Chapter 2
Techniques of Analysis: Optimization
2.1OPTIMIZATION AND OTHER TECHNIQUES OF ANALYSIS IN MANAGERIAL ECONOMICS
We have seen in Chapter 1 that managerial economics examines how an organization can achieve its objectives most efficiently. For example, the optimal behavior of a firm is to maximize profits or the present value of the firm. Thus, optimization analysis, the subject of this chapter, is one of the most important techniques of analysis in managerial economics. Other important techniques of analysis in managerial economics are risk analysis and estimation, which are examined in the next two chapters.
EXAMPLE 1. The total profit of a firm is equal to total revenue minus total cost. Thus, the firm maximizes its profit when the positive difference between total revenue and total cost is greatest. The total profit of the firm will increase as long as the extra or marginal revenue that the firm receives from producing and selling each extra unit of output exceeds the extra or marginal cost that the firm incurs to produce the extra unit of output. The process of profit maximization, as a most important example of optimization by the firm, is examined in this chapter, using total and marginal revenue and cost curves. Since most managerial decisions are made in the face of risk or uncertainty, it is also very important to examine risk and to show how it can be incorporated into the managerial decision process. Finally, it is very important in the managerial decision process to estimate the quantitative relationship among economic variables. Optimization analysis, risk analysis, and estimation are techniques used throughout the rest of the book in the study of managerial economics.
2.2TOTAL, AVERAGE, AND MARGINAL ECONOMIC RELATIONSHIPS
To study the process of optimization by the firm, we begin by examining the relationship between total revenue, average revenue, and marginal revenue on the one hand, and total cost, average cost, and marginal cost on the other. In the next section, we bring together the revenue and cost concepts and curves presented in this section to examine the process of optimization by the firm.
Average revenue (AR) equals total revenue (TR) divided by quantity (Q), and marginal revenue (MR) is the change in total revenue (i.e., TR) per unit change in output (Q). Similarly, average cost (AC) equals total cost (TC) divided by quantity (Q), and marginal cost (MC) is the change in total cost (i.e., TC) per unit change in output (Q).
EXAMPLE 2. Suppose that the total revenue function of a firm is given by
TR = 8Q - Q2. By substituting various values of Q into the TR function, we generate the TR schedule shown in the first and the third columns of Table 2.1. From the TR schedule, we can then derive the AR and MR schedules (fourth and fifth columns, respectively, of Table 2.1): AR = TR/Q. MR = TR/Q. By plotting the TR, AR, and MR schedules in Table 2.1, we get the TR, AR, and MR curves in Fig. 2-1. Note that AR is equal to the slope of a ray from the origin to the TR curve. In Fig. 2-1, AR declines continuously but remains positive as long as TR is positive. Since MR is the change in TR per unit change in output, the MR values of Table 2.1 are plotted halfway between successive levels of output in Fig. 2-I. At any particular point on the TR curve, MR is equal to the slope of the TR curve at that point. MR declines continuously and is less than AR at every level of output. At Q = 4, TR is at its maximum and MR is zero. At larger outputs, TR declines and MR is negative.
Table 2.1
Q / 8Q – Q2 / TR / AR / MR0 / 8(0) – (0)2 / $0 / –– / ––
1 / 8(1) – (1)2 / 7 / $7 / $7
2 / 8(2) – (2)2 / 12 / 6 / 5
3 / 8(3) – (3)2 / 15 / 5 / 3
4 / 8(4) – (4)2 / 16 / 4 / 1
5 / 8(5) – (5)2 / 15 / 3 / –1
EXAMPLE 3. Suppose that the total cost schedule of a firm is the one given in the first two columns of Table 2.2. We can then derive the corresponding average and marginal cost schedules in the third and fourth columns, respectively: AC = TC/Q. MC = TC/Q. By plotting the schedules of Table 2.2, we get the TC, AC, and MC curves in Fig. 2-2. AC is equal to the slope of a ray from the origin to the TC curve. This declines to point B and then rises. The slope of the TC curve, or MC, falls to point A (the point of inflection) and then rises, and is equal to AC at point B. Note that when the AC curve falls, the MC curve is below it. When the AC curve rises, the MC curve is above it, and when AC is lowest, MC = AC.
