enhanced by the use of jargon.
Many decisions involve a flow of revenues extending beyond the present period. In choosing between or among various alternatives, it is important to distinguish between revenues that are received immediately and those that are received at some later date. A dollar received today is worth more than a dollar received next year, which in turn is worth more than a dollar received the following year. The reason for this is that a dollar held today may be deposited in a bank or other interest-earning security, and at the end of one year it will be worth the original dollar plus the interest earned on that dollar. Hence, if the interest rate is, say, 10 percent, a dollar today will be worth $1.10 one year from today. Looking at this from the reverse aspect, a dollar earned one year from today is worth less than a dollar that is held today. Thus the future earnings must be discounted by the interest rate they could have earned had they been held today.
In this section we introduce and examine the concept of the present value of cash flows that occur beyond the present period. Both revenues to be received and costs to be incurred in the future must be reduced to present-value terms for proper evaluation of decision alternatives. Net present value is found by subtracting the present value of costs from the present value of revenues and can be expressed alternatively as the present value of profits. The major convention used for discounting future cash flows to their present value is to treat all flows as if they arrive, or are incurred, at the end of year one, year two, and so on. We look also at other cash flow patterns, such as the receipt of funds as an annuity, or on a daily, weekly, or monthly basis.
■ Definition: The future value in one year of $1.00 presently held is equal to $1.00 plus the annual rate of interest times $1.00. That is,
FV = PV( 1 + r)
where FV denotes future value, PV denotes present value, and r is the rate of interest available.
Example: If the interest rate is 10 percent, r — 0.1, the future value of a principal sum of $1.00 is
FV = $1 (1 + 0.1)
= $ 1.10
Now suppose we could leave this money in the bank for a second year, also at 10 percent interest. The future value would be
FV = $1.10 (1 Hr 0.1)
= $ 1.21
If a third year were possible, we would find
FV = $1.21 (1 + 0.1)
= $1,331
and so on for future years. Note that the principal sum each year was simply multiplied by (1 + r). In effect, the dollar was multiplied by (1 + r) initially, then the resulting product was multiplied by (1 + r), and then the product of that was multiplied by (1 + r) again.
That is, for the three-year deposit
FV = PV(1 + r) (1 + r) (1 Hr r)
which simplifies to
FV = PV( 1 + r) 3
Generalizing for any number of periods into the future, we have
FV=PV(l + r)" (1-1)
where n represents the number of years into the future that the principal sum plus interest will be returned.
Example: Suppose we lend $2,500 for a period of five years at 8.5 percent interest. What is the future value of the presently held $2,500?
FV = $2,500 (1 + 0.085) 5 = 2,500 (1.085) 5 = 2,500 (1.50366)
= $3,759.15
Thus $2,500 saved at 8.5 percent for five years will return the saver $3,759.15 at the end of the five-year period.
Note: The above process is known as compounding the principal sum plus annual interest over the period of the loan. It tells us that $2,500 held today is worth $3,759.15 in five years if we can obtain 8.5 percent interest compounded annually. The compound factor which we used to multiply the $2,500 to obtain $3,759.15 was 1.50366. This compound factor effectively says that $1.00 today is worth $1.50366 in five years if the interest rate is 8.5 percent.
Now let’s do it in the reverse direction. The present value of a. future value can be found by manipulating equation (1-1). Dividing both sides by (1 + r)" we find
FV
(1 + r)"
( 1 - 2 )
Example: Although we already know the answer, let’s find the present value of $3,759.15 available in five years during which the available interest rate is 8.5 percent.
$3,759.15
py =- - -
(1 + 0.085) 5
3,759.15
~ 1.50366
= $2,500.00
What we have just done is to discount $3,759.15 (future value) back to present- value terms and demonstrate that the discounting process is simply the inverse of the compounding process. Whereas future value equals present value multiplied by the compound factor, present value equals future value divided by the compound factor. Alternatively, let us call the reciprocal of the compound factor the discount factor and say equivalently that present value is equal to future value multiplied by the discount factor. The reciprocal of 1.50366 is 0.66504, which is the discount factor when the interest rate is 8.5 percent. (Note that $3,759.15 multiplied by 0.66504 equals $2,500.00.)
■ Definition: The present value of a sum of money to be received or disbursed in the future is the value of that future sum when discounted at the appropriate discount rate. Alternatively, it is the sum which would grow to that future sum when compounded at the appropriate interest rate.
