ࡱ>  w#` 0bjbjmm 8Ir,r,r,8,^-<W..2"P2P2P2P2243rVtVtVtVtVtVtV$Xh][tV3P2P233VP2P2V45=5=5=3P2P2rV5=3rV5=5=PsRP2. `$Y}Br,i:hFR$RV<WjR$[<[HR[R *3>35=L3 X3S*3*3*3VV<d*3*3*3W3333d%r,r, CHAPTER 7 THE COST OF PRODUCTION QUESTIONS FOR REVIEW 1. A firm pays its accountant an annual retainer of $10,000. Is this an economic cost? Explicit costs are actual outlays. They include all costs that involve a monetary transaction. An implicit cost is an economic cost that does not necessarily involve a monetary transaction, but still involves the use of resources. When a firm pays an annual retainer of $10,000, there is a monetary transaction. The accountant trades his or her time in return for money. Therefore, an annual retainer is an explicit cost. 2. The owner of a small retail store does her own accounting work. How would you measure the opportunity cost of her work? Opportunity costs are measured by comparing the use of a resource with its alternative uses. The opportunity cost of doing accounting work is the time not spent in other ways, i.e., time such as running a small business or participating in leisure activity. The economic, or opportunity, cost of doing accounting work is measured by computing the monetary amount that the owners time would be worth in its next best use. 3. Please explain whether the following statements are true or false. If the owner of a business pays himself no salary, then the accounting cost is zero, but the economic cost is positive. True. Since there is no monetary transaction, there is no accounting, or explicit, cost. However, since the owner of the business could be employed elsewhere, there is an economic cost. The economic cost is positive, reflecting the opportunity cost of the owners time. The economic cost is the value of the next best alternative, or the amount that the owner would earn if he took the next best job. A firm that has positive accounting profit does not necessarily have positive economic profit. True. Accounting profit considers only the explicit, monetary costs. Since there may be some opportunity costs that were not fully realized as explicit monetary costs, it is possible that when the opportunity costs are added in, economic profit will become negative. This indicates that the firms resources are not being put to their best use. If a firm hires a currently unemployed worker, the opportunity cost of utilizing the workers services is zero. False. The opportunity cost measures the value of the workers time, which is unlikely to be zero. Though the worker was temporarily unemployed, the worker still possesses skills, which have a value and make the opportunity cost of hiring the worker greater than zero. In addition, since opportunity cost is the equivalent of the workers next best option, it is possible that the worker might have been able to get a better job that utilizes his skills more efficiently. Alternatively, the worker could have been doing unpaid work, such as care of a child or elderly person at home, which would have had a value to those receiving the service. 4. Suppose that labor is the only variable input to the production process. If the marginal cost of production is diminishing as more units of output are produced, what can you say about the marginal product of labor ? The marginal product of labor must be increasing. The marginal cost of production measures the extra cost of producing one more unit of output. If this cost is diminishing, then it must be taking fewer units of labor to produce the extra unit of output, since the extra cost refers to the extra cost of the labor. If fewer units of labor are required to produce a unit of output, then the marginal product (extra output produced by an extra unit of labor) must be increasing. Note also, that MC=w/MPL, so that if MC is diminishing then MPL must be increasing for any given w. 5. Suppose a chair manufacturer finds that the marginal rate of technical substitution of capital for labor in his production process is substantially greater than the ratio of the rental rate on machinery to the wage rate for assembly-line labor. How should he alter his use of capital and labor to minimize the cost of production? To minimize cost, the manufacturer should use a combination of capital and labor so the rate at which he can trade capital for labor in his production process is the same as the rate at which he can trade capital for labor in external markets. The manufacturer would be better off if he increased his use of capital and decreased his use of labor, decreasing the marginal rate of technical substitution, MRTS. He should continue this substitution until his MRTS equals the ratio of the rental rate to the wage rate. The MRTS in this case is equal to MPK/MPL. As the manufacturer uses more K and less L, the MPK will diminish and the MPL will increase, both of which will decrease the MRTS until it is equal to the ratio of the input prices (rental rate on capital divided by wage rate). 6. Why are isocost lines straight lines? The isocost line represents all possible combinations of labor and capital that may be purchased for a given total cost. The slope of the isocost line is the ratio of the input prices of labor and capital. If input prices are fixed, then the ratio of these prices is clearly fixed and the isocost line is straight. Only when the ratio or factor prices change as the quantities of inputs change is the isocost line not straight. 7. Assume the marginal cost of production is increasing. Can you determine whether the average variable cost is increasing or decreasing? Explain. Marginal cost can be increasing while average variable cost is either increasing or decreasing. If marginal cost is less (greater) than average variable cost, then each additional unit is adding less (more) to total cost than previous units added to the total cost, which implies that the AVC declines (increases). Therefore, we need to know whether marginal cost is greater than average variable cost to determine whether the AVC is increasing or decreasing. 