MATH MODULE 11. Maximizing Total Net Benefit. 1. Discussion M111


 Lindsay Long
 1 years ago
 Views:
Transcription
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 decisionmaking 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 M111
2 M112 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.111 indicates. TABLE M.111 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.111(a), and the MNB(x) function is graphed immediately below it in Figure M.111(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 M113 FIGURE M.111 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 M114 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.111 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 M101 to M104. 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.111 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.112 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.112(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.112(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.112(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.112(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.112(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 M115 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.112
6 M116 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.112 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 M117 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, pricetaking 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.111 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, pricetaking 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, pricetaking 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.113(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 M118 MAXIMIZING TOTAL NET BENEFIT FIGURE M.113 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 M119 Figure M.113(b) is the analogue to Figure M.113(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 downwardsloping and the Marginal cost curves are upwardsloping. They do not have to take this form, however, for there to be a welldefined 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 downwardsloping, and it must intersect the horizontal axis at a single point. But this can occur even if the Marginal Benefit curves are upwardsloping or the Marginal Cost curves are downwardsloping. 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 upwardsloping, while the MC(x) function in Case 4.iii is downwardsloping. 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 M1110 MAXIMIZING TOTAL NET BENEFIT TABLE M.112 Maximizing Total Net Benefit with Nonstandard 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.114 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 M1111 is no level of x at which MB(x) = MC(x). With x restricted to nonnegative 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.114 Example 2: Kinked Total Benefit and Total Cost curves: Figure M.115 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.115(a) have the following form: Total Cost C(x) = 4x Total Benefit B(x) = 6x, 0 x 50. B(x) = x, 50 < x.
12 M1112 MAXIMIZING TOTAL NET BENEFIT The corresponding Marginal curves, depicted in Figure M.115(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.115
13 MATH MODULE 11: MAXIMIZING TOTAL NET BENEFIT M1113 As Figure M.115(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.115, 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.116 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.116(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.116(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.116, 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 M1114 MAXIMIZING TOTAL NET BENEFIT FIGURE M.116 Example 4: When MB = MC at Minimum Total Net Benefit: the SecondOrder 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.117 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 M1115 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.117 Example 5: When setting x = 0 beats picking the level of x where MB = MC: In Figure M.117, (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 M1116 MAXIMIZING TOTAL NET BENEFIT FIGURE M.118 > 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.118 shows that there can be multiple equilibria, or local maximum points, where MB = MC and the secondorder condition (that the MNB curve is negatively sloped) are both satisfied. In Figure M.118, 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 wellbehaved 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 M1117 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 M1118 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.
Review of Fundamental Mathematics
Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decisionmaking tools
More informationElasticity. I. What is Elasticity?
Elasticity I. What is Elasticity? The purpose of this section is to develop some general rules about elasticity, which may them be applied to the four different specific types of elasticity discussed in
More informationManagerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay. Lecture  13 Consumer Behaviour (Contd )
(Refer Slide Time: 00:28) Managerial Economics Prof. Trupti Mishra S.J.M. School of Management Indian Institute of Technology, Bombay Lecture  13 Consumer Behaviour (Contd ) We will continue our discussion
More informationECON 600 Lecture 3: Profit Maximization Π = TR TC
ECON 600 Lecture 3: Profit Maximization I. The Concept of Profit Maximization Profit is defined as total revenue minus total cost. Π = TR TC (We use Π to stand for profit because we use P for something
More informationChapter 27: Taxation. 27.1: Introduction. 27.2: The Two Prices with a Tax. 27.2: The PreTax Position
Chapter 27: Taxation 27.1: Introduction We consider the effect of taxation on some good on the market for that good. We ask the questions: who pays the tax? what effect does it have on the equilibrium
More informationEconomics 101 Fall 2013 Answers to Homework 5 Due Tuesday, November 19, 2013
Economics 101 Fall 2013 Answers to Homework 5 Due Tuesday, November 19, 2013 Directions: The homework will be collected in a box before the lecture. Please place your name, TA name and section number on
More informationEconomics 165 Winter 2002 Problem Set #2
Economics 165 Winter 2002 Problem Set #2 Problem 1: Consider the monopolistic competition model. Say we are looking at sailboat producers. Each producer has fixed costs of 10 million and marginal costs
More informationAn increase in the number of students attending college. shifts to the left. An increase in the wage rate of refinery workers.