Fig. 2-1
Table 2.2
Q / TC / AC / MC0 / $6 / –– / ––
1 / 8 / $8 / $2
2 / 9 / 4.50 / 1
3 / 12 / 4 / 3
4 / 20 / 5 / 8
Fig. 2-2
2.3OPTIMIZATION ANALYSIS: PROFIT MAXIMIZATION
Optimization analysis can best be explained by examining the process by which the firm determines the output level at which it maximizes total profit. A firm maximizes total profit at the level of output at which the positive difference between total revenue and total cost is greatest. At this level of output, the firm's total revenue curve and total cost curve are parallel and marginal revenue equals marginal cost. This is one of the most important concepts in managerial economics. That is, according to marginal analysis, as long as the marginal benefit of an activity (such as expanding output or sales) exceeds the marginal cost, it pays for the organization (firm) gains by increasing the activity (output or sales). The total benefit (profit) is maximized when the marginal benefit (revenue) equals the marginal cost.
Fig. 2-3
EXAMPLE 4.The determination of the output level at which the firm maximizes total profit is shown in Fig. 2-3. In the top panel of Fig. 2-3, the TR and MR curves are those of Fig. 2-1, while the TC and MC curves are those of Fig. 2-2. The total profit () curve in the bottom panel of Fig. 2-3 is derived by subtracting the TC curve vertically from the TR curve. The maximum r (of $4) occurs at Q = 2.5, where the r curve has zero slope. At Q = 2.5, the TR curve is above and parallel to the TC curve, so that the vertical distance between them () is greatest. This is also the output level at which the slope of the TR curve, or MR, equals the slope of the TC curve, or MC. AT Q < 2.5, MR > MC and the firm would be adding more to TR than to TC, so that would increase by expanding output and sales. (The symbols < and > mean "smaller than" and "larger than," respectively.) However, at Q > 2.5, MR < MC and the firm would be adding less to TR than to TC, so that would increase by reducing output. As a result, is maximized at Q = 2.5, where MR = MC. This is a most important example of optimization by marginal analysis.
Glossary
Average cost (AC) Total cost divided by output (i.e., TC/Q).
Average revenue (AR) Total revenue divided by output or sales (i.e., TR/Q).
Estimation techniques Methods of calculating quantitative relationships among economic variables.
Marginal analysis An analytical technique which postulates that an activity should be carried out until the marginal benefit of the activity equals the marginal cost.
Marginal cost (MC) The change in total cost per unit change in output (i.e., TC/Q).
Marginal revenue (MR) The change in total revenue per unit change in output or sales (i.e., TR/Q).
Optimization analysis The process whereby an organization can achieve its objective most efficiently; for a firm this involves maximizing profits or the value of the firm.
Risk analysis The study of how risk and uncertainty can be incorporated into the managerial decision process.
Total cost (TC) The total expenditures of the firm to hire the inputs, or resources, required to produce and sell its output.
Total revenue (TR) The earnings of the firm in selling its output; price times quantity sold.
Review Questions
1.The most important technique of analysis in managerial economics is
(a)optimization analysis.
(b)risk analysis.
(c)estimation techniques.
(d)all of the above.
Ans. (d)See Section 2.1.
2.Which of the following is false with respect to optimization analysis?
(a)It refers to the process whereby an organization achieves its objectives most efficiently.
(b)For a business firm, this usually involves maximizing profits or the value of the firm.
(c)It relies on the relationship among total, average, and marginal concepts or measures.
(d)None of the above.
Ans. (d)See Section 2.1.
3.The firm maximizes profits when
(a)the positive difference between total revenue and total cost is at the maximum.
(b)total revenue is at its maximum.
(c)total cost is at its minimum.
(d)all of the above.
Ans. (a) See Section 2.1 and Example 1.
4.Which of the following statements is false with respect to risk?
(a)Most managerial decisions are made in the face of risk.
(b)Risk and uncertainty arise when there is more than one possible outcome of a decision.
(c)Risk analysis cannot be incorporated into the optimization calculations of the firm.
(d)None of the above.
Ans. (c)See Section 2.1 and Example 1.
5.If the price of the commodity declines as the firm sells more units of the commodity, the total revenue (TR) curve of the firm
(a)is negatively sloped.
(b)is a positively sloped straight line.