The decision maker must choose the rate of discount quite carefully, since use of the “wrong” discount factor could cause a poor decision to be made in cases where the time profiles of future profit streams associated with alternative decisions differ markedly. The appropriate discount rate is the “opportunity” discount rate.
■ Definition: The opportunity discount rate is the rate of interest or return the decision maker could earn in his or her best alternative use of the funds at the same level of risk. Note that we require that the alternative use of the funds must involve the same level of risk or uncertainty, since many other alternative uses of the funds will be more or less risky or uncertain and are thus not strictly comparable with the present proposal.
Example: Suppose that a firm intends to invest $10,000 in an expansion of its facilities but might otherwise invest the funds in a bond issue, which is considered to have similar risk and which would pay 12 percent interest compounded annually. The opportunity discount rate to be used when evaluating the future returns from the project under consideration is, therefore, 12 percent.
How do we ascertain whether the alternative investment or savings opportunities have similar risk? This issue is discussed in detail in Chapter 2. It is enough to say here that the risk in any decision lies in the dispersion of possible outcomes. Finding the equal-risk, best-alternative use of the funds, therefore, involves finding the alternative savings and investment opportunities which have the same or very similar dispersions of possible outcomes, and noting the highest rate of interest or rate of return on investment available within this subset. If the project or decision is risk free, the appropriate comparison would be treasury bills or guaranteed bank deposits, which are also risk free. More likely, the project or decision does involve risk, and we must look around
for other investment opportunities that involve similar risk and note the highest rate available among these investments in the same risk class. That rate is the opportunity discount rate to be used when discounting future cash flows associated with the decision under consideration.
It is important to see that the higher is the opportunity rate of interest (or opportunity discount rate), and the longer the time period, the lower is the discount factor. In Table 1-1 we show the discount factors associated with several different opportunity interest rates and several different periods of time. Each discount factor is calculated using the expression 1/(1 + r)", and you can calculate the discount factor appropriate to any other opportunity interest rate and time period using the same expression. Note that the discount factor is inversely related both to the length of the time period and the opportunity interest rate. Discount factors effectively tell you the value of $1.00 at the end of a given period for any opportunity interest rate. Thus, at 10 percent opportunity interest rate, $1.00 is worth $0.9091 if received after one year; $0.6209 if received after five years; and $0.0923 if received after twenty-five years. Similarly, with a 20 percent opportunity interest rate, $1.00 is worth only $0.4019 if received after five years; $0.0649 after fifteen years; and only a fraction over one cent if received after twenty-five years!
Example: Which would you prefer, $5,000 now, $20,000 in ten years, or $100,000 in 25 years? If the opportunity discount rate is 10 percent, the present values of these alternatives are $5,000, $7,710, and $9,230, respectively. Thus, you would prefer to take $100,000 in 25 years over the other opportunities. But note that if the opportunity discount rate is 15 percent, the present values become $5,000, $4,944, and $3,040, respectively. In the latter case it is preferable to take the $5,000 now. This example demonstrates the powerful effect that higher opportunity discount rates have upon future cash flows, and it shows that the selection of the appropriate discount rate is critical for the effective evaluation of decision alternatives.
Table captionTABLE 1-1. Discount Factors for Several Different Opportunity Interest Rates and Time Periods
Time Period (years hence) | OPPORTUNITY INTEREST RATE (%) | |||
5 | 10 | 15 | 20 | |
0 | 1.0000 | 1.0000 | 1.0000 | 1.0000 |
1 | 0.9524 | 0.9091 | 0.8696 | 0.8333 |
2 | 0.9070 | 0.8264 | 0.7561 | 0.6944 |
3 | 0.8638 | 0.7513 | 0.6575 | 0.5787 |
4 | 0.8227 | 0.6830 | 0.5718 | 0.4823 |
5 | 0.7835 | 0.6209 | 0.4972 | 0.4019 |
10 | 0.6139 | 0.3855 | 0.2472 | 0.1615 |
15 | 0.4810 | 0.2394 | 0.1229 | 0.0649 |
20 | 0.3769 | 0.1486 | 0.0611 | 0.0261 |
25 | 0.2953 | 0.0923 | 0.0304 | 0.0105 |
Table 1-1 is excerpted from Table B-l in Appendix B of this book. The table in the appendix shows the discount factors for all opportunity interest rates from 1 percent to 28 percent over periods from one year to 25 years. This table is provided for your convenience, although with the aid of a calculator you can find the discount factor for any opportunity interest rate over any period of time using the formula supplied by equation (1-2). For discount rates involving fractions, such as 16.5 percent, you will have to use your calculator, although an approximation can be obtained by a simple interpolation between the discount factors given in the appendix. 3
When a decision involves both revenues and costs in future periods, we net the costs of each year against the revenues of each year to find the present value of the net revenues, or net costs. Since profits are the excess of revenues over costs, and losses are the excess of costs over revenues, this procedure amounts to finding the present value of the profits (or losses) associated with the decision in future years.