8. Assume the marginal cost of production is greater than the average variable cost. Can you determine whether the average variable cost is increasing or decreasing? Explain. If the average variable cost is increasing (decreasing), then the last unit produced is adding more (less) to total variable cost than the previous units did, on average. Therefore, marginal cost is above (below) average variable cost. In fact, the point where marginal cost exceeds average variable cost is also the point where average variable cost starts to rise. 9. If the firms average cost curves are U-shaped, why does its average variable cost curve achieve its minimum at a lower level of output than the average total cost curve? Total cost is equal to fixed plus variable cost. Average total cost is equal to average fixed plus average variable cost. When graphed, the difference between the U-shaped total cost and average variable cost curves is the average fixed cost curve. Thus, falling average variable cost and average fixed cost sum up to a falling average total cost curve. Since average fixed cost continues to fall as more output is produced, average total cost will continue to fall even after average variable cost has reached its minimum because the drop in average fixed cost exceeds the increase in the average variable cost. Eventually, the fall in average fixed cost becomes small enough so that the rise in average variable cost causes average total cost to begin to rise. 10. If a firm enjoys economies of scale up to a certain output level, and then cost increases proportionately with output, what can you say about the shape of the long-run average cost curve? When the firm experiences increasing returns to scale, its long-run average cost curve is downward sloping. When the firm experiences constant returns to scale, its long-run average cost curve is horizontal. If the firm experiences increasing returns to scale, then constant returns to scale, its long-run average cost curve falls, then becomes horizontal. 11. How does a change in the price of one input change the firms long-run expansion path? The expansion path describes the combination of inputs that the firm chooses to minimize cost for every output level. This combination depends on the ratio of input prices: if the price of one input changes, the price ratio also changes. For example, if the price of an input increases, less of the input can be purchased for the same total cost, and the intercept of the isocost line on that inputs axis moves closer to the origin. Also, the slope of the isocost line, the price ratio, changes. As the price ratio changes, the firm substitutes away from the now more expensive input toward the cheaper input. Thus, the expansion path bends toward the axis of the now cheaper input. 12. Distinguish between economies of scale and economies of scope. Why can one be present without the other? Economies of scale refer to the production of one good and occur when proportionate increases in all inputs lead to a more-than-proportionate increase in output. Economies of scope refer to the production of more than one good and occur when joint output is less costly than the sum of the costs of producing each good or service separately. There is no direct relationship between increasing returns to scale and economies of scope, so production can exhibit one without the other. See Exercise (14) for a case with constant product-specific returns to scale and multiproduct economies of scope. 13. Is the firms expansion path always a straight line? No. If the long run expansion path is a straight line this means that the firm always uses capital and labor in the same proportion. If the capital labor ratio changes as output is increased then the expansion path is not a straight line. 14. What is the difference between economies of scale and returns to scale? Economies of scale measures the relationship between cost and output, i.e., when output is doubled, does cost double, less then double, or more than double. Returns to scale measures what happens to output when all inputs are doubled. EXERCISES 1. Joe quits his computer-programming job, where he was earning a salary of $50,000 per year to start his own computer software business in a building that he owns and was previously renting out for $24,000 per year. In his first year of business he has the following expenses: salary paid to himself $40,000, rent, $0, and other expenses $25,000. Find the accounting cost and the economic cost associated with Joes computer software business. The accounting cost represents the actual expenses, which are $40,000+$0 + $25,000=$65,000. The economic cost includes accounting cost, but also takes into account opportunity cost. Therefore, economic will include, in addition to accounting cost, an extra $24,000 because Joe gave up $24,000 by not renting the building , and an extra $10,000 because he paid himself a salary $10,000 below market ($50,000-$40,000). Economic cost is then $99,000. 2. a. Fill in the blanks in the following table. Units of OutputFixed CostVariable CostTotal CostMarginal CostAverage Fixed CostAverage Variable CostAverage Total Cost01000100----0--1100251252510025125210045145205022.572.53100571571233.31952.3410077177202519.2544.255100102202252020.440.461001362363416.6722.6739.371001702703414.324.338.681002263265612.528.2540.7591002983987211.133.144.21010039049092103949 Draw a graph that shows marginal cost, average variable cost, and average total cost, with cost on the vertical axis and quantity on the horizontal axis. Average total cost is u-shaped and reaches a minimum at an output of 7, based on the above table. Average variable cost is u-shaped also and reaches a minimum at an output of 3. Notice from the table that average variable cost is always below average total cost. The difference between the two costs is the average fixed cost. Marginal cost is first diminishing, to a quantity of 3 based on the table, and then increases as q increases. Marginal cost should intersect average variable cost and average total cost at their respective minimum points, though this is not accurately reflected in the numbers in the table. If the specific functions had been given in the problem instead of just a series of numbers, then it would be possible to find the exact point of intersection between marginal and average total cost and marginal and average variable cost. The curves are likely to intersect at a quantity that is not a whole number, and hence are not listed in the above table. 