1. Which of the following would shift the demand curve for new textbooks to the right? a. A fall in the price of paper used in publishing texts. b. A fall in the price of equivalent used text books. c.
More informationNumber of Workers Number of Chairs 1 10 2 18 3 24 4 28 5 30 6 28 7 25
Intermediate Microeconomics Economics 435/735 Fall 0 Answers for Practice Problem Set, Chapters 68 Chapter 6. Suppose a chair manufacturer is producing in the short run (with its existing plant and euipment).
More informationShortRun Production and Costs
ShortRun Production and Costs The purpose of this section is to discuss the underlying work of firms in the shortrun the production of goods and services. Why is understanding production important to
More informationModule 2 Lecture 5 Topics
Module 2 Lecture 5 Topics 2.13 Recap of Relevant Concepts 2.13.1 Social Welfare 2.13.2 Demand Curves 2.14 Elasticity of Demand 2.14.1 Perfectly Inelastic 2.14.2 Perfectly Elastic 2.15 Production & Cost
More informationUnit 7. Firm behaviour and market structure: monopoly
Unit 7. Firm behaviour and market structure: monopoly Learning objectives: to identify and examine the sources of monopoly power; to understand the relationship between a monopolist s demand curve and
More informationMARKETS WITHOUT POWER Microeconomics in Context (Goodwin, et al.), 3 rd Edition
Chapter 16 MARKETS WITHOUT POWER Microeconomics in Context (Goodwin, et al.), 3 rd Edition Chapter Summary This chapter presents the traditional, idealized model of perfect competition. In it, you will
More informationLecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization
Lecture 2. Marginal Functions, Average Functions, Elasticity, the Marginal Principle, and Constrained Optimization 2.1. Introduction Suppose that an economic relationship can be described by a realvalued
More informationChapter 3. The Concept of Elasticity and Consumer and Producer Surplus. Chapter Objectives. Chapter Outline
Chapter 3 The Concept of Elasticity and Consumer and roducer Surplus Chapter Objectives After reading this chapter you should be able to Understand that elasticity, the responsiveness of quantity to changes
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MPP 801 Perfect Competition K. Wainwright Study Questions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Refer to Figure 91. If the price a perfectly
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
MBA 640 Survey of Microeconomics Fall 2006, Quiz 6 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A monopoly is best defined as a firm that
More informationCompetitive Market Equilibrium
Chapter 14 Competitive Market Equilibrium We have spent the bulk of our time up to now developing relationships between economic variables and the behavior of agents such as consumers, workers and producers.
More informationCHAPTER 10 MARKET POWER: MONOPOLY AND MONOPSONY
CHAPTER 10 MARKET POWER: MONOPOLY AND MONOPSONY EXERCISES 3. A monopolist firm faces a demand with constant elasticity of .0. It has a constant marginal cost of $0 per unit and sets a price to maximize
More informationA Detailed Price Discrimination Example
A Detailed Price Discrimination Example Suppose that there are two different types of customers for a monopolist s product. Customers of type 1 have demand curves as follows. These demand curves include
More informationor, put slightly differently, the profit maximizing condition is for marginal revenue to equal marginal cost:
Chapter 9 Lecture Notes 1 Economics 35: Intermediate Microeconomics Notes and Sample Questions Chapter 9: Profit Maximization Profit Maximization The basic assumption here is that firms are profit maximizing.
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Chapter 11 Perfect Competition  Sample Questions MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) Perfect competition is an industry with A) a
More informationEquations and Inequalities
Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.