Using symbols, we can write
NPV = C 0 -
D
1 = 1
FVt
(1 + r) 1
where NPV is the net present value, C 0 is the initial (present period) cost of the project or decision, and the TV, are the future net revenues (profits) to be received in each of the n years over which cash flows are expected. The E (sigma) connotes the sum of the discounted future values from i — 1 (the first year) to i = n (the last year).
Example: Suppose we are considering the installation of equipment to manufacture a gift item, and we anticipate the cost and revenue streams shown in Table 1-2. The cost stream includes the initial capital cost and the annual operating costs of the equipment and associated labor and material costs. The revenue stream shows the revenue expected from sales of the product and includes the salvage value of the equipment in the last year, when we assume the market will become saturated and production will cease. Since there is some uncertainty about the future revenue and cost streams, we must identify the opportunity discount rate. Suppose we investigate and conclude that it is 18 percent. We can then calculate the present value of the profits from the project as shown in Table 1-2 using the discount factors from Table B-l in Appendix B.
Note that this example includes year zero, or the present period, in the calculations, since the capital expenditure on the equipment ($744,850) must be made in the
•’Interpolation is the imputing of values between two points in a data set. We stress that this interpolation is approximate because it is a linear approximation of a curvilinear relationship. Considerations of accuracy aside, for more complex interpolations (such as for 16.335 percent), you would most likely use a calculator and would find it more simple to compute the exact discount factor using equation (1-2).
Table captionTABLE 1-2. Net Present Value of a Proposed Decision
Year | Revenue Stream ($000) | Cost Stream ($000) | Profit (Loss) Stream ($000) | Discount Factor (@ 18%) | Present Value ($000) |
0 | — | 744.85 | (744.85) | 1.0000 | -744.85 |
1 | 400.00 | 224.62 | 175.38 | 0.8475 | 148.63 |
2 | 1,085.00 | 648.22 | 436.78 | 0.7182 | 313.70 |
3 | 872.50 | 456.98 | 415.52 | 0.6086 | 252.89 |
4 | 220.00 | 131.43 | 88.57 | 0.5158 | 45.68 |
5 | 380.00 | 58.35 | 321.65 | 0.4371 Net Present Value | 140.59 156.64 |
present period. Thus, the appropriate discount factor is one, since this sum is already in present-value terms. For each year in which there are both revenues and costs, the net revenues (profits) are discounted by the appropriate discount factor to find the present value of each year’s profits from the proposed decision. Adding up the present values of each year’s profits or losses results in a net present value of the decision of $156,640. If this is the best alternative open to the firm, we would certainly advise them to go ahead with it, since it represents $156,640 more than the firm could earn if it invested the funds elsewhere at the opportunity interest rate. 4
Annuities. A simplification can be applied if the stream of cash flows is regular and uniform for a number of periods. Such a uniform cash flow over several consecutive time periods is known as an annuity. You are probably aware of the concept of annuities. Retirement savings can be used to purchase an annuity which pays the owner a given sum of money every year or every month for a prescribed number of years. Any regular and uniform stream of payments can be treated like an annuity. For example, a firm expecting to earn $50,000 each year for five years as the result of a particular decision, is, in effect, expecting an annuity.
It can be shown that the present value of an annuity can be calculated as the amount to be received in each period times the sum of the discount factors for each period. That is,
FV, , , FV » = v FV ‘
(1 + rY i= l (1 + r)‘
PV =
FV ,
+
FVi
+
(1 + r)' ' (1 + r) 2 ' (1 + r) 3
1
= FV H -
/=i (1 + r) 1
since all the FV s have the same value.