3. A firm has a fixed production cost of $5,000 and a constant marginal cost of production of $500 per unit produced. What is the firms total cost function? Average cost? The variable cost of producing an additional unit, marginal cost, is constant at $500, so  EMBED Equation , and  EMBED Equation  Fixed cost is $5,000 and average fixed cost is  EMBED Equation . The total cost function is fixed cost plus variable cost or TC=$5,000+$500q. Average total cost is the sum of average variable cost and average fixed cost:  EMBED Equation  If the firm wanted to minimize the average total cost, would it choose to be very large or very small? Explain. The firm should choose a very large output because average total cost will continue to decrease as q is increased. As q becomes infinitely large, ATC will equal $500. 4. Suppose a firm must pay an annual tax, which is a fixed sum, independent of whether it produces any output. a. How does this tax affect the firms fixed, marginal, and average costs? Total cost, TC, is equal to fixed cost, FC, plus variable cost, VC. Fixed costs do not vary with the quantity of output. Because the franchise fee, FF, is a fixed sum, the firms fixed costs increase by this fee. Thus, average cost, equal to  EMBED Equation , and average fixed cost, equal to  EMBED Equation , increase by the average franchise fee  EMBED Equation . Note that the franchise fee does not affect average variable cost. Also, because marginal cost is the change in total cost with the production of an additional unit and because the fee is constant, marginal cost is unchanged. Now suppose the firm is charged a tax that is proportional to the number of items it produces. Again, how does this tax affect the firms fixed, marginal, and average costs? Let t equal the per unit tax. When a tax is imposed on each unit produced, variable costs increase by tq. Average variable costs increase by t, and because fixed costs are constant, average (total) costs also increase by t. Further, because total cost increases by t with each additional unit, marginal costs increase by t. 5. A recent issue of Business Week reported the following: During the recent auto sales slump, GM, Ford, and Chrysler decided it was cheaper to sell cars to rental companies at a loss than to lay off workers. Thats because closing and reopening plants is expensive, partly because the auto makers current union contracts obligate them to pay many workers even if theyre not working. When the article discusses selling cars at a loss, is it referring to accounting profit or economic profit? How will the two differ in this case? Explain briefly. When the article refers to the car companies selling at a loss, it is referring to accounting profit. The article is stating that the price obtained for the sale of the cars to the rental companies was less than their accounting cost. Economic profit would be measured by the difference of the price with the opportunity cost of the cars. This opportunity cost represents the market value of all the inputs used by the companies to produce the cars. The article mentions that the car companies must pay workers even if they are not working (and thus producing cars). This implies that the wages paid to these workers are sunk and are thus not part of the opportunity cost of production. On the other hand, the wages would still be included in the accounting costs. These accounting costs would then be higher than the opportunity costs and would make the accounting profit lower than the economic profit. 6. Suppose the economy takes a downturn, and that labor costs fall by 50 percent and are expected to stay at that level for a long time. Show graphically how this change in the relative price of labor and capital affects the firms expansion path. Figure 7.6 shows a family of isoquants and two isocost curves. Units of capital are on the vertical axis and units of labor are on the horizontal axis. (Note: In drawing this figure we have assumed that the production function underlying the isoquants exhibits constant returns to scale, resulting in linear expansion paths. However, the results do not depend on this assumption.) If the price of labor decreases while the price of capital is constant, the isocost curve pivots outward around its intersection with the capital axis. Because the expansion path is the set of points where the MRTS is equal to the ratio of prices, as the isocost curves pivot outward, the expansion path pivots toward the labor axis. As the price of labor falls relative to capital, the firm uses more labor as output increases.  EMBED PowerPoint.Show.4  Figure 7.6 7. The cost of flying a passenger plane from point A to point B is $50,000. The airline flies this route four times per day at 7am, 10am, 1pm, and 4pm. The first and last flights are filled to capacity with 240 people. The second and third flights are only half full. Find the average cost per passenger for each flight. Suppose the airline hires you as a marketing consultant and wants to know which type of customer it should try to attract, the off-peak customer (the middle two flights) or the rush-hour customer (the first and last flights). What advice would you offer? The average cost per passenger is $50,000/240 for the full flights and $50,000/120 for the half full flights. The airline should focus on attracting more off-peak customers in order to reduce the average cost per passenger on those flights. The average cost per passenger is already minimized for the two peak time flights. 8. You manage a plant that mass produces engines by teams of workers using assembly machines. The technology is summarized by the production function.  