# MATH MODULE 11. Maximizing Total Net Benefit. 1. Discussion M11-1

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1 MATH MODULE 11 Maximizing Total Net Benefit 1. Discussion In one sense, this Module is the culminating module of this Basic Mathematics Review. In another sense, it is the starting point for all of the other modules. It shows how most forms of economic decision-making can be understood as ways of maximizing the Total Net Benefit of an activity. The Module also shows how the optimizing rule that appears in many different contexts in the text that of setting the Marginal Benefit of x, MB(x), equal to the Marginal Cost of x, MC(x), to determine the optimal level of x is just another way of finding the level of x that maximizes the Total Net Benefit of x. Finally, it analyzes some situations in which the simple Set MB MC rule can break down when the objective is to maximize Total Net Benefit. Chapter 1 of the text sets out a basic decision rule that underlies virtually all of mainstream microeconomics. We can express the Rule as follows: Rule : If the level of activity x can be varied continuously, then we reach the optimal level of x if we choose the level of x so that the additional benefit from the last unit of x the Marginal Benefit of x, MB(x) equals the additional cost of the last unit the Marginal Cost of x, MC(x). The Rule in its simplest form doesn t always work. In this Module, we set out the conditions under which it will work, and then analyze the exceptions. 1.1 WHEN DOES SETTING MB MC MAXIMIZE TOTAL NET BENEFIT? Why do we have the Rule? In a world of scarcity that is, in the real world many resources have a cost, and hence achieving our objectives generally entails a cost. As a result, the optimizing behaviour of rational economic agents is typically not aimed at maximizing the Total Benefit B(x) from an activity x. Nor is it aimed at minimizing the M11-1

2 M11-2 MAXIMIZING TOTAL NET BENEFIT Total Cost C(x) of the activity. Rather, it is aimed at maximizing the difference between the Total Benefit and the Total Cost of the activity, which is known as the Total Net Benefit of the activity, NB(x). The general form of the economic decision problem is thus: Maximize NB(x) = B(x) C(x). (M.11.1) Economists interpret the actions of both firms and households in terms of their efforts to maximize their Total Net Benefit, as Table M.11-1 indicates. TABLE M.11-1 Maximizing Total Net Benefit AGENT Max Total Net Benefit = Total Benefit Total Cost Maximize NB(x) = B(x) C(x) Households Max Consumer Surplus = Total Utility (Willingness to Pay) Total Expenditure (P X Q X ) Firms Max Producer Surplus = Total Revenue (P X Q X ) Total Variable Cost (VC) Max Profit = Total Revenue (P X Q X ) Total Cost (= Fixed Cost + Variable Cost) As we saw in Module 10, if the Total Net Benefit or NB(x) function has a maximum, then at the level of x where NB(x) reaches its maximum value, the slope of the tangent to the NB(x) function will be zero. The slope of the Total Net Benefit function is called the Marginal Net Benefit function, MNB(x). It equals the difference between Marginal Benefit MB(x), the slope of the Total Benefit or B(x) function, and Marginal Cost MC(x), the slope of the Total Cost or C(x) function: MNB(x) = MB(x) MC(x) (M.11.2) When the Marginal Net Benefit function equals zero, MNB(x) = MB(x) MC(x) = 0. Hence, at this point MB(x) = MC(x), the Marginal Benefit of x equals the Marginal Cost of x. This is the basic logic behind the Rule in Chapter 1. The following examples illustrate the major links between Total and Marginal Net Benefit. Example 1: Total Net Benefit NB(x) and Marginal Net Benefit MNB(x): Suppose the Total Net Benefit function has the form NB(x) = 10x 0.25x 2. Then, using either calculus or The Rules from Module 5, we know that the slope or derivative of the NB(x) function the Marginal Net Benefit function has the form MNB(x) = x. We measure x in units, NB(x) in dollars, and MNB(x) in dollars per unit. The NB(x) function is graphed in Figure M.11-1(a), and the MNB(x) function is graphed immediately below it in Figure M.11-1(b). NB(x) reaches a maximum when x = x* = 20 units. At this point, NB(x) = 10(20) 0.25(20) 2 = \$100, and MNB(x) = (20) = zero.