More informationEfficiency, Optimality, and Competitive Market Allocations
Efficiency, Optimality, and Competitive Market Allocations areto Efficient Allocations An allocation of resources is areto efficient if it is not possible to reallocate resources in economy to make one
More informationEconomics 352: Intermediate Microeconomics
Economics 35: Intermediate Microeconomics Notes and Sample Questions Chapter Twelve: The Partial Equilibrium Competitive Model and Applied Competitive Analysis This chapter will investigate perfect competition
More informationWeek 1: Functions and Equations
Week 1: Functions and Equations Goals: Review functions Introduce modeling using linear and quadratic functions Solving equations and systems Suggested Textbook Readings: Chapter 2: 2.12.2, and Chapter
More informationLearning Objectives. After reading Chapter 11 and working the problems for Chapter 11 in the textbook and in this Workbook, you should be able to:
Learning Objectives After reading Chapter 11 and working the problems for Chapter 11 in the textbook and in this Workbook, you should be able to: Discuss three characteristics of perfectly competitive
More informationSession 7 Bivariate Data and Analysis
Session 7 Bivariate Data and Analysis Key Terms for This Session Previously Introduced mean standard deviation New in This Session association bivariate analysis contingency table covariation least squares
More informationManagerial Economics Prof. Trupti Mishra S.J.M School of Management Indian Institute of Technology, Bombay. Lecture  14 Elasticity of Supply
Managerial Economics Prof. Trupti Mishra S.J.M School of Management Indian Institute of Technology, Bombay Lecture  14 Elasticity of Supply We will continue our discussion today, on few more concept of
More informationMarket Supply in the Short Run
Equilibrium in Perfectly Competitive Markets (Assume for simplicity that all firms have access to the same technology and input markets, so they all have the same cost curves.) Market Supply in the Short
More informationLimit processes are the basis of calculus. For example, the derivative. f f (x + h) f (x)
SEC. 4.1 TAYLOR SERIES AND CALCULATION OF FUNCTIONS 187 Taylor Series 4.1 Taylor Series and Calculation of Functions Limit processes are the basis of calculus. For example, the derivative f f (x + h) f
More informationWhen the price of a good rises (falls) and all other variables affecting the
ONLINE APPENDIX 1 Substitution and Income Effects of a Price Change When the price of a good rises (falls) and all other variables affecting the consumer remain the same, the law of demand tells us that
More informationTaylor Polynomials and Taylor Series Math 126
Taylor Polynomials and Taylor Series Math 26 In many problems in science and engineering we have a function f(x) which is too complicated to answer the questions we d like to ask. In this chapter, we will
More informationD) Marginal revenue is the rate at which total revenue changes with respect to changes in output.
Ch. 9 1. Which of the following is not an assumption of a perfectly competitive market? A) Fragmented industry B) Differentiated product C) Perfect information D) Equal access to resources 2. Which of
More informationLinear Programming. Solving LP Models Using MS Excel, 18
SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting
More informationSummary Chapter 12 Monopoly
Summary Chapter 12 Monopoly Defining Monopoly  A monopoly is a market structure in which a single seller of a product with no close substitutes serves the entire market  One practical measure for deciding
More informationPrinciples of Economics: Micro: Exam #2: Chapters 110 Page 1 of 9
Principles of Economics: Micro: Exam #2: Chapters 110 Page 1 of 9 print name on the line above as your signature INSTRUCTIONS: 1. This Exam #2 must be completed within the allocated time (i.e., between
More informationMicroeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS
DUSP 11.203 Frank Levy Microeconomics Sept. 16, 2010 NOTES ON CALCULUS AND UTILITY FUNCTIONS These notes have three purposes: 1) To explain why some simple calculus formulae are useful in understanding
More informationchapter >> Consumer and Producer Surplus Section 3: Consumer Surplus, Producer Surplus, and the Gains from Trade
chapter 6 >> Consumer and Producer Surplus Section 3: Consumer Surplus, Producer Surplus, and the Gains from Trade One of the nine core principles of economics we introduced in Chapter 1 is that markets
More informationChapter 23. Monopoly