EMBED Equation.2  EMBED Equation.2 q = 5 KL where q is the number of engines per week, K is the number of assembly machines, and L is the number of labor teams. Each assembly machine rents for r = $10,000 per week and each team costs w = $5,000 per week. Engine costs are given by the cost of labor teams and machines, plus $2,000 per engine for raw materials. Your plant has a fixed installation of 5 assembly machines as part of its design. a. What is the cost function for your plant namely, how much would it cost to produce q engines? What are average and marginal costs for producing q engines? How do average costs vary with output? K is fixed at 5. The short-run production function then becomes q = 25L. This implies that for any level of output q, the number of labor teams hired will be  EMBED Equation . The total cost function is thus given by the sum of the costs of capital, labor, and raw materials:  EMBED Equation.DSMT36  The average cost function is then given by:  EMBED Equation.DSMT36  and the marginal cost function is given by:  EMBED Equation.DSMT36  Marginal costs are constant and average costs will decrease as quantity increases (due to the fixed cost of capital). b. How many teams are required to produce 250 engines? What is the average cost per engine? To produce q = 250 engines we need labor teams  EMBED Equation or L=10. Average costs are given by  EMBED Equation.DSMT36  c. You are asked to make recommendations for the design of a new production facility. What capital/labor (K/L) ratio should the new plant accommodate if it wants to minimize the total cost of producing any level of output q? We no longer assume that K is fixed at 5. We need to find the combination of K and L that minimizes costs at any level of output q. The cost-minimization rule is given by  EMBED Equation  To find the marginal product of capital, observe that increasing K by 1 unit increases q by 5L, so MPK = 5L. Similarly, observe that increasing L by 1 unit increases Q by 5K, so MPL = 5K. Mathematically,  EMBED Equation . Using these formulas in the cost-minimization rule, we obtain:  EMBED Equation . The new plant should accommodate a capital to labor ratio of 1 to 2. Note that the current firm is presently operating at this capital-labor ratio. 9. The short-run cost function of a company is given by the equation TC=200+55q, where TC is the total cost and q is the total quantity of output, both measured in thousands. a. What is the companys fixed cost? When q = 0, TC = 200, so fixed cost is equal to 200 (or $200,000). b. If the company produced 100,000 units of goods, what is its average variable cost? With 100,000 units, q = 100. Variable cost is 55q = (55)(100) = 5500 (or $5,500,000). Average variable cost is  EMBED Equation or $55,000. c. What is its marginal cost per unit produced? With constant average variable cost, marginal cost is equal to average variable cost, $55 (or $55,000). d. What is its average fixed cost? At q = 100, average fixed cost is  EMBED Equation or ($2,000). e. Suppose the company borrows money and expands its factory. Its fixed cost rises by $50,000, but its variable cost falls to $45,000 per 1,000 units. The cost of interest (i) also enters into the equation. Each one-point increase in the interest rate raises costs by $3,000. Write the new cost equation. Fixed cost changes from 200 to 250, measured in thousands. Variable cost decreases from 55 to 45, also measured in thousands. Fixed cost also includes interest charges: 3i. The cost equation is C = 250 + 45q + 3i. 10. A chair manufacturer hires its assembly-line labor for $30 an hour and calculates that the rental cost of its machinery is $15 per hour. Suppose that a chair can be produced using 4 hours of labor or machinery in any combination. If the firm is currently using 3 hours of labor for each hour of machine time, is it minimizing its costs of production? If so, why? If not, how can it improve the situation? Graphically illustrate the isoquant and the two isocost lines, for the current combination of labor and capital and the optimal combination of labor and capital. If the firm can produce one chair with either four hours of labor or four hours of capital, machinery, or any combination, then the isoquant is a straight line with a slope of -1 and intercept at K = 4 and L = 4, as depicted in figure 7.10. The isocost line, TC = 30L + 15K has a slope of  EMBED Equation  when plotted with capital on the vertical axis and has intercepts at  EMBED Equation  and  EMBED Equation . The cost minimizing point is a corner solution, where L = 0 and K = 4. At that point, total cost is $60. Two isocost lines are illustrated on the graph. The first one is further from the origin and represents the higher cost ($105) of using 3 labor and 1 capital. The firm will find it optimal to move to the second isocost line which is closer to the origin, and which represents a lower cost ($60). In general, the firm wants to be on the lowest isocost line possible, which is the lowest isocost line that still intersects the given isoquant.  EMBED Word.Picture.8  Figure 7.10 11. Suppose that a firms production function is  EMBED Equation.DSMT36 . The cost of a unit of labor is $20 and the cost of a unit of capital is $80. The firm is currently producing 100 units of output, and has determined that the cost-minimizing quantities of labor and capital are 20 and 5 respectively. Graphically illustrate this situation on a graph using isoquants and isocost lines. The isoquant is convex. The optimal quantities of labor and capital are given by the point where the isocost line is tangent to the isoquant. The isocost line has a slope of 1/4, given labor is on the horizontal axis. The total cost is TC=$20*20+$80*5=$800, so the isocost line has the equation $800=20L+80K. On the graph, the optimal point is point A.  EMBED Word.Picture.8  The firm now wants to increase output to 140 units. If capital is fixed in the short run, how much labor will the firm require? Illustrate this point on your graph and find the new cost. The new level of labor is 39.2. To find this, use the production function  EMBED Equation.DSMT36  and substitute 140 in for output and 5 in for capital. The new cost is TC=$20*39.2+$80*5=$1184. The new isoquant for an output of 140 is above and to the right of the old isoquant for an output of 100. Since capital is fixed in the short run, the firm will move out horizontally to the new isoquant and new level of labor. This is point B on the graph below. This is not likely to be the cost minimizing point. Given the firm wants to produce more output, they are likely to want to hire more capital in the long run. Notice also that there are points on the new isoquant that are below the new isocost line. These points all involve hiring more capital.  