3 MATH MODULE 11: MAXIMIZING TOTAL NET BENEFIT M11-3 FIGURE M.11-1 The economic sense of the fact that MNB(x) is zero at the point of maximum Total Net Benefit is straightforward. To the left of x*, the marginal benefit of an additional unit of x exceeds the marginal cost of an additional unit of x, since MNB(x) = MB(x) MC(x) > 0. Hence if we choose a level of x less than x*, we are not maximizing NB(x), because increasing x would increase NB(x). By the same token, to the right of x*, the marginal cost of an additional unit of x exceeds the marginal benefit of an additional unit of x, since MNB(x) = MB(x) MC(x) < 0. Hence if we choose a level of x greater

4 M11-4 MAXIMIZING TOTAL NET BENEFIT than x*, we are not maximizing NB(x), because decreasing x would increase NB(x). Only where MB(x) = MC(x), and MNB(x) = 0, are we maximizing Total Net Benefit. Figure M.11-1 also reveals some additional aspects of the links between the Total and Marginal Net Benefit functions. Note that the height of the NB(x) function at x* is \$100, which is equal to the area of triangle AOB under the MNB curve: (\$10/unit x 20 units)/2 = \$100. (The reason they are equal is related to Example 7 in Module 4 and is explained in the discussion of Module 10, pages M10-1 to M10-4. The sum or integral, in calculus terms of all of the marginal additions to Total Net Benefit up to any level of x equals the Total Net Benefit at that level of x.) The second relationship shown by Figure M.11-1 can be seen where x = 40. At this point, Total Net Benefit = 0. (If we set NB(x) = 10x 0.25x 2 = 0 and solve for x, we get x = 10/0.25 = 40 units.) Note that to the right of x* = 20 units, Total Net Benefit NB(x) is decreasing, the NB(x) curve has a negative slope, and the MNB(x) curve is negative: it lies below the horizontal axis. When x = 40, MNB(x) = (40) = 10, and the area of triangle DCB = ( 10)(40 20)/2 = \$100. Hence the area of the two triangles AOB + DCB = (100) + ( 100) = 0 = NB(x = 40). Example 2: Four Related Diagrams: Suppose that the Total Benefit Function has the form B(x) = 12x 0.5x 2, and the Total Cost function has the form C(x) = 3x x 2. What are the formulas for the Total and Marginal Net Benefit functions, what level of x maximizes NB(x), and what is the value of NB(x) at this point? Figure M.11-2 contains four related graphs containing the answers to these questions. We get the formula for Total Net Benefit by substituting into the equation NB(x) = B(x) C(x) = (12x 0.5x 2 ) (3x x 2 ) = 9x 0.75x 2, from which we can calculate that Marginal Net Benefit MNB(x) = 9 1.5x. Alternatively, we can derive Marginal Benefit MB(x) = 12 x and Marginal Cost MC(x) = x, and then calculate MNB(x) = MB(x) MC(x) = (12 x) ( x) = 9 1.5x. We set MNB(x) = 9 1.5x = 0 to find the value of x (x*) that maximizes Total Net Benefit: x* = 9/1.5 = 6 units. With x* = 6 units, we have B(x*) = 12(6) 0.5(6) 2 = \$54, C(x*) = 3(6) (6) 2 = \$27, and NB(x*) = B(x*) C(x*) = 9(6) 0.75(6) 2 = \$27. Figure M.11-2(a) shows the B(x) and C(x) curves for this case. The vertical distance between them, B(x) C(x), is equal to the Total Net Benefit, NB(x), which is graphed in Figure M.11-2(b). NB(x) reaches a maximum at x = 6 units, and equals zero at the origin and at x = 12, where B(x) = C(x). For x > 12, NB(x) is negative. The slope of the Total Benefit curve (the Marginal Benefit curve) continuously declines as x increases, and for x > 12 it becomes negative. The slope of the Total Cost curve, on the other hand, continuously increases as x increases. The slopes, MB(x) and MC(x), are graphed in Figure M.11-2(c). They are equal when x = 6 units: both MB(x) and MC(x) equal \$6/unit. In other words, the Total Benefit and Total Cost curves in Figure M.11-2(a) are parallel (and their slopes are equal) at x* = 6 units. To the left of x*, the slope of the B(x) curve is greater than that of the C(x) curve, or in other words, the Marginal Benefit of an additional unit of x exceeds its Marginal Cost. When MB(x) > MC(x), necessarily MNB(x) is positive, and therefore, to maximize Total Net Benefit, the level of x should be increased. To the right of x*, the slope of the C(x) curve is greater than that of the B(x) curve, and the Marginal Cost of an additional unit of x exceeds its Marginal Benefit. When MB(x) < MC(x), necessarily MNB(x) is negative, and therefore the level of x should be decreased. Figure M.11-2(d) shows the Marginal Net Benefit or MNB(x) function. Just as the Total Net Benefit function is the difference, or vertical distance, between the B(x) and C(x) functions, so the MNB(x) function is the difference, or vertical distance, between