Chapter 23 Monopoly We will now turn toward an analysis of the polar opposite of the extreme assumption of perfect competition that we have employed thus far. 1 Under perfect competition, we have assumed
More informationChapter 4 Online Appendix: The Mathematics of Utility Functions
Chapter 4 Online Appendix: The Mathematics of Utility Functions We saw in the text that utility functions and indifference curves are different ways to represent a consumer s preferences. Calculus can
More informationSubstitution and Income Effect, Individual and Market Demand, Consumer Surplus. 1 Substitution Effect, Income Effect, Giffen Goods
Substitution Effect, Income Effect, Giffen Goods 4.0 Principles of Microeconomics, Fall 007 ChiaHui Chen September 9, 007 Lecture 7 Substitution and Income Effect, Individual and Market Demand, Consumer
More informationChapter 3 Consumer Behavior
Chapter 3 Consumer Behavior Read Pindyck and Rubinfeld (2013), Chapter 3 Microeconomics, 8 h Edition by R.S. Pindyck and D.L. Rubinfeld Adapted by Chairat Aemkulwat for Econ I: 2900111 1/29/2015 CHAPTER
More informationc. Given your answer in part (b), what do you anticipate will happen in this market in the longrun?
Perfect Competition Questions Question 1 Suppose there is a perfectly competitive industry where all the firms are identical with identical cost curves. Furthermore, suppose that a representative firm
More informationMicroeconomics Instructor Miller Practice Problems Monopolistic Competition
Microeconomics Instructor Miller Practice Problems Monopolistic Competition 1. A monopolistically competitive market is described as one in which there are A) a few firms producing an identical product.
More informationThe Keynesian Cross. A Fixed Price Level. The Simplest KeynesianCross Model: Autonomous Consumption Only
The Keynesian Cross Some instructors like to develop a more detailed macroeconomic model than is presented in the textbook. This supplemental material provides a concise description of the Keynesiancross
More informationDEMAND AND SUPPLY IN FACTOR MARKETS
Chapter 14 DEMAND AND SUPPLY IN FACTOR MARKETS Key Concepts Prices and Incomes in Competitive Factor Markets Factors of production (labor, capital, land, and entrepreneurship) are used to produce output.
More informationLet s explore the content and skills assessed by Heart of Algebra questions.
Chapter 9 Heart of Algebra Heart of Algebra focuses on the mastery of linear equations, systems of linear equations, and linear functions. The ability to analyze and create linear equations, inequalities,
More informationMidterm Exam #1  Answers
Page 1 of 9 Midterm Exam #1 Answers Instructions: Answer all questions directly on these sheets. Points for each part of each question are indicated, and there are 1 points total. Budget your time. 1.
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Chapter 11 Monopoly practice Davidson spring2007 MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 1) A monopoly industry is characterized by 1) A)
More informationFinal Exam 15 December 2006
Eco 301 Name Final Exam 15 December 2006 120 points. Please write all answers in ink. You may use pencil and a straight edge to draw graphs. Allocate your time efficiently. Part 1 (10 points each) 1. As
More informationEQUATIONS and INEQUALITIES
EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line
More informationTaxes and Subsidies PRINCIPLES OF ECONOMICS (ECON 210) BEN VAN KAMMEN, PHD
Taxes and Subsidies PRINCIPLES OF ECONOMICS (ECON 210) BEN VAN KAMMEN, PHD Introduction We have already established that taxes are one of the reasons that supply decreases. Subsidies, which could be called
More informationCHAPTER 11 PRICE AND OUTPUT IN MONOPOLY, MONOPOLISTIC COMPETITION, AND PERFECT COMPETITION
CHAPTER 11 PRICE AND OUTPUT IN MONOPOLY, MONOPOLISTIC COMPETITION, AND PERFECT COMPETITION Chapter in a Nutshell Now that we understand the characteristics of different market structures, we ask the question
More informationThe Graphical Method: An Example
The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,
More informationSolving Quadratic Equations
9.3 Solving Quadratic Equations by Using the Quadratic Formula 9.3 OBJECTIVES 1. Solve a quadratic equation by using the quadratic formula 2. Determine the nature of the solutions of a quadratic equation
More informationchapter Behind the Supply Curve: >> Inputs and Costs Section 2: Two Key Concepts: Marginal Cost and Average Cost
chapter 8 Behind the Supply Curve: >> Inputs and Costs Section 2: Two Key Concepts: Marginal Cost and Average Cost We ve just seen how to derive a firm s total cost curve from its production function.