(c)first rises, reaches a maximum, and then declines.
(d)is convex.
Ans. (c)See Example 2 and Fig. 2-1.
6.Which of the following statements about average revenue (AR) is false?
(a)If the TR curve is concave, the AR curve declines continuously.
(b)When the TR curve begins to decline, AR becomes negative.
(c)If the TR curve is a positively sloped straight line, the AR curve is horizontal.
(d)AR is given by the slope of a ray from the origin to the TR curve.
Ans. (b)See Section 2.2, Example 2, and Fig. 2-1.
7.Which of the following statements about marginal revenue is false?
(a)It is positive when total revenue rises.
(b)It is zero when total revenue is zero.
(c)It is negative when total revenue declines.
(d)None of the above.
Ans. (b)See Section 2.2.
8.Which of the following statements about the relationship between marginal revenue and average revenue is false?
(a)When the AR curve is falling, MR is negative.
(b)When the AR curve is falling, the MR curve is below the AR curve.
(c)When AR is positive, MR can be positive or negative.
(d)None of the above.
Ans. (a) See Section 2.2 and Fig. 2-1.
9.Which of the following statements about the total cost (TC) curve is true?
(a)The TC curve is always positively sloped.
(b)The TC curve can be positive at zero output.
(c)The TC curve can first be concave and then convex to the origin.
(d)All of the above.
Ans. (d)See Fig. 2-2.
10.Which of the following statements about the average and marginal cost curves is true?
(a)The AC curve falls when the MC curve falls.
(b)The AC curve rises when the MC curve rises.
(c)At the lowest point on the AC curve, MC = AC.
(d)All of the above.
Ans. (c) See Section 2.2 and Fig. 2-2.
11.A firm maximizes total profits when
(a)MR = MC and the MC curve intersects the MR curve from below.
(b)the positive difference between TR and TC is at the maximum.
(c)the TR and TC curves have equal slopes, and the TR curve is above the TC curve.
(d)all of the above.
Ans. (d)See Section 2.3.
12.At the point of profit maximization
(a)MR is at its maximum.
(b)profit per unit is at its maximum.
(c)MC is at its minimum.
(d)none of the above.
Ans. (d)See Section 2.3 and Fig. 2-3.
Solved Problems
OPTIMIZATION AND OTHER TECHNIQUES OF ANALYSIS IN MANAGERIAL ECONOMICS
2.1(a)Explain what is meant by optimization by the firm. (b)How is optimization analysis conducted?
(a)Optimization by the firm refers to the process by which the firm seeks to achieve its objectives most efficiently. The optimal behavior of the firm is usually taken to be profit maximization, or the maximization of the present value of the firm. Sometimes, optimization may involve minimizing the total costs of the firm. This would be the case if the firm had a contract to supply a specific quantity of a commodity at a given price. By minimizing total costs for the given total revenue (since price per unit and sales are given). the firm would also be maximizing total profits. Thus, optimization may involve the maximization of profits or the minimization of costs by the firm.
(b)To a large extent, optimization analysis rests on the relationship among total, average, and marginal concepts and measures. The average is equal to the total divided by the quantity. while the marginal value is equal to the change in the total per unit change in quantity. The firm maximizes profits when the positive difference between total revenue and total cost is greatest. The total profits of the firm will increase as long as the extra or marginal revenue that the firm receives from producing and selling one extra unit of output exceeds the extra or marginal cost that the firm incurs to produce the extra unit of output. The total profit of the firm increases as long as marginal revenue exceeds marginal cost and until they are equal.
2.2(a) What is meant by risk? (b)Why is it important to incorporate risk in managerial decision making?
(a)Risk refers to the situation in which there is more than one possible outcome to a decision. For example, a firm may be anticipating sales of $100 million next year. However, depending on the general economic conditions that will prevail in the economy next year, the behavior of competitors, and other conditions, the sales of the firm could be as high as $150 million or as low as $75 million. If the firm can attach some probabilities to the occurrence of the events that affect its sales (e.g., if the firm knows or can forecast the probability of a recession next year), we say that the firm faces a risk.