5 MATH MODULE 11: MAXIMIZING TOTAL NET BENEFIT M11-5 the MB(x) and the MC(x) functions. When x = 0, on the vertical axis, MB(x) = \$12/unit, MC(x) = \$3/unit, and MNB(x) = MB(x) MC(x) = \$9/unit. When x = 6 units, MB(x) = \$6/unit, MC(x) = \$6/unit, and MNB(x) = MB(x) MC(x) = 0. Thus, as we saw in Example 1, when Total Net Benefit NB(x) is at a maximum, the tangent to the NB(x) curve is horizontal, or equivalently, MNB(x) = 0. FIGURE M.11-2

6 M11-6 MAXIMIZING TOTAL NET BENEFIT The maximum value of Total Net Benefit, which equals \$27 when x = 6 units, appears in all four parts of Figure M In (a), it is the vertical distance between the B(x) and C(x) curves, = \$27. In (b), it is the height of the NB(x) curve at x = 6. In (c), it is the shaded triangular area between the MB(x) and the MC(x) curves, and in (d), it is the shaded triangle between the MNB(x) curve and the horizontal axis. Parts (a) and (b) should be clear, and we saw a version of (d) in Figure M Part (c), however, deserves a closer look. Note that the area between the MB(x) curve and the horizontal axis at x = 6 equals 6(12 + 6)/2 = \$54, which is the height of the B(x) curve at that point in (a). Similarly, the area between the MC(x) curve and the horizontal axis at x = 6 equals 6(3 + 6)/2 = \$27, which is the height of the C(x) curve at x = 6 in (a). The shaded triangle is thus the difference B(x) C(x), which is the definition of Total Net Benefit: NB(x) = B(x) C(x) = = \$27. Although they have different shapes, the shaded triangles in (c) and (d) have the same area at x = 6 units, since they both have the same base (12 3 = 9) and the same height (= 6). Example 3: Maximizing Total Net Benefit with Fixed Costs: Suppose that the Total Benefit Function has the form B(x) = 12x 0.5x 2, as in Example 2 above. The Total Cost function, however, now has the form C(x) = F + 3x x 2, where F is a certain number of dollars of Fixed Cost that must be paid regardless of the level of x that is, whether x = 0 or x > 0. What change occurs in the graphs of the Total and Marginal functions, what level of x now maximizes NB(x), and what is the value of NB(x) at this point? We shall consider two cases: i. Fixed Cost F = \$15, C(x) = x x 2. ii. Fixed Cost F = \$30, C(x) = x x 2. Case 3.i. In this case, using Figure M.11-2 as a base, the graph of the Total Cost function in (a) will be \$15 higher at every level of x, and therefore in (b) the Total Net Benefit function will be \$15 lower. Its equation will now be NB(x) = B(x) C(x) = (12x 0.5x 2 ) (15 + 3x x 2 ) = x 0.75x 2. Instead of having NB(x) = 0 when x = 0 and when x = 12, now when x = 0, NB(x) = 15, and NB(x) will equal zero when x = 2 and when x = 10. (You should verify these numbers for yourself.) What is most interesting, however, is what doesn t change. Because the C(x) function has increased by a constant \$15 at every level of x, the slope of the C(x) function the Marginal Cost function MC(x) is the same as it was before: MC(x) = x! Since the Marginal Benefit function is also unchanged, the Marginal Net Benefit function remains MNB(x) = 9 1.5x, and so the level of x that maximizes Total Net Benefit, x*, is still 6 units. At x* = 6, however, NB(x) = x 0.75x 2 = (6).75(6) 2 = \$12, that is, \$15 less than the \$27 Total Net Benefit in Example 2. (If you have calculus, one of the simplest ways of seeing why this result occurs is to note that the derivative of a constant in this case, 15 or 15 is zero, since the slope of a horizontal line is zero.) Case 3.ii. This case should offer no surprises, since you have already worked through Case 3.i. Here the graph of the Total Cost function in (a) will be \$30 higher at every level of x, and in (b) the Total Net Benefit function will everywhere be \$30 lower, than in Figure M Again, since the Marginal curves do not change, x* = 6 units as before, but in the present case, maximum Total Net Benefit = \$3. This is the best we can do. If we set x = 0, for instance, then NB(x) = \$30, which is much worse. Sometimes, maximizing Total Net Benefit means minimizing Total Net Losses!