More informationIntroduction. Agents have preferences over the two goods which are determined by a utility function. Speci cally, type 1 agents utility is given by
Introduction General equilibrium analysis looks at how multiple markets come into equilibrium simultaneously. With many markets, equilibrium analysis must take explicit account of the fact that changes
More informationIn early 2002, with New Jersey facing a $5.3 billion budget gap, Governor
19.1 The Three Rules of Tax Incidence The Equity Implications of Taxation: Tax Incidence 19 19.2 Tax Incidence Extensions 19.3 General Equilibrium Tax Incidence 19.4 The Incidence of Taxation in the United
More informationRecitation #5 Week 02/08/2009 to 02/14/2009. Chapter 6  Elasticity
Recitation #5 Week 02/08/2009 to 02/14/2009 Chapter 6  Elasticity 1. This problem explores the midpoint method of calculating percentages and why this method is the preferred method when calculating price
More information2. THE xy PLANE 7 C7
2. THE xy PLANE 2.1. The Real Line When we plot quantities on a graph we can plot not only integer values like 1, 2 and 3 but also fractions, like 3½ or 4¾. In fact we can, in principle, plot any real
More informationChapter. Perfect Competition CHAPTER IN PERSPECTIVE
Perfect Competition Chapter 10 CHAPTER IN PERSPECTIVE In Chapter 10 we study perfect competition, the market that arises when the demand for a product is large relative to the output of a single producer.
More informationLab 17: Consumer and Producer Surplus
Lab 17: Consumer and Producer Surplus Who benefits from rent controls? Who loses with price controls? How do taxes and subsidies affect the economy? Some of these questions can be analyzed using the concepts
More informationAP MICROECONOMICS 2015 SCORING GUIDELINES
AP MICROECONOMICS 2015 SCORING GUIDELINES Question 1 10 points (1+5+1+3) (a) 1 point: One point is earned for stating that the firm s price is equal to the market price because the firm is a price taker.
More informationMarginal Decisions and Externalities  Examples 1
Marginal Decisions and Externalities  Examples 1 Externalities drive the free market away from the socially efficient equilibrium. As microeconomists, we re interested in maximizing social welfare, so
More informationNotes on indifference curve analysis of the choice between leisure and labor, and the deadweight loss of taxation. Jon Bakija
Notes on indifference curve analysis of the choice between leisure and labor, and the deadweight loss of taxation Jon Bakija This example shows how to use a budget constraint and indifference curve diagram
More informationchapter >> Making Decisions Section 2: Making How Much Decisions: The Role of Marginal Analysis
chapter 7 >> Making Decisions Section : Making How Much Decisions: The Role of Marginal Analysis As the story of the two wars at the beginning of this chapter demonstrated, there are two types of decisions:
More informationCHAPTER 9 MAXIMIZING PROFIT
CHAPTER 9 MAXIMIZING PROFIT Chapter in a Nutshell In Chapter 8, we hinted at how you might determine whether a firm is making a profit or a loss by comparing the price of a good with its average total
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More information1. Briefly explain what an indifference curve is and how it can be graphically derived.