EMBED Word.Picture.8  Graphically identify the cost-minimizing level of capital and labor in the long run if the firm wants to produce 140 units. This is point C on the graph above. When the firm is at point B they are not minimizing cost. The firm will find it optimal to hire more capital and less labor and move to the new lower isocost line. All three isocost lines above are parallel and have the same slope. If the marginal rate of technical substitution is  EMBED Equation.DSMT36 , find the optimal level of capital and labor required to produce the 140 units of output. Set the marginal rate of technical substitution equal to the ratio of the input costs so that  EMBED Equation.DSMT36  Now substitute this into the production function for K, set q equal to 140, and solve for L:  EMBED Equation.DSMT36  The new cost is TC=$20*28+$80*7 or $1120. 12. A computer companys cost function, which relates its average cost of production AC to its cumulative output in thousands of computers Q and its plant size in terms of thousands of computers produced per year q, within the production range of 10,000 to 50,000 computers is given by AC = 10 - 0.1Q + 0.3q. a. Is there a learning curve effect? The learning curve describes the relationship between the cumulative output and the inputs required to produce a unit of output. Average cost measures the input requirements per unit of output. Learning curve effects exist if average cost falls with increases in cumulative output. Here, average cost decreases as cumulative output, Q, increases. Therefore, there are learning curve effects. b. Are there economies or diseconomies of scale? Economies of scale can be measured by calculating the cost-output elasticity, which measures the percentage change in the cost of production resulting from a one percentage increase in output. There are economies of scale if the firm can double its output for less than double the cost. There are economies of scale because the average cost of production declines as more output is produced, due to the learning effect. c. During its existence, the firm has produced a total of 40,000 computers and is producing 10,000 computers this year. Next year it plans to increase its production to 12,000 computers. Will its average cost of production increase or decrease? Explain. First, calculate average cost this year: AC1 = 10 - 0.1Q + 0.3q = 10 - (0.1)(40) + (0.3)(10) = 9. Second, calculate the average cost next year: AC2 = 10 - (0.1)(50) + (0.3)(12) = 8.6. (Note: Cumulative output has increased from 40,000 to 50,000.) The average cost will decrease because of the learning effect. 13. Suppose the long-run total cost function for an industry is given by the cubic equation TC = a + bQ + cQ2 + dQ3. Show (using calculus) that this total cost function is consistent with a U-shaped average cost curve for at least some values of a, b, c, d. To show that the cubic cost equation implies a U-shaped average cost curve, we use algebra, calculus, and economic reasoning to place sign restrictions on the parameters of the equation. These techniques are illustrated by the example below. First, if output is equal to zero, then TC = a, where a represents fixed costs. In the short run, fixed costs are positive, a > 0, but in the long run, where all inputs are variable a = 0. Therefore, we restrict a to be zero. Next, we know that average cost must be positive. Dividing TC by Q: AC = b + cQ + dQ2. This equation is simply a quadratic function. When graphed, it has two basic shapes: a U shape and a hill shape. We want the U shape, i.e., a curve with a minimum (minimum average cost), rather than a hill shape with a maximum. At the minimum, the slope should be zero, thus the first derivative of the average cost curve with respect to Q must be equal to zero. For a U-shaped AC curve, the second derivative of the average cost curve must be positive. The first derivative is c + 2dQ; the second derivative is 2d. If the second derivative is to be positive, then d > 0. If the first derivative is equal to zero, then solving for c as a function of Q and d yields: c = -2dQ. If d and Q are both positive, then c must be negative: c < 0. To restrict b, we know that at its minimum, average cost must be positive. The minimum occurs when c + 2dQ = 0. We solve for Q as a function of c and d:  EMBED Equation . Next, substituting this value for Q into our expression for average cost, and simplifying the equation:  EMBED Equation , or  EMBED Equation  implying  EMBED Equation . Because c2 >0 and d > 0, b must be positive. In summary, for U-shaped long-run average cost curves, a must be zero, b and d must be positive, c must be negative, and 4db > c2. However, the conditions do not insure that marginal cost is positive. To insure that marginal cost has a U shape and that its minimum is positive, using the same procedure, i.e., solving for Q at minimum marginal cost  EMBED Equation  and substituting into the expression for marginal cost b + 2cQ + 3dQ2, we find that c2 must be less than 3bd. Notice that parameter values that satisfy this condition also satisfy 4db > c2, but not the reverse. For example, let a = 0, b = 1, c = -1, d = 1. Total cost is Q - Q2 + Q3; average cost is 1 - Q + Q2; and marginal cost is 1 - 2Q + 3Q2. Minimum average cost is Q = 1/2 and minimum marginal cost is 1/3 (think of Q as dozens of units, so no fractional units are produced). See Figure 7.13.  EMBED PowerPoint.Show.4  Figure 7.13 *14. A computer company produces hardware and software using the same plant and labor. The total cost of producing computer processing units H and software programs S is given by TC = aH + bS - cHS, where a, b, and c are positive. Is this total cost function consistent with the presence of economies or diseconomies of scale? With economies or diseconomies of scope? There are two types of scale economies to consider: multiproduct economies of scale and product-specific returns to scale. From Section 7.5 we know that multiproduct economies of scale for the two-product case, SH,S, are  EMBED Equation  where MCH is the marginal cost of producing hardware and MCS is the marginal cost of producing software. The product-specific returns to scale are:  EMBED Equation  and  EMBED Equation  where TC(0,S) implies no hardware production and TC(H,0) implies no software production. We know that the marginal cost of an input is the slope of the total cost with respect to that input. Since  EMBED Equation  we have MCH = a - cS and MCS = b - cH. Substituting these expressions into our formulas for SH,S, SH, and SS:  EMBED Equation  or  EMBED Equation , because cHS > 0. Also,  EMBED Equation , or  EMBED Equation  and similarly  EMBED Equation  There are multiproduct economies of scale, SH,S > 1, but constant product-specific returns to scale, SH = SC = 1. Economies of scope exist if SC > 0, where (from equation (7.8) in the text):  EMBED Equation , or,  EMBED Equation , or  EMBED Equation  Because cHS and TC are both positive, there are economies of scope.     