7 MATH MODULE 11: MAXIMIZING TOTAL NET BENEFIT M11-7 Note: Examples 2 and 3 provide us with an important reminder. All three cases we looked at have identical Marginal Benefit, Marginal Cost, and Marginal Net Benefit functions, but the corresponding Total functions are different. We can go from the Total functions to the corresponding Marginal functions, where they exist (by taking the derivative of the Total function or by other means). As we saw in Modules 5 and 10, however, if we know only the Marginal curves, we can calculate x*, but we cannot calculate the Total Net Benefit NB(x) at x* unless we also know what NB(x) is when x = 0. This information gives us the fixed or constant component of NB(x), what mathematicians call the constant of integration. Example 4: Perfectly Competitive Equilibrium Maximizes Total Social Net Benefit, the Sum of Consumer Surplus and Producer Surplus: Chapters 2 and 11 of the text develop one of the fundamental welfare theorems of neoclassical economics: Theorem: Under certain conditions (such as no transactions costs, no externalities, perfect information, price-taking behaviour, and no trading at disequilibrium prices, for starters), the equilibrium price set in a perfectly competitive market will result in the optimal quantity exchanged, in that it maximizes the sum of producer surplus and consumer surplus. This theorem is actually an application of the framework in this Module. Table M.11-1 gives the objectives for firms and households in terms of total net benefit maximization. Households are assumed to have the objective of maximizing consumer surplus, the difference between the total utility U(x) or B(x) that they receive from a quantity of x (as measured by the maximum amount they would be willing to pay for it) and their total expenditure on x. The demand curve measures the Marginal Benefit of the last unit of x purchased, MB(x). To maximize consumer surplus, price-taking households set MB(x) equal to the Marginal Cost of x, which for them is the market price, P x, to determine the optimal quantity of x to purchase. Firms are assumed to have the objective of maximizing their profit, the difference between their total revenue from selling a given quantity of x and their total cost of producing it, or (equivalently) of maximizing their producer surplus, the difference between their total revenue from x and the variable cost of producing it. If there are no fixed costs of production, profit and producer surplus are equal in value. To maximize their producer surplus, price-taking firms set the Marginal Benefit from x (which for the firms is the market price, P x ) equal to the Marginal Cost of producing the last unit of x, to determine the optimal quantity of x to produce. We can summarize these relations as follows: Households: Maximize NB(x) = B(x) C(x) (M.11.3) or Consumer Surplus = U(x) P x x by setting MB(x) MU(x) = P x MC(x) to choose x*. Firms: Maximize NB(x) = B(x) C(x) or Producer Surplus = P x x C(x) by setting MB(x) P x = MC(x) to choose x*. (M.11.4) Note that the Total Expenditure on x (P x x) functions as a Cost for households but as a Benefit for firms. Figure M.11-3(a) depicts the familiar supply and demand curve diagram, with equilibrium price P x* and equilibrium quantity x*. For simplicity we shall assume that there are no fixed costs of production. For households, the Total Benefit is the sum of the areas under the demand curve ( ). Their Total Cost is their expenditure on

8 M11-8 MAXIMIZING TOTAL NET BENEFIT FIGURE M.11-3 x (P x* x*), given by areas (2 + 3). Hence their Total Net Benefit or Consumer Surplus is given by ( ) (2 + 3) = area 1. For firms, the Total Benefit (Total Revenue) is given by P x* x*, areas (2 + 3), and the Total Cost is given by area 3. Hence their Total Net Benefit or Producer Surplus (which with no fixed costs is equal to Profit) is given by (2 + 3) (3) = area 2. The Total Social Net Benefit at x* is the sum of Consumer Surplus and Producer Surplus, or areas (1 + 2).