Chapter 2: Consumer Choice Short Answer Questions 1. Briefly explain what an indifference curve is and how it can be graphically derived. Answer: An indifference curve shows the set of consumption bundles
More information5. The supply curve of a monopolist is A) upward sloping. B) nonexistent. C) perfectly inelastic. D) horizontal.
Chapter 12 monopoly 1. A monopoly firm is different from a competitive firm in that A) there are many substitutes for a monopolist's product but there are no substitutes for a competitive firm's product.
More informationChapter 4. Elasticity
Chapter 4 Elasticity comparative static exercises in the supply and demand model give us the direction of changes in equilibrium prices and quantities sometimes we want to know more we want to know about
More informationCopyright 1997, David G. Messerschmitt. All rights reserved. Economics tutorial. David G. Messerschmitt CS 2946, EE 290X, BA 296.
Economics tutorial David G. Messerschmitt CS 2946, EE 290X, BA 296.5 Copyright 1997, David G. Messerschmitt 3/5/97 1 Here we give a quick tutorial on the economics material that has been covered in class.
More information1 Mathematical Models of Cost, Revenue and Profit
Section 1.: Mathematical Modeling Math 14 Business Mathematics II Minh Kha Goals: to understand what a mathematical model is, and some of its examples in business. Definition 0.1. Mathematical Modeling
More informationIndifference Curves and the Marginal Rate of Substitution
Introduction Introduction to Microeconomics Indifference Curves and the Marginal Rate of Substitution In microeconomics we study the decisions and allocative outcomes of firms, consumers, households and
More informationPractice Questions Week 8 Day 1
Practice Questions Week 8 Day 1 Multiple Choice Identify the choice that best completes the statement or answers the question. 1. The characteristics of a market that influence the behavior of market participants
More informationChapter 12: Cost Curves
Chapter 12: Cost Curves 12.1: Introduction In chapter 11 we found how to minimise the cost of producing any given level of output. This enables us to find the cheapest cost of producing any given level
More informationUnit 3 Practice Exam Answer the questions on a separate sheet of paperplease do not write on this practice test.
Unit 3 Practice Exam Answer the questions on a separate sheet of paperplease do not write on this practice test. 1. Which of the following items is most likely to be an implicit cost of production? a.
More informationOn Lexicographic (Dictionary) Preference
MICROECONOMICS LECTURE SUPPLEMENTS Hajime Miyazaki File Name: lexico95.usc/lexico99.dok DEPARTMENT OF ECONOMICS OHIO STATE UNIVERSITY Fall 993/994/995 Miyazaki.@osu.edu On Lexicographic (Dictionary) Preference
More informationChapter 13 Perfect Competition
Chapter 13 Perfect Competition 13.1 A Firm's ProfitMaximizing Choices 1) What is the difference between perfect competition and monopolistic competition? A) Perfect competition has a large number of small
More informationPreTest Chapter 22 ed17
PreTest Chapter 22 ed17 Multiple Choice Questions 1. Refer to the above diagram. At the profitmaximizing level of output, total revenue will be: A. NM times 0M. B. 0AJE. C. 0EGC. D. 0EHB. 2. For a pure
More informationChapter 14: Firms in Competitive Markets. Total revenue = price per unit sold number of units sold = p q
Chapter 14: Firms in Competitive Markets Profit and Revenue The firm's goal is to maximize profit. Profit = total revenue  total cost (opportunity cost) Total revenue = price per unit sold number of units
More informationCHAPTER 12 MARKETS WITH MARKET POWER Microeconomics in Context (Goodwin, et al.), 2 nd Edition
CHAPTER 12 MARKETS WITH MARKET POWER Microeconomics in Context (Goodwin, et al.), 2 nd Edition Chapter Summary Now that you understand the model of a perfectly competitive market, this chapter complicates
More informationI. Introduction to Taxation
University of PacificEconomics 53 Lecture Notes #17 I. Introduction to Taxation Government plays an important role in most modern economies. In the United States, the role of the government extends from
More informationMATH MODULE 5. Total, Average, and Marginal Functions. 1. Discussion M51
MATH MODULE Total, Average, and Marginal Functions 1. Discussion A very important skill for economists is the ability to relate total, average, and marginal curves. Much of standard microeconomics involves
More informationTeaching and Learning Guide 3: Linear Equations Further Topics
Guide 3: Linear Equations Further Topics Table of Contents Section 1: Introduction to the guide...3 Section : Solving simultaneous equations using graphs...4 1. The concept of solving simultaneous equations
More informationChapter 8. Competitive Firms and Markets
Chapter 8. Competitive Firms and Markets We have learned the production function and cost function, the question now is: how much to produce such that firm can maximize his profit? To solve this question,
More informationCHAPTER 3 CONSUMER BEHAVIOR
CHAPTER 3 CONSUMER BEHAVIOR EXERCISES 2. Draw the indifference curves for the following individuals preferences for two goods: hamburgers and beer. a. Al likes beer but hates hamburgers. He always prefers
More information7.5 SYSTEMS OF INEQUALITIES. Copyright Cengage Learning. All rights reserved.