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L=Q40mFMathType Equation 3.6+DNQEEquation.DSMT36! L=Q40 Equation! L=q25FMathType Equation 3.6+DNQEOle10FmtProgID a Equation Native =_988017159dF\Y}B\Y}BOle CompObjceWObjInfofEquation Native !_9880170298biF\Y}B\Y}BEquation.DSMT36 TC(q) = rK+wL+2000q = (10,000)(5) + (5,000)(q25) + 2,000 qTC(q)= 50,000 +2200q.Ole CompObjhjWObjInfokEquation Native FMathType Equation 3.6+DNQEEquation.DSMT36l AC(q)=TC(q)q=50,000+2200qq.FMathType Equation 3.6+DNQE_988017186gnF\Y}B\Y}BOle CompObjmoWObjInfop      #$')*+,-./0134568;<?ABCDEFGHIJKLMNOPQRSTUVWXY[^_bcfhijklmnopqrstuvwxyz{|}~Equation.DSMT36E MC(q)=TCq=2200.L0%dxpr MTHU %GrphbjEquation Native a_988017300uF\Y}B\Y}BOle PIC rtLPICT CompObjswWObjInfoOle10Nativevx% %"% currentpoint " +L) =( Q(40"  $/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1184 div 896 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 643 403 moveto 434 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (L) 12 502 sh (Q) 722 261 sh 384 /Symbol f1 (=) 325 502 sh 384 /Times-Roman f1 (40) 671 795 sh end MTsave restore d-MATH!. L=Q40mFMathType Equation 3.6+DNQEEquation.DSMT36! L=Q40 Equation! L=q25FMathType Equation 3.6+DNQEOle10FmtProgID y Equation Native =_988017342q|F\Y}B\Y}BOle CompObj{}WObjInfo~!Equation Native "_923414663 F\Y}B\Y}BEquation.DSMT36{ AC(q=250)=50,000+2200(250)250=2400.L^ hOOle %PIC &LMETA (HPICT 2^ hW   .1  &@ & MathType "-@ P Century Schoolbook- 2 `MP- 2 dr 2 x= 2 MP- 2 }w 2  .WW`Century Schoolbookx- 2 DK{ 2 Lj & "Systemn-4!*@@C C C""%dPPNTCentury Schoolbook,Century Schoolbook .+ MP+r(=( &MP+w(< .dPPNTCentury Schoolbook (K)"LdPPNT"System FMathType Equation Equation    )  !#"$%'&+(*-,/.01T:3456789S<=>?@ABDEFGHIJLMNOPQRU^WXYZ[\]`abcdefhijklmnpqrstuvyz{|}~CompObj7\ObjInfo9Ole10Native:_981783226IYF\Y}B\Y}BEquation9q ` @MP @K @r=MP @L @w . L8DOle =PIC >LPICT @wCompObjZWwdxpr MTHU Grphbj " currentpoint " +MP +K (=( %)Q(%)K"%(8=) 4)L) and )MP +L (~=( )Q()L"(=) 4)K/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5824 div 928 3 -1 roll exch div scale currentpoint translate 64 49 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1120 428 moveto 513 0 rlineto stroke 4314 428 moveto 505 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (MP) 9 527 sh (Q) 1336 286 sh (K) 1332 820 sh (L) 2278 527 sh (MP) 3237 527 sh (Q) 4526 286 sh (L) 4554 820 sh (K) 5457 527 sh 224 ns (K) 505 623 sh (L) 3737 623 sh 384 /Symbol f1 (=) 802 527 sh (\266) 1147 286 sh (\266) 1143 820 sh (=) 1741 527 sh (=) 3996 527 sh (\266) 4337 286 sh (\266) 4365 820 sh (=) 4927 527 sh 384 /Times-Roman f1 (4) 2055 527 sh (4) 5241 527 sh 384 /Times-Roman f1 ( and ) 2491 527 sh end MTsave restore dMATH MP K =QK=4L and MP L =QL=4KowFMathType Equation 3.6+DNQEEquation.DSMT36 MP K =QK=4L and MP L =QL=4KObjInfo\Ole10Native]Ole10FmtProgID ` Equation Native a Equation MP K =QK=5L and MP L =QL=5KLJ4 Ddxpr MTHU Grphbj_981783243F\Y}B\Y}BOle dPIC eLPICT g " currentpoint " + 4)L(r" (=( 4)K(#w" (2( BK+L" A (O=( Zw+r" Y (f=( u3000(p12) ,)000" p!(=( 1*4" /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5408 div 928 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 403 moveto 488 0 rlineto stroke 914 403 moveto 545 0 rlineto stroke 2032 403 moveto 338 0 rlineto stroke 2796 403 moveto 308 0 rlineto stroke 3530 403 moveto 1100 0 rlineto stroke 5056 403 moveto 241 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Roman f1 (4) 28 261 sh (4) 942 261 sh (3000) 3693 261 sh (12) 3520 795 sh (000) 4032 795 sh (1) 5080 261 sh (4) 5084 795 sh 384 /Times-Italic f1 (L) 251 261 sh (r) 165 795 sh (K) 1158 261 sh (w) 1059 795 sh (K) 2069 261 sh (L) 2105 795 sh (w) 2823 261 sh (r) 2871 795 sh 384 /Symbol f1 (=) 596 502 sh (\336) 1554 502 sh (=) 2478 502 sh (=) 3212 502 sh (=) 4738 502 sh 384 /Times-Roman f1 (,) 3900 795 sh end MTsave restore dMATH 4Lr=4KwKL=wr=300012,000=14O`FMathType Equation 3.6+DNQEEquation.DSMT36 4Lr=4KwKCompObjWObjInfoOle10NativeOle10FmtProgID  L=wr=300012,000=14 Equation 5Lr=5KwKL=wr=500010,000=12L FlEquation Native _988013346F\Y}B\Y}BOle PIC Lg FD 5  .1  @ & I & MathType-@vU Century Schoolbook- 2 QBTVC 2 QSymbol- 2 k= 2 =Century SchoolbookxMETA PICT CompObjWObjInfo- 2 Q$530 2 $53 2  .Y 2 )10 & "System-bdxpr MTHU bGrphbj b"b currentpoint " + TVC+Q"(=( '$)530(,10"&(C=) $)53) ,/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3136 div 992 3 -1 roll exch div scale currentpoint translate 64 56 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 0 421 moveto 749 0 rlineto stroke 1175 421 moveto 806 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (TVC) 20 279 sh (Q) 236 813 sh 384 /Symbol f1 (=) 857 520 sh (=) 2089 520 sh 384 /Times-Roman f1 ($) 1191 279 sh ($) 2391 520 sh (,) 2946 520 sh 384 /Times-Roman f1 (530) 1383 279 sh (10) 1370 813 sh (53) 2583 520 sh end MTsave restore dWMATHK  TVCQ=$53010=$53,32FMathType Equation 3.6+DNQEEquation.DSMT36K TVCQ=$53010=$53, EquationQ TVCq=Ole10NativeOOle10FmtProgID  Equation Native m_988017690F\Y}BY}B$5500100=$55,L D D ,  .1   & I & MathType-@iH CenturOle PIC LMETA PICT By Schoolbook- 2 QBTFC 2 QSymbol- 2 ^= 2 =Century Schoolbookx- 2 Qu$190 2 $19 2 10 & "System-'<3FP(>>W2HGHHt HBW W W""#dPPNTCentury Schoolbook,Century Schoolbook .+ TFC+QdPPNTSymbol, Symbol(=)"=dPPNTCentury Schoolbook( $$190+!$19()10dPPNT"SystemFMathType Equation 3.6+DNQEEquation.DSMT36CompObjWObjInfoOle10NativeEquation Native d TFCQ=$19010=$19 DH TFCq=$200100=$2Lt_981784362FY}BY}BOle PIC LMETA (~G   .1  ` &  & MathType-$> SymbolN- 2 =- 2 = 2 -Century Schoolbookx- 2 Q22 2 110 2 R0 2 82 2 .Y & "System-9 9 9" dPPNTSymbol, Symbol .