9 MATH MODULE 11: MAXIMIZING TOTAL NET BENEFIT M11-9 Figure M.11-3(b) is the analogue to Figure M.11-3(a), using the corresponding Total variables: B(x), C(x), and Total Expenditure/Revenue (P x* x). The slope of the (P x* x) curve is simply P x*. At x*, the slopes of all three curves are parallel: MB(x*) = MC(x*) = P x*. Since MB(x*) = MC(x*), we know that at x*, Total Social Net Benefit is being maximized. For households, Total Benefit equals the height of the B(x) curve (segments ), while the Total Cost to (Expenditure by) households C(x) = P x* x* equals segments Hence Consumer Surplus is given by B(x) C(x) = segment 1. For firms, Total Benefit equals Total Revenue = P x* x* (segments 2 + 3), while the Total Cost to firms of producing x* equals segment 3. Hence Producer Surplus is given by B(x) C(x) = segment 2. Total Social Net Benefit is the sum of Consumer Surplus and Producer Surplus, given by the combined length of segments As explained above, we know that Total Social Net Benefit is maximized at x*, with P x = P x*, because at that point, MB(x*) = MC(x*). Consider what would happen if P x were greater than P x*. In (b), the P x x line would rotate upward and its slope (= P x ) would be greater. Households, still trying to maximize Consumer Surplus by setting MB(x) = P x, would have to reduce the level of x they demanded to reach a steeper part of the B(x) curve. In contrast, firms would want to increase the level of x they supplied to reach a point on the C(x) curve where MB(x) = P x was equal to the slope of the C(x) function. The result would be a situation of excess supply, not an equilibrium. (You can explain what would happen if P x were lower than P x*, and show that the result would be a situation of excess demand.) Only at x*, where MB(x*) = MC(x*) = P x*, is there an equilibrium of quantity demanded and quantity supplied at a price where competitive households are maximizing their consumer surplus, competitive firms are maximizing their producer surplus, and Total Social Net Benefit is at its maximum level. Example 5: Maximizing Total Net Benefit with Rising Marginal Benefit or Declining Marginal Cost Functions: In Examples 2, 3, and 4, the Marginal Benefit curves are downward-sloping and the Marginal cost curves are upward-sloping. They do not have to take this form, however, for there to be a well-defined level of x that maximizes Total Net Benefit. What is required for there to be a unique level of x that maximizes NB(x)? The Marginal Net Benefit curve must be downward-sloping, and it must intersect the horizontal axis at a single point. But this can occur even if the Marginal Benefit curves are upward-sloping or the Marginal Cost curves are downward-sloping. To convince yourself that this is so, you will want to complete Table M Put the appropriate formulas in Rows 3 6, and the correct numerical solutions in Rows Then for each case draw two graphs, one with B(x) and C(x), and one with MB(x) and MC(x), directly below the first graph. The three cases have apparently quite different B(x) and C(x) functions, and the MB(x) function in Case 4.ii is upward-sloping, while the MC(x) function in Case 4.iii is downward-sloping. Yet as your calculations have shown, all three NB(x) and MNB(x) functions are identical! Hence x*, the optimal level of x, is identical in all three cases, and so is the value of maximum Total Net Benefit, at x*. The crucial factor is that everywhere to the left of x*, MB(x) > MC(x), and so MNB(x) > 0, while everywhere to the right of x*, MB(x) < MC(x), and so MNB(x) < 0. Therefore, only at x* will MB(x) = MC(x), MNB(x) = 0, and NB(x) be at its unique maximum value. When this condition holds, the Set MB(x) = MC(x) Rule for maximizing Total Net Benefit will generate the proper results.