7.5 SYSTEMS OF INEQUALITIES Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of inequalities in two variables. Solve systems of inequalities. Use systems of inequalities
More information13 Externalities. Microeconomics I  Lecture #13, May 12, 2009
Microeconomics I  Lecture #13, May 12, 2009 13 Externalities Up until now we have implicitly assumed that each agent could make consumption or production decisions without worrying about what other agents
More informationc 2008 Je rey A. Miron We have described the constraints that a consumer faces, i.e., discussed the budget constraint.
Lecture 2b: Utility c 2008 Je rey A. Miron Outline: 1. Introduction 2. Utility: A De nition 3. Monotonic Transformations 4. Cardinal Utility 5. Constructing a Utility Function 6. Examples of Utility Functions
More informationPractice Problem Set 2 (ANSWERS)
Economics 370 Professor H.J. Schuetze Practice Problem Set 2 (ANSWERS) 1. See the figure below, where the initial budget constraint is given by ACE. After the new legislation is passed, the budget constraint
More information5.4 The Quadratic Formula
Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function
More informationThe Revenue of a Competitive In perfect competition, average revenue equals the price of the good. Total revenue Average Revenue = = The Revenue of a
In this chapter, look for the answers to these questions: What is a perfectly competitive market? What is marginal revenue? How is it related to total and average revenue? How does a competitive firm determine
More informationProblem Set 2  Answers. Gains from Trade and the Ricardian Model
Page 1 of 11 Gains from Trade and the Ricardian Model 1. Use community indifference curves as your indicator of national welfare in order to evaluate the following claim: An improvement in the terms of
More informationThe PointSlope Form
7. The PointSlope Form 7. OBJECTIVES 1. Given a point and a slope, find the graph of a line. Given a point and the slope, find the equation of a line. Given two points, find the equation of a line y Slope
More informationOpen Economy Macroeconomics: The ISLMBP Model
Open Economy Macroeconomics: The ISLMBP Model When we open the economy to international transactions we have to take into account the effects of trade in goods and services (i.e. items in the current
More informationGraphical Integration Exercises Part Four: Reverse Graphical Integration
D4603 1 Graphical Integration Exercises Part Four: Reverse Graphical Integration Prepared for the MIT System Dynamics in Education Project Under the Supervision of Dr. Jay W. Forrester by Laughton Stanley
More informationECON 103, 20082 ANSWERS TO HOME WORK ASSIGNMENTS
ECON 103, 20082 ANSWERS TO HOME WORK ASSIGNMENTS Due the Week of June 23 Chapter 8 WRITE [4] Use the demand schedule that follows to calculate total revenue and marginal revenue at each quantity. Plot
More informationPerman et al.: Ch. 4. Welfare economics and the environment
Perman et al.: Ch. 4 Welfare economics and the environment Objectives of lecture Derive the conditions for allocative efficiency of resources Show that a perfect market provides efficient allocation of
More information