(b)Most managerial decisions are made in the face of risk. Thus, it is extremely important that the firm consider the effect of risk in optimizing its behavior. For example, a firm may have a choice of drilling in one of two oil fields, one with very high profitability but with a low probability of finding petroleum and the other with a large probability of finding less petroleum. The entire future of the firm may depend on the choice it makes. Thus, it is crucial for the firm to be able to incorporate (i.e., take into consideration) the information about the amount of oil and the probability of striking it in determining the optimal course of action. Risk analysis is one of the most important techniques of analysis in managerial economics. Because of the importance of risk analysis and because it is used throughout the book, we devote all of Chapter 3 to the topic before beginning to study the formal subject matter of managerial economics (i.e., the analysis of demand, supply, costs, prices, market structures, government regulations, and so on).
2.3(a) What is meant by estimation techniques? (b)Why are these important to managerial economics?
(a)Estimation techniques are the methods of determining the quantitative relationship between a dependent variable, or variable that we seek to explain, and one or more independent or explanatory variables. For example, economic theory postulates that the quantity demanded of a commodity is a function of, or depends on, the price of the commodity, the income of consumers, and the price of related commodities (complements and substitutes). One of the most important estimation techniques used in managerial economics is regression analysis. This can be used, for example, to estimate how much the quantity demanded of a commodity declines, given a specific increase in the price of the commodity and in the price of the complementary commodity, and how much the quantity demanded of a commodity increases upon a specific increase in consumers' income and in the price of a substitute commodity. Estimation techniques (mostly regression analysis) are examined in Chapter 4.
(b)In order for a firm to maximize its total profits, or its value, the firm must have some estimates of the quantitative relationships among the variables that affect its revenues and costs. It is not sufficient to know, for example, that the quantity demanded of a commodity is inversely related to the price of the commodity and the price of complementary commodities, and directly related to consumers' income and the price of substitute commodities. In order to maximize profits, a firm needs also to know how much the quantity demanded of the commodity changes for a given change in the commodity price, in consumers income, and in the price of related commodities.
TOTAL, AVERAGE, AND MARGINAL ECONOMIC RELATIONSHIPS
2.4Indicate (a) the different forms in which economic relationships can be expressed and (b)the advantage of each.
(a)Economic relationships can be expressed in equational, tabular, and graphical forms. Example 2 gave a hypothetical TR function (equation). By substituting various values of Q into the TR function, we generated the TR schedule in Table 2.1 and, from it, the AR and MR schedules. By then plotting the TR, AR, and MR schedules of Table 2. 1~ we obtained the corresponding TR, AR, and MR curves shown in Fig. 2-1.
(b)The advantage of expressing economic relationships in tabular form is that a table can readily summarize a large amount of data. The advantage of a figure is that it provides a quick visual overview of changes in the data over time or across units of observation. Expressing an economic relationship in functional or equational form allows us to utilize the powerful techniques of mathematics (e.g., differential calculus) in determining the optimal solution to problems.
2.5Given the total revenue lunation TR 12 - 2Q2 (a) derive the total revenue, average revenue, and marginal revenue schedules. (b)On the same set of axes, plot the total revenue, the average revenue, and the marginal revenue schedules of part (a). (c)Using the figure you drew for part (b), explain the relationship among the total, average, and marginal revenue curves.
(a)The total, average, and marginal revenue schedules are derived in Table 2.3.
Table 2.3
Q / 12 – 2Q2 / TR / AR / MR0 / 12(0) – 2(0)2 / $0 / –– / ––
1 / 12(1) – 2(1)2 / 10 / $10 / $ 10
2 / 12(2) – 2(2)2 / 16 / 8 / 6
3 / 12(3) – 2(3)2 / 18 / 6 / 2
4 / 12(4) – 2(4)2 / 16 / 4 / –2
(b)The TR, AR, and MR schedules of Table 2.3 are plotted in Fig. 2~. The AR curve is the demand curve (d)for the product that the firm faces. Note that the MR values are plotted halfway between successive levels of output because MR represents the change in TR per unit change in output.
Fig. 2-4
(c)The slope of a ray from the origin to the TR curve, (i.e., the average revenue), falls continuously and is positive as long as TR is positive. Since the TR curve is concave, its slope, or MR, also falls continuously. MR is positive as long as TR increases, MR = 0 when TR is at its maximum, and MR is negative when TR declines. Since the AR, or D, curve falls continuously, the MR curve is always below it.