10 M11-10 MAXIMIZING TOTAL NET BENEFIT TABLE M.11-2 Maximizing Total Net Benefit with Non-standard Marginal Curves CASE 4.i 4.ii 4.iii 1. B(x) 12x 0.125x 2 6x x 2 18x 0.5x 2 2. C(x) 8x x 2 2x + 0.5x 2 14x 0.25x 2 3. NB(x) = B(x) C(x) 4. MB(x) 5. MC(x) 6. MNB(x) = MB(x) MC(x) 7. Optimal level of x (x*) 8. MB(x = x*) = MC(x = x*) 9. NB(x= x*) 10. NB(x= 16) 1.2 SOME SITUATIONS WHERE SETTING MB = MC DOESN T WORK In some circumstances, the Set MB = MC method of choosing x* to maximize Total Net Benefit poses problems. This subsection gives examples of some of the most important of these circumstances, and indicates what (if anything) needs to be done to solve the problems when they arise. Example 1: Situations where MB never equals MC: Figure M.11-4 can be used to depict two situations where the MB = MC Rule doesn t work. Line T 1 is the graph of the equation T 1 = 6x. Line T 2 is the graph of the equation T 2 = 4x. These two Total curves have corresponding Marginal curves, M 1 = 6 and M 2 = 4, respectively. Suppose that T 1 and M 1 represent Total and Marginal Benefit, while T 2 and M 2 represent Total and Marginal Cost, respectively. Then at every level of x, each additional unit of x adds \$2 to Total Net Benefit, and the Marginal Net Benefit curve is a horizontal line with the equation MNB(x) = +2. Anyone with this MNB(x) function would want to have x at an infinite level, because the MNB never equals zero, but is always positive. Hence NB(x) constantly increases as x increases, and therefore never reaches a maximum! As a practical matter, in a world of scarce resources we are not likely to encounter an activity that can be carried out at an infinite level and that promises infinite Total Net Benefits. Hence the concern is a theoretical one rather than a practical one: at some finite level, the cost of an additional unit of x is likely to rise. At a theoretical level, however, it does provide one qualification to the MB = MC Rule. Suppose instead that T 1 and M 1 represent Total and Marginal Cost, while T 2 and M 2 represent Total and Marginal Benefit, respectively. In this case, each additional unit of x subtracts \$2 from Total Net Benefit, which is zero when x is zero, and the Marginal Net Benefit curve is thus a horizontal line with the equation MNB(x) = 2. Anyone with this MNB(x) function would want to set x at a zero level, because Total Net Benefit is zero when x equals zero and becomes increasingly more negative as x increases. Again, there

11 MATH MODULE 11: MAXIMIZING TOTAL NET BENEFIT M11-11 is no level of x at which MB(x) = MC(x). With x restricted to non-negative values, however, x* = 0 is a corner solution (as discussed in the text) at which Total Net Benefit is maximized, even though MB(x) < MC(x) and MNB(x) < 0 at this point. FIGURE M.11-4 Example 2: Kinked Total Benefit and Total Cost curves: Figure M.11-5 depicts a kinked Total Benefit curve and the discontinuous Marginal Benefit curve that is derived from it. The Total Cost and Total Benefit curves depicted in Figure M.11-5(a) have the following form: Total Cost C(x) = 4x Total Benefit B(x) = 6x, 0 x 50. B(x) = x, 50 < x.

12 M11-12 MAXIMIZING TOTAL NET BENEFIT The corresponding Marginal curves, depicted in Figure M.11-5(b), have the following form: Marginal Cost MC(x) = 4 Marginal Benefit MB(x) = 6, 0 x 50. MB(x) = 2, 50 < x. FIGURE M.11-5

13 MATH MODULE 11: MAXIMIZING TOTAL NET BENEFIT M11-13 As Figure M.11-5(b) shows, MB(x) > MC(x) to the left of and including x* = 50 units, and MB(x) < MC(x) to the right of x* = 50 units, but at no point does MB(x) actually equal MC(x). Because MNB(x) > 0 to the left of x* = 50 units, however, Total Net Benefit is continuously increasing over the interval 0 < x 50. And because MNB(x) < 0 to the right of x* = 50 units, Total Net Benefit is continuously decreasing over the interval x > 50. Hence at x* = 50 units, Total Net Benefit is maximized. At x* in Figure M.11-5, B(x) = 6(50) = \$300, C(x) = 4(50) = \$200, and NB(x) = B(x) C(x) = \$300 \$200 = \$100. Total Net Benefit appears in both parts of the Figure. In the upper diagram, B(x) is the vertical distance (1 + 2), C(x) is the vertical distance 2, and NB(x) = B(x) C(x) is the vertical distance 1. In the lower diagram, B(x) is the area (1 + 2), C(x) is area 2, and NB(x) = B(x) C(x) is area 1. Example 3: Total Net Benefit is maximized over a range, not at a single point: Figure M.11-6 illustrates a case where the MB = MC Rule works, but where it does not identify a unique level of x at which Total Net Benefit is maximized. In Figure M.11-6(a), between x A and x B, the B(x) and C(x) curves are parallel: their slopes are equal, or in other words MB(x) = MC(x), as shown in Figure M.11-6(b). This means that the Total Net Benefit at x A equals Total Net Benefit at x B : NB(x A ) = NB(x B ). To the left of x A, MB(x) > MC(x), and to the right of x B, MB(x) < MC(x). This fact means that Total Net Benefit is increasing up to x A, constant at its maximum level between x A and x B, and decreasing to the right of x B. Hence the MB = MC Rule tells us that to maximize Total Net Benefit NB(x), we need to pick x so that x A x x B, but the Rule does not identify a unique optimal level of x. Any level of x between and including x A and x B will yield maximum Total Net Benefit. In Figure M.11-6, as we have seen before, areas 1, 2, and 3 in (b) correspond to distances 1, 2, and 3 in (a), respectively. At the origin and at x C, Total Net Benefit is zero, since B(x) = C(x) at both points. This fact also means that in (b), area 1, where MB(x) > MC(x) and Total Net Benefit NB(x) is therefore increasing, must be equal to area 4, where MB(x) < MC(x) and Total Net Benefit NB(x) is therefore decreasing. For x > x C, Total Net Benefit is actually negative, since C(x) is rising more rapidly than B(x). In (b), going from x A to x B, both Total Benefit and Total Cost increase by the same amount (area 3), and so there is no change in Total Net Benefit over this range. Correspondingly, in (a), both Total Benefit and Total Cost are greater at x B than at x A by the vertical distance 3.