Example: A firm expects to receive $50,000 at the end of each year for five years as a result of a decision it is about to implement. What is the present value of this annuity if the firm’s opportunity discount rate is 16 percent? In Table 1-3 we find the present value of the revenue stream of $50,000 each year for five years, given 16 percent discount factors. The sum of the present values of the annual payments is $163,720. Now, notice that the sum of the discount factors is 3.2744 and that this figure multiplied by $50,000 equals $163,720. The sum of the discount factors is known as the present- value factor for an annuity over that period of time.
Table B-2 in Appendix B shows the present-value factors for annuities up to periods of 25 years and for opportunity interest rates of 1 percent through 28 percent. It is certainly much quicker to find the present value of an annuity in a single calculation rather than to find the present value of each annual cash flow and then add these to find the present value of the revenue stream. Referring to Table B-2, we find the present-value factors for a 16 percent discount rate over five years to be 3.2743. (This is the more accurate figure, since the sum of the discount factors used above includes a discrepancy caused by rounding off at the fourth decimal place.) Thus, the present value of the annuity is 3.2743 times the periodic payment of $50,000, or $163,715.
Discounting Using Daily Interest Rates. Most of the cash flows paid and received by firms do not occur in lump sums at the end of the year, but instead occur at more frequent intervals. Cash receipts from customers may be received daily, wages may be
TABLE 1-3. | Present Value of an Annuity | ||
Revenue | Discount | Present | |
Stream | Factor | Value | |
Year | $ | (@ 16%) | $ |
1 | 50,000 | 0.8621 | 43,105 |
2 | 50,000 | 0.7432 | 37,160 |
3 | 50,000 | 0.6407 | 32,035 |
4 | 50,000 | 0.5523 | 27,615 |
5 | 50,000 | 0.4761 | 23,805 |
Totals | 3.2744 | $163,720 |
paid weekly, and management salary, payments to suppliers, and other expenses usually occur monthly. It makes a significant difference to the present value if the payments and receipts are received and disbursed frequently during the year, rather than being delayed until year end. Money received during the year can be put in the bank and will earn interest between now and the end of the year. Daily-interest savings and checking accounts are now common, and our analysis should reflect the greater present value of a sum if it is received during the year as a stream of payments rather than being received at the end of the year as a lump sum. The end-of-year convention is a simplification that is often quite appropriate. In many cases, the cash flows do occur annually in lump sums. For example, the firm’s cash flow may be highly seasonal (such as around Christmas), and the end-of-year assumption is a tolerable approximation. In other cases, however, the cash flows of a decision are expected to occur more or less evenly throughout the year, and it is more appropriate to use discount factors based on daily interest rates.
The present-value formula is easily modified to reflect higher frequencies of receipt or disbursement of funds. Let m be the number of times that the payment period fits into the year, that is m — 365 /d where d is the number of days in the payment period. For example, m = 365 for daily cash flows, m — 52 for weekly cash flows, m — 12 for monthly cash flows, and m — 4 for quarterly cash flows. The modified present-value formula would then read
pv _ FV/m | FV/m | FV/m | | FV/m
(1+r/m) (1+r/m) (1+r/m) (1 + r/m)
_ " FV/m /=i (1+r/m)
where FV again represents the future sum received during the year but is received in a series of payments, each equal to FV/m. In effect, the firm receives an annuity of 1/m of the FV value every m days during the year. For the daily interest case, and assuming FV - $1, that dollar is presumed to arrive in 365 equal instalments, each equal to 1/365 of one dollar. Note that the above expression for PV is also the formula for the daily interest rate discount factor when FV = $1.
Discount factors based on daily interest rates are provided in Table B-3 in Appendix B. We can see the impact of using daily interest discount factors if we redo the NPV calculations of the cashflows shown in Table 1-2, now assuming that the cash flows occur on a more or less daily basis throughout each year. 5 The results are shown in Table 1-4.