+-)=)-dPPNTCentury Schoolbook,Century Schoolbook( 22( 110(+0)2(/.dPPNT"SystemFMathType Equation 3.6+DNQEEquation.DSMT36PICT CompObjWObjInfoOle10Natived` -22110=-0.2* -3015=-2L4Equation Native F_981784393FY}BY}BOle PIC LMETA (PICT  CompObjWObjInfoG   .1  ` &  & MathType- Century Schoolbook- 2 JK 2 QTCSymbol- 2 =Century Schoolbookx- 2 110 & "System- ) ) )"dPPNTCentury Schoolbook,Century Schoolbook .+K( TCdPPNTSymbol, Symbol(=dPPNTCentury Schoolbook+110dPPNT"SystemFMathType Equation 3.6+DNQEEquation.DSMT36      #%(,147:;>@ABCDEFGHJKLMORUWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|~` K=TC1101-$ K=TC15LF   Ole10NativedEquation Native @_981784405FY}BY}BOle PIC LMETA (PICT  CompObjW.1  `&`  & MathType-fS Century Schoolbook- 2 6L 2 QhTCSymbol- 2 [=Century Schoolbookx- 2 22 & "System- % % %"dPPNTCentury Schoolbook,Century Schoolbook .+L( TCdPPNTSymbol, Symbol( =dPPNTCentury 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Timesw@ [wdw0-!2 & 'dxpr MTHU Grphbj " currentpoint " +AC)=) b) +)cQ)+) dQ ( S2 +=) b)+) c ( -)c(2)d"  (y *  ( *  (+) d ( -)c(2)d"  ( *  ( *   (21/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6176 div 960 3 -1 roll exch div scale currentpoint translate 64 49 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 4023 460 moveto 399 0 rlineto stroke 5372 460 moveto 399 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 352 /Times-Italic f1 (AC) 15 559 sh (b) 861 559 sh (cQ) 1395 559 sh (dQ) 2163 559 sh (b) 3130 559 sh (c) 3664 559 sh (d) 4977 559 sh 320 ns (c) 4249 318 sh (d) 4224 852 sh (c) 5598 318 sh (d) 5573 852 sh 352 /Symbol f1 (=) 568 559 sh (+) 1121 559 sh (+) 1883 559 sh (=) 2837 559 sh (+) 3390 559 sh (+) 4697 559 sh 320 ns (-) 4058 318 sh (-) 5407 318 sh 384 ns (\346) 3826 412 sh (\350) 3826 751 sh (\366) 4432 412 sh (\370) 4432 751 sh (\346) 5175 412 sh (\350) 5175 751 sh (\366) 5781 412 sh (\370) 5781 751 sh 192 /Times-Roman f1 (2) 2605 387 sh (2) 5980 152 sh 320 ns (2) 4046 852 sh (2) 5395 852 sh end MTsave restore dMATHw  `AC=b+cQ+dQ `2 `=b+c @-c2d() `+d @-c2d() ` `2m FMathType Equation1ELO Equation9q  `AC=b+cQ+dQ `2 `=CompObj\ObjInfoOle10NativeOle10FmtProgID  b+c @-c2d() `+d @-c2d() ` `2 EquationdX l   .  & _892710735A& FY}BY}BOle PIC dMETA  Timesw@ :[wdw0-!ACPSymbolw@ d[wdw0-!=Timesw@ ;[wdw0-!bPSymbolw@ e[wdw0-!-'Timesw@ <[wdw0-!c 2Timesw@ f[wdw0-!27Timesw@ =[wdw0-!21Timesw@ g[wdw0-!d7-0=PSymbolw@ >[wdw0-!+@Timesw@ h[wdw0-!c KTimesw@ ?[wdw0-!2QTimesw@ i[wdw0-!4JTimesw@ @[wdw0-!dPIVPSymbolw@ j[wdw0-!=ZTimesw@ A[wdw0-!bdPSymbolw@ k[wdw0-!-mTimesw@ B[wdw0-!2 wTimesw@ l[wdw0-!c |Timesw@ C[wdw0-!2Timesw@ m[wdw0-!4xTimesw@ D[wdw0-!d~vPSymbolw@ n[wdw0-!+Timesw@ E[wdw0-!c Timesw@ o[wdw0-!2Timesw@ F[wdw0-!4Timesw@ p[wdw0-!dPSymbolw@ G[wdw0-!=Timesw@ q[wdw0-!bPSymbolw@ H[wdw0-!-Timesw@ r[wdw0-!c Timesw@ I[wdw0-!2Timesw@ s[wdw0-!4Timesw@ J[wdw0-!dPSymbolw@ t[wdw0-!>Timesw@ K[wdw0-!0!. & ' dxpr MTHU Grphbj " currentpoint " +AC)=) b) - ( 2c (72 (12)d"0 (@+( Kc (Q2 (J4)d"I  (Z=) b) - ( w2)c (2 (x4)d"v (+ PICT   CompObj\ObjInfo  Ole10Native  ( c (2 (4)d"  (=) b)- ( c (2 (4)d"  (>) 0)./MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 7296 div 960 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1493 466 moveto 429 0 rlineto stroke 2300 466 moveto 434 0 rlineto stroke 3726 466 moveto 537 0 rlineto stroke 4658 466 moveto 434 0 rlineto stroke 6084 466 moveto 434 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (AC) 17 565 sh (b) 918 565 sh (b) 3151 565 sh (b) 5509 565 sh 352 ns (c) 1553 324 sh (d) 1707 858 sh (c) 2363 324 sh (d) 2519 858 sh (c) 3934 324 sh (d) 3996 858 sh (c) 4721 324 sh (d) 4877 858 sh (c) 6147 324 sh (d) 6303 858 sh 384 /Symbol f1 (=) 609 565 sh (-) 1192 565 sh (=) 2842 565 sh (-) 3425 565 sh (+) 4356 565 sh (=) 5200 565 sh (-) 5783 565 sh (>) 6621 565 sh 352 ns (+) 2015 565 sh 192 /Times-Roman f1 (2) 1727 152 sh (2) 2537 152 sh (2) 4108 152 sh (2) 4895 152 sh (2) 6321 152 sh 352 ns (2) 1515 858 sh (4) 2328 858 sh (2) 3748 324 sh (4) 3805 858 sh (4) 4686 858 sh (4) 6112 858 sh 384 ns (0) 6924 565 sh 384 /Times-Roman f1 (.) 7116 565 sh end MTsave restore dMATH% AC=b- `c `2 `2d+c `2 `4d =b- `2c `2 `4d + `c `2 `4d =b- `c `2 `4d >0.۠ FMathType Equation1ELO Eq     !"#$%&')/12345679:;=@CEFGHIJKLMNOPQRSTUVX\^_`abcdefghijklmnopqrstuvwxyz{|}~uation9q AC=b- `c `2 `2d+c `2 `4d =b- `2c `2 `4d + `c `2 `4d =b- `c `2 `4d >0. Equation     76 !"#$%&'()*+,-./012345A89:<O=>?@CBKDEFGHIJLMNSPeQRTYUVWXZ[\b]^_`acdfnzghijklmpotqrsvu~wxy{|}Ole10FmtProgID  _889516336 FY}BY}BOle PIC ddX ^  $ .  & Timesw@4 [wdw0-!bPSymbolw@! [wdw0-!> Timesw@4 [wdw0-!c META  PICT UCompObj(PObjInfo*Timesw@! [wdw0-!2Timesw@4 [wdw0-!4Timesw@! [wdw0-!d-! & 'U$dxpr MTHU $Grphbj $"$ currentpoint " +b) >( c (2 (4)d" H/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 1152 div 960 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 588 466 moveto 466 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (b) -9 565 sh (c) 653 324 sh (d) 822 858 sh 384 /Symbol f1 (>) 276 565 sh 224 /Times-Roman f1 (2) 840 152 sh 384 ns (4) 616 858 sh end MTsave restore d:MATH. b>c 2 4d } FMathType Equation1ELO Equation. b>c 2 4d EquationLOle10Native+2Ole10FmtProgID , _892710861 FY}BY}BOle -PIC .LMETA 0PICT 8CompObj<\   .1  ` &e & MathType0 PSymbol#- 2 -TimesF- 2 c 2 dTimes#- 2 /Y 2 ,P 2 _3 & "Systemn- ! ! dPPNTPSymbol !, Symbol .+-dPPNTTimes,Times)c)ddPPNTTimes(/),(3dPPNT"System FMathType Equation Equation Equation9q@ @-c/3d,ObjInfo>Ole10Native?D_8927159918 QHFY}B@ Y}BOle AL1" h hdPPNT h 4h $$!DdPPNTCentury Schoolbook,Century Schoolbook .+CostsdPPPIC "BLMETA PICT !$DCompObjWo1"f\ 1 1& @  & &$TNPPMicrosoft PowerPoint & TNPPf & &TNPP   145--A &M--HM---' & & --/ ---' & &ce_Century Schoolbook-.  2 .CoststPJ>J & &--"System-I-(c--' & &?KY--Y-;G--' & &S--S?-{--' & &Y--Y ---' & & S--SK---' & &yY--Y9-u--' & &9ES--S-5A--' & &pB#D_Century Schoolbook-.   2 0.17Y,YY & &7]9_Century Schoolbook-.   2 0.33Y,YY & &=?_Century Schoolbook-.   2 $0.50Y,YY & &679_Century Schoolbook-.   2 p0.67Y,YY & &j,._Century Schoolbook-.   2 0.83Y,YY & &,]._Century Schoolbook-.   2 1.00Y,YY & &Z _Century Schoolbook-.  2 Quantity|bYb>3>V_Century Schoolbook-.  