14 M11-14 MAXIMIZING TOTAL NET BENEFIT FIGURE M.11-6 Example 4: When MB = MC at Minimum Total Net Benefit: the Second-Order Condition for a Maximum: In Module 10, we saw that if the slope of the tangent to a function is zero, the function may be at a relative or local maximum or at a relative or local minimum position. At a relative maximum, the second derivative of the function (the derivative, or slope, of the derivative) is negative, while at a relative minimum, the second derivative is positive. Parts (a) and (b) of Figure M.11-7 depict a case where MB = MC at two points: x 1 and x*. At x 1, Total Net Benefit NB(x) is at a local minimum: decreasing or increasing the level of x will increase NB(x). In contrast, x* is a local maximum: decreasing or increasing the level of x will decrease NB(x). The difference between the two points is that at x 1, the slope of the MC curve is negative while the slope of the MB curve is zero. Hence the slope of the Marginal Net Benefit (MNB) curve, which equals the slope of the MB curve minus the slope of the MC curve, is

15 MATH MODULE 11: MAXIMIZING TOTAL NET BENEFIT M11-15 positive. In contrast, at x* the slope of the MNB curve is negative. Hence x 1 is a local minimum and x* is a local maximum. Using (b), we can say that if the MC curve cuts the MB curve from above, as at x 1, we are at a local minimum, and if it cuts the MB curve from below, as at x*, we are at a local maximum. In this case, x* is not only a local maximum but also the point at which globally (for all values of x) Total Net Benefit is maximized. Example 5 shows, however, that this condition is necessary but not sufficient for the MB = MC Rule to work. FIGURE M.11-7 Example 5: When setting x = 0 beats picking the level of x where MB = MC: In Figure M.11-7, (c) and (d) look very similar to (a) and (b). In (d), x 1 is a local minimum and x 2 is a local maximum. Yet x 2 is not the optimal level of x, despite the fact that it is a local maximum value for NB(x), with MB = MC. Instead, the level of x that maximizes Total Net Benefit is x* = 0! The reason is clear when we look at (c) immediately above. At the origin, B(x) = C(x), and so Total Net Benefit is zero. For all x