Table captionTABLE 1-4. Net Present Value of Proposed Decision Using
Table captionDaily Discount Factors
Year | Profit Stream ($000) | Discount Factor (@ 18%) | Present Value ($000) |
0 | -744.85 | 1.0000 | -744.850 |
1 | 175.38 | 0.9150 | 160.473 |
2 | 436.78 | 0.7643 | 333.831 |
3 | 415.52 | 0.6384 | 265.268 |
4 | 88.57 | 0.5333 | 47.234 |
5 | 321.65 | 0.4454 | 143.263 |
Net Present Value | 205.219 |
Note that each discount factor is slightly higher than the year-end discount factor for the same year and opportunity rate, and it reflects the greater value of a dollar received as a stream of payments during the year rather than as a simple payment at the end of the year. Consequently, the NPV is higher (by $48,579) when we account for the daily cash flow pattern rather than assuming year-end payments and receipts. This could be a critical consideration in deciding between two decision alternatives. If they both have the same annual cash flows, but one series of cash flows occurs at the end of each year, while the other series occurs more or less evenly throughout each year, then the latter will be the preferred option. NPV analysis should use the discount factors that are appropriate for the cash flow pattern expected. End-of-year discount factors will be appropriate for some decision alternatives, whereas daily discount factors will be more appropriate for others.
■ Definition: The firm’s planning period is the period of time over which the firm takes into account the cost and revenue implications of its decisions. The firm’s time horizon is the point in the future at which the firm no longer considers the cost and revenue implications of its decisions. The time horizon is, therefore, the end of the firm’s planning period.
Example: Suppose a firm is considering investing in a new building and calculates the present value of the initial cost and future maintenance costs for the next fifteen years and sets against these costs the present value of the revenues it expects to receive from the building over the same fifteen-year period. This firm’s planning period is, thus, fifteen years, since it has taken into account the cost and revenue implications of its decision to buy the new building only up to the fifteenth year into the future. The sixteenth and subsequent years’ costs and revenues are ignored. The firm, in effect, considers them to occur so far into the future that they are insignificant to the decision.
A firm’s time horizon, and, thus, the length of its planning period, is likely to vary among firms for various reasons. Firms involved in intense competition for their day-to-day survival are less likely to worry about the longer-term cost and revenue implications of their decisions. Immediate or short-term costs and revenues may be given full weight in their decision making, to the complete exclusion of the longer-term profit implications of their actions. Conversely, a firm that is well established and secure in its market, without the constant pressure of day-to-day price competition, can afford the luxury of taking into consideration the future profit implications of its current decisions.
Another reason for differing planning periods among firms is the motivation of the manager. A manager who expects to retire in six months, or who is actively seeking a promotion within the year, may be expected to prefer actions which have greater short-term profit implications over those that promise greater longer-term profits. Conversely, managers who wish to make a career out of their position, or who prefer longer-term stability of employment and salary income, may be expected to take the longer view, preferring to take actions which promise greater profits over an extended planning period.
Finally, the firm’s planning period may be inversely related to the general level of interest rates, because the present value of future profits declines as the opportunity discount rate increases. This presumes that the decision maker decides to suspend the search for information (concerning future costs and revenues associated with a particular decision) when the present value of $1.00 earned at the time horizon falls below a predetermined level.
Example: A firm that places its time horizon at fifteen years when the opportunity discount rate is 10 percent may shorten its planning period to ten years when the opportunity discount rate is 15 percent, since the present value of $1.00 falls below $0.25 at the time horizon in each case. (You may confirm this in Table 1-1.)
The reason a firm may adopt a cutoff rule like this is related to the cost of obtaining information, or search costs, which we examine in Chapter 2. It will suffice to say here that if the firm expects to pay $0.25 now to find out that it will earn $1.00 in ten years, it is profit maximizing to suspend search activity at that point if the present value of the $1.00 is less than $0.25.
The firm’s time horizon will occur either within the present period or in some future period. If it occurs in the present period, we can treat all cash flows at face value, since they will already be in present-value terms. If the firm’s time horizon lies beyond the present period, we must discount future cash flows back to present-value terms in order to compare them properly with present-period costs and revenues.
How long is the present period? Strictly it should be very short indeed, and should end before the value of a dollar to be received in the present period depreciates at all. Note that $1,000 to be received next week is worth fractionally less than $1,000 to be received today, because a lesser sum could grow to $1,000 (in a daily-interest
savings account) by next week. Realistically, however, there is a cost involved in converting every cash flow into its present-value equivalent, and it is not worth doing for most cash flows that will arrive or will be disbursed in the near future. For our purposes it is probably sufficient to treat cash flows that occur within the coming year in nominal terms, ignoring the fact that doing so will slightly overstate their present value. But when the dollar amounts are very large and the interest rates are relatively high, we should certainly discount the nominal values to present values for cash flows occurring within the present year. 6
In this section we consider risk and uncertainty, and we describe the method for evaluating decision alternatives when the decision maker faces risk and uncertainty. We first clarify what is meant by risk and uncertainty, and note that the phrase implies a probability distribution of outcomes related to each decision alternative. Expected- present-value (EPV) analysis is then introduced to summarize each probability distribution of outcomes into a single number such that the decision alternatives may be compared and an optimal decision may be selected.
The state of information under which a decision is made has important implications for the predictability of the outcome of that decision. If there is full information, the outcome of a decision will be foreseen clearly and unambiguously. In this situation (of certainty) the firm can accurately predict the outcome of each of its decisions. When there is less than full information, however, the decision maker may foresee several potential outcomes to a decision and, therefore, will be unable to predict consistently which outcome will actually occur. In this case we say that the individual or firm is operating under conditions of risk and uncertainty.
■ Definition: Certainty exists if the outcome of a decision is known in advance without a shadow of a doubt. Under conditions of certainty, a decision leads to a single possible outcome, which is perfectly foreseen. Risk and uncertainty are involved when a decision leads to one of several possible outcomes but the exact outcome is not known in
To discount a cash flow to be received within the current year, use the same formula
FV
PV =-
(1 + r)"
where n is the number of years as before, except that it can take fractional values. For example, a lump-sum payment due in six months would have n = 0.5, and a lump-sum payment due in two and a half years would have n = 2.5.
Introduction to Managerial Economics 19
advance. Instead, there will be a probability distribution of possible outcomes, which the decision maker must identify. 7
Risk is involved when one flips a coin, throws dice, or plays a hand of poker. The probability of flipping a coin and having it land “heads” is V 2 , since there are only two possible outcomes (ruling out the coin landing on its edge), and each is equally likely to occur, given an unbiased coin. Similarly, when one throws two dice, the probability that they will turn up “snake eyes,” or any other pair of numbers, is >A X 'A = >/36. The probability of drawing a “royal flush” in poker, or any other combination of cards, can likewise be calculated.
In each of the illustrations above the probability of each outcome is known a priori. That is, on the basis of known mathematical and physical principles, we can deduce—prior to the act—the proportion of the total number of outcomes that should be attained by each particular outcome. We can confirm this calculation by undertaking a number of trials. Although “heads” might appear three or even four times out of the first four tosses of a coin, given a sufficiently large number of trials, the proportions will converge upon V 2 for each of the two possible outcomes.
A second class of risk situations is that in which probabilities are assigned a posteriori, or on the basis of past experience under similar circumstances. The business of insurance is based upon this type of risk situation. The possible outcomes are known: for example, a particular vehicle will or will not be involved in an accident. Insurance companies keep extensive data on previous policies and claims and other pertinent data; from these they compile actuarial tables, which show the relative incidence of the various outcomes in past situations or trials. On the presumption that a particular driver and vehicle are similar in all important respects to those of the data base, the companies are able to form an expectation (or assign a probability) of the chances of that particular driver having an accident in that vehicle.
In perhaps the great majority of decison-making problems, the precise nature of the potential outcomes cannot be foreseen clearly in advance. Instead the decision maker must estimate a range of potential outcomes. Similarly, there is typically no data bank of past decisions that are sufficiently similar to the one at hand to allow the assignment of probabilities based on past observations. In these situations, the decision maker must assign the probabilities subjectively, based on experience, intuition and judgment.
Under conditions of risk and uncertainty the decision maker looks at each of the po- tential solutions to a problem and, for each of these possible decisions foresees a probability distribution of outcomes. That is, several different levels of profit (or loss) are perceived as possible, and each of these is assigned a probability of occurring. How does the decision maker summarize all this data so that they can be compared with other potential solutions to the same problem?
■ Definition: The expected value of an outcome is the value of that outcome multiplied by the probability of that outcome occurring. Since several outcomes are possible under risk and uncertainty, the expected value of a decision is the sum of the expected values of all the possible outcomes that may follow the decision. 8 The expected value ot a decision thus allows the probability distribution of possible outcomes to be characterized by a single number, which can then be compared with the expected values of other potential solutions to the problem.