16 M11-16 MAXIMIZING TOTAL NET BENEFIT FIGURE M.11-8 > 0, including at x 2, B(x) < C(x), and so Total Net Benefit is negative everywhere except at x = 0! Therefore x = 0 is the level of x that maximizes Total Net Benefit. By comparing (b) and (d), we can identify the reason for the difference. In both diagrams, area 1 is equal to the Total Net Benefit at x 1 ; it is negative, since between 0 and x 1, MC(x) > MB(x). Between x 1 and x 2, MB(x) > MC(x), and so over this interval, area 2 is added to Total Net Benefit. In (b), area 2 is greater than area 1, and so at x*, Total Net Benefit is positive. In (d), however, area 1 is greater than area 2, and so at x 2, Total Net Benefit is negative. Hence the zero Total Net Benefit at x* = 0 is preferable to the negative Total Net Benefit at x 2. Example 6: The Case of Multiple Equilibria: Examples 4 and 5 showed that not all points at which MB = MC are ones that maximize Total Net Benefit. Figure M.11-8 shows that there can be multiple equilibria, or local maximum points, where MB = MC and the second-order condition (that the MNB curve is negatively sloped) are both satisfied. In Figure M.11-8, Total Net Benefit NB(x) reaches a relative or local maximum at points A, C, E, and G. Points B, D, and F are all local minimum points. Note that the maximum and minimum points alternate. Of the local maximum points, how do we pick the one that is the global or overall maximum? In some cases, where there are potentially an infinite number of local maximum points, choosing between them can be effectively impossible. In the present case, however, and in similar cases, it is fairly easy. Here we have to compare and rank only four points, A, C, E, and G. Taking the absolute values of the areas 1 to 7, we proceed as follows. At A, NB(x) is equal to area 1. NB(x) at C is greater than at A, because (since area 3 > area 2) area 1 + (3 2) > area 1. Similarly, NB(x) at E is greater than at C, since area 5 > area 4, and therefore area 1 + (3 2) + (5 4) > area 1 + (3 2). Since area 6 > area 7, however, NB(x) at E is greater than at G: area 1 + (3 2) + (5 4) > area 1 + (3 2) + (5 4) + (7 6). Hence Total Net Benefit is maximized when x is at the level of E. This pairwise comparison method does not require a complete ordering of all the local maximum points. We do not have to determine, for instance, whether Total Net Benefit is greater at C or at G, merely that it is greater at E than at either C or G. When the number of local maximum points is considerable, however, it can be a resourceconsuming job to pick out the point with the highest Total Net Benefit from all of the candidates, each of which satisfies the MB = MC criterion. As a practical matter, most of the cases you will likely encounter in your microeconomics course will be well-behaved ones not because the problem cases outlined in this Module do not occur in the real world, but solely for mathematical simplicity. Yet it is a good idea to recognize that behind the simple decision rule To

17 MATH MODULE 11: MAXIMIZING TOTAL NET BENEFIT M11-17 maximize the Total Net Benefit of x, set Marginal Benefit equal to Marginal Cost lurk a number of implicit conditions and potential exceptions. 2. Exercises 1. For an activity x that can be carried on at a range of levels, the Total Cost and Total Benefit curves have the following form, with x in units and B(x) and C(x) in dollars: Total Cost C(x) 4x Total Benefit B(x) 10x, 0 x 10. B(x) 100, 10 x. (a) Calculate the levels of x at which Total Net Benefit NB(x) 0; the level of x at which NB(x) is maximized (x*); and the value of B(x), C(x), and NB(x) at x*. (b) Give the formulas for the MB(x) and MC(x) curves corresponding to these Total Cost and Total Benefit functions, and explain the relationship between them at x* and where NB(x) In the following three perfectly competitive markets, the Total Benefit B(x) function for all consumer households and the Total Cost C(x) function for all producer firms are given, where x is in tonnes and B(x) and C(x) are in dollars. For each case, give the equations for the demand and supply curves; the equilibrium quantity of x (x*); the equilibrium price P x* : Total Expenditure by households Total Revenue of firms P x* x*; Total Social Net Benefit (the sum of consumer and producer surplus); Consumer Surplus; and Producer Surplus. Briefly compare your results. (a) Households: B(x) 32x x 2 ; Firms: 8x 2x 2. (b) Households: B(x) 32x 2x 2 ; Firms: 8x x 2. (c) Households: B(x) 28x 0.25x 2 ; Firms: 16x 0.5x Given the following diagram, state whether propositions (a) to (j) about the diagram are True or False: (a) Total Cost is constant. (b) Total Benefit is increasing between D and E. (c) Total Benefit is increasing between E and F.

18 M11-18 MAXIMIZING TOTAL NET BENEFIT (d) Total Benefit is increasing between F and G. (e) Marginal Net Benefit is increasing between O and B and decreasing between B and D. (f) Total Net Benefit is decreasing between B and C and increasing between C and D. (g) Marginal Net Benefit is negative between B and C and positive between C and E. (h) Total Net Benefit is increasing between F and H. (i) Total Benefit is maximized at E. (j) Total Net Benefit is maximized at J. **4. [This question is a chance to practise your calculus, using the Table in Module 10 if you need it.] For an activity x, the Total Benefit is given by the formula B(x) = 54x 1/2, and the Total Cost is given by C(x) = 2x 3/2, where x is measured in units and B(x) and C(x) are measured in dollars. (a) Give the formulas for the Marginal Benefit, Marginal Cost, and Total Net Benefit functions. (b) Calculate the value of x (x*) at which Total Net Benefit is maximized, and give the values for B(x*), C(x*), NB(x*) MB(x*), and MC(x*). (c) Calculate the positive level of x at which Total Net Benefit is equal to zero, and give the values for Total Benefit and Total Cost at this level of x.

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