Example: In Table 1-5 we show a hypothetical probability distribution of profit levels that are expected to be possible outcomes of a decision to invest in a particular investment project. The first column shows the possible profit levels, ranging from a loss of $50,000 to a profit of $250,000. The second column shows the probability of each profit (or loss) level, as assigned by the decision maker. Thus there is considered
TABLE 1-5. | The Expected Value of | a Decision |
Possible | Probability | Expected Value |
Profit | of Each | of Each |
Levels | Occurring | Profit Level |
($) | (P) | ($) |
-50,000 | 0.05 | -2,500 |
0 | 0.10 | 0 |
50,000 | 0.15 | 7,500 |
100,000 | 0.20 | 20,000 |
150,000 | 0.25 | 37,500 |
200,000 | 0.15 | 30,000 |
250,000 | 0.10 | 25,000 |
Totals | 1.00 | 117,500 |
8 Formally, we define the expected value of a decision as
II
EV= E R,p,
i = i
where L connotes “the sum of”; /?, is the return of the ;'th outcome; i = 1, 2, 3,. . . , n identifies each separate possible outcome; /; is the total number of possible outcomes; and P, the probability of the fth outcome occurring.
to be a 5 percent chance of losing $50,000, a 10 percent chance of only breaking even, a 15 percent chance of making $50,000 in profits, and so on. Note that the probabilities must total 1.00, since all possible outcomes are included and these outcomes are mutually exclusive.
The third column in the table is the product of columns one and two. The expected value of each possible outcome is equal to the possible profit (or loss) associated with each outcome, multiplied by the probability of that outcome’s occurring. The sum of the expected values of all the possible outcomes is $117,500. This is the expected value of the decision to invest in this particular investment opportunity. Note that the actual outcome will not be known until after the investment is made and all returns are in. The expected value is an a priori measure of the decision that allows the probability distribution of outcomes to be summarized as a single number. This expected value is actually a weighted average of the possible profit levels, with each possible outcome weighted by the probability that it will occur. 9
Thus, if a firm is faced with a decision problem, the decision maker(s) should investigate all potential solutions to that problem and evaluate the probability distribution of each potential solution in terms of its expected value. The firm should choose that alternative which promises the highest expected value, subject to adjustment for risk, which we examine in Chapter 2.
Any decision problem should be approached with the firm’s objectives clearly in mind, so that the actual decision taken will best serve the firm’s objectives. What are the objectives of the business firm? Is it profit maximization, or would the firm sacrifice some current profitability for an enlarged market share? Does the firm wish to maximize its rate of growth, or is management content to attain profit, market share, and growth targets, while maximizing their own benefits and the quality of their lives?
At this point we shall confine ourselves to the assumption, which is well supported in the literature and the related disciplines of finance and accounting, that the decision maker’s objective is to maximize the net worth of the firm over its time horizon, subject to considerations of risk and uncertainty. The other objectives mentioned above will be incorporated into the analysis and reconciled with this objective in later chapters. 10
“The actual outcome might be, for example, $67,964.45, a value that does not appear as a possible outcome in the table. The potential outcomes each represent a range of outcomes, and this (actual) outcome falls within one of those ranges. In this case the potential outcome of $50,000 represents the range from $25,000 to $75,000, in which the actual outcome fell. The probability distribution is continuous in reality, but we have “summarized” it using, in this case, seven points along the continuum.
l0 We shall see in Chapter 9 that sales, or market share maximization, growth maximization, managerial utility maximization, and even “satisficing,” can be regarded as real-world approximations for the maximization of the expected present value of profits.
22
Introduction
■ Definition: Net worth, also known as owner's equity, is measured as the excess of the firm’s assets (cash, securities, land, buildings, plant and equipment, and so on) over its liabilities (amounts owed to creditors, short-term and long-term loans, and so on). There are three groups of items on the firm’s balance sheet, namely, assets, liabilities, and owner’s equity. Assets are equal to the sum of the other two. Thus, maximization of the net worth of the firm requires maximizing the difference between assets and liabilities, or, what amounts to the same thing, maximizing the owners equity.
The preceding simple statement of the firm’s objective function implies slightly different decision criteria for each of the different scenarios in which a firm may find itself. In the foregoing we have established two separate dichotomies, one referring to the time horizon (the present period versus future periods) and the second referring to the state of information (certainty versus uncertainty). There are four combinations of these circumstances, and we now proceed to specify the decision criterion which will allow the maximization of net worth under each scenario.