2 W in Dozens 2b-|PMPbJ, & &---.-(c--' & &_Century Schoolbook-.   2 1Y & &ce_Century Schoolbook-.   2 2Y & &[- [[ & &[  Z[d & &  & &    | & &CeEaCentury Schoolbook-.   2 MCt & &FaCentury Schoolbook-.   2 ACqs & &TNPP & ---NT"System""H"l""""dPPNTCentury Schoolbook+(0.17dPPNTCentury Schoolbook(b0.33dPPNTCentury Schoolbook+$0.50dPPNTCentury Schoolbook(0.67dPPNTCentury Schoolbook)#0.83dPPNTCentury Schoolbook)$1.00dPPNTCentury Schoolbook(QuantitydPPNTCentury Schoolbook* in Dozens dPPNT"System"<dPPNTCentury Schoolbook(1dPPNTCentury Schoolbook(?2`Z$Z`ZZ`h$Z`iZZdPPNTCentury Schoolbook+MCdPPNTCentury Schoolbook+F:ACdPPNTdPPNT"System QHFMS PowerPoint 4.0 PresentationPowerPoint.Show.4PowerPoint.Show.4di@ i 1  ! .  & Timesw@v [wdw0-!ObjInfo#%YOle10Native$1_892714579/* F@ Y}B Y}BOle Z 1ޭ 20f1(yU@\\\\\\\\\\\\\\\\@\\\\\\\\\\\\\\\\\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \\\\\\\\\\\\U@mL& "! A!a!w!@!!!!&#@$Q$Q$NQ$&%"%%%%%%& &"3&U&m&m&m&m&&{&"&&&&&'H7'@'v'v5(((",,?.V0 HMH c]@HH?QCHKHQHKHyQ}H9K=pB<,@7U1@=7@6714@j,&@,U&@R <@H@c]@CS@ CdS`C@ C|`C]=l @F @L9.@ddCosts dd9_dd0.17 dd9Tdd0.33 dd9$Z[dd0.50 dd9pTdd0.67 dd9I dd0.83 dd9I dd1.00 dd9 ddQuantity in Dozens  dd dd9#]dd1 dd9i@dd2 dd9`# ddMC dd9dddAC dd@_ @ AGpn >LO_ Ap \O_rV !,dd Click to edit Master title style  ,  dd2V ddRClick to edit Master text styles Second level Third level Fourth level Fifth level R!   R!dd dd dd dd dd   Ap0 O_A]2 s ddRClick to edit Master text styles Second level Third level Fourth level Fifth level RR R!dd dd dd dd dd@ 7__ @  @!ʅ?@_# OJ@  @@``@ @`@'?@7_%&(dddddHcXYhhhYDGa[3(dddddH&aXYhhhYDGa[3(?@?_*+d@@T@r,,,,,dddddddddd2 dddddddddd2     dddddddddd2dddddddddd3dddddddddd00?@@/_Times New Roman`NRKR "Arial_NRL_=N_EB:`KCentury Schoolbook`'M4 H3'Century SchoolbookwK@^d6*XXj|VDHP LaserJet 4/4MHPPCL5ELPT1: uy50<)4A>'#~1^ 0:/lG""!0+n:k+)(B. f/|1HP LaserJet 4/4M D.X|t  @ p@ "@B)!$',).-/0'X!ޭ PIC ')[dMETA ]jPICT (,CompObj\STimesw@ [wdw0-!HTimesw@v [wdw0-!, Timesw@ [wdw0-!SPSymbolw@v [wdw0-!=Timesw@ [wdw0-!TC @Timesw@v [wdw0-! OTimesw@ [wdw0-!H WTimesw@v [wdw0-!, `Timesw@ [wdw0-!S ePSymbolw@v [wdw0-!( R!) kTimesw@ [wdw0-!H&PSymbolw@v [wdw0-!(!!)0Timesw@ [wdw0-!MC8Timesw@v [wdw0-!HJPSymbolw@ [wdw0-!(3!)QPSymbolw@v [wdw0-!+WTimesw@ [wdw0-!SdPSymbolw@v [wdw0-!(`!)kTimesw@ [wdw0-!MCsTimesw@v [wdw0-!SPSymbolw@ [wdw0-!(n!)-! & '!dxpr MTHU!Grphbj !"! currentpoint " +S +H),)S (=( @TC) )H) ,)S ( R())(&H (!()))MC +H  (3())(W+) S (`() ))MC +S  (n())"!l/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4608 div 1056 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1008 434 moveto 3487 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (S) -3 533 sh (TC) 2012 292 sh (H) 2741 292 sh (S) 3169 292 sh (H) 1174 826 sh (MC) 1747 826 sh (S) 3159 826 sh (MC) 3633 826 sh 224 ns (H) 185 629 sh (S) 442 629 sh (H) 2319 922 sh (S) 4198 922 sh 224 /Times-Roman f1 (,) 367 629 sh 384 ns (,) 3031 292 sh 384 /Symbol f1 (=) 690 533 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 2587 302 sh (\)) 3373 302 sh (\() 1020 836 sh (\)) 1477 836 sh 384 1000 1369 /Symbol f3 (\() 1593 854 sh (\)) 2536 854 sh 384 /Symbol f1 (+) 2735 826 sh 384 1000 1171 /Symbol f3 (\() 3017 836 sh (\)) 3363 836 sh 384 1000 1393 /Symbol f3 (\() 3479 856 sh (\)) 4357 856 sh 384 /Times-Roman f1 ( ) 2481 292 sh end MTsave restore dMATH" S H,S =TC H,S()H()MC H ()+S()MC S () FMathType Equation1ELO Equation9q S H,S =TC H,S()H()MC H ()+S()MC S () EquationObjInfo+-Ole10Native.Ole10FmtProgID  _8927146663 F Y}B Y}BOle PIC 02dMETA vPICT 15dp i( p i   ! .  & Timesw@s F[wdw0-!STimesw@ [wdw0-!HPSymbolw@s G[wdw0-!=Timesw@ [wdw0-!TC Timesw@s H[wdw0-! *Timesw@ [wdw0-!H 2Timesw@s I[wdw0-!, ;Timesw@ [wdw0-!S @PSymbolw@s J[wdw0-!( -!) FPSymbolw@ [wdw0-!- LTimesw@s K[wdw0-!TC UTimesw@ [wdw0-! d!0 k!, qTimesw@s L[wdw0-!S vPSymbolw@ [wdw0-!( g!) |Timesw@s M[wdw0-!H9PSymbolw@ [wdw0-!(4!)BTimesw@s N[wdw0-!MCKTimesw@ [wdw0-!H\PSymbolw@s O[wdw0-!(F!)c- & '!dxpr MTHU!Grphbj !"! currentpoint " +S +H (=( TC) )H) ,)S ( -()))-) TC) )0),)S ( g())(9H (4())) MC +H  (F())"df/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4160 div 1056 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 809 434 moveto 3247 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (S) -3 533 sh (TC) 829 292 sh (H) 1558 292 sh (S) 1986 292 sh (TC) 2677 292 sh (S) 3714 292 sh (H) 1765 826 sh (MC) 2338 826 sh 224 ns (H) 185 629 sh (H) 2910 922 sh 384 /Symbol f1 (=) 491 533 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 1404 302 sh (\)) 2190 302 sh 384 /Symbol f1 (-) 2388 292 sh 384 1000 1171 /Symbol f3 (\() 3252 302 sh (\)) 3918 302 sh (\() 1611 836 sh (\)) 2068 836 sh 384 1000 1369 /Symbol f3 (\() 2184 854 sh (\)) 3127 854 sh 384 /Times-Roman f1 ( ) 1298 292 sh ( ) 3146 292 sh 384 /Times-Roman f1 (,) 1848 292 sh (,) 3576 292 sh 384 /Times-Roman f1 (0) 3388 292 sh end MTsave restore dMATH S H =TC H,S()-TC 0,S()H()MC H () FMathType Equation1ELO EqCompObj\ObjInfo46Ole10Native7Ole10FmtProgID  uation9q S H =TC H,S()-TC 0,S()H()MC H () Equationd iP _892714736[< F Y}B !Y}BOle PIC 9;dMETA  i   ! .  & Timesw@! [wdw0-!STimesw@" 7[wdw0-!SPSymbolw@! [wdw0-!=Timesw@" 8[wdw0-      !"#$%&'()*+,-.0348:;<=>?@ABCDEFGHIJKLMNOPQRSUVWXYZ[\]^_`abcdefghijklnquwxyz{|}~!TC Timesw@! [wdw0-! (Timesw@" 9[wdw0-!H 0Timesw@! [wdw0-!, 9Timesw@" :[wdw0-!S >PSymbolw@! [wdw0-!( +!) DPSymbolw@" ;[wdw0-!- JTimesw@! [wdw0-!TC STimesw@" <[wdw0-! bTimesw@! [wdw0-!H jTimesw@" =[wdw0-!, s!0 wPSymbolw@! [wdw0-!( e!) }Timesw@" >[wdw0-!S;PSymbolw@! [wdw0-!(6!)ATimesw@" ?[wdw0-!MCJTimesw@! [wdw0-!S[PSymbolw@" @[wdw0-!(E!)`- & 'PICT :>CompObj/\ObjInfo=?1Ole10Native@2!dxpr MTHU!Grphbj !"! currentpoint " +S +S (=( TC) )H) ,)S ( +()))-) TC) )H) ,)0 ( e())(;S (6() )) MC +S  (E())"hf/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4224 div 1056 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 744 434 moveto 3358 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (S) -3 533 sh (TC) 764 292 sh (H) 1493 292 sh (S) 1921 292 sh (TC) 2612 292 sh (H) 3341 292 sh (S) 1832 826 sh (MC) 2306 826 sh 224 ns (S) 178 629 sh (S) 2871 922 sh 384 /Symbol f1 (=) 426 533 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 1339 302 sh (\)) 2125 302 sh 384 /Symbol f1 (-) 2323 292 sh 384 1000 1171 /Symbol f3 (\() 3187 302 sh (\)) 3964 302 sh (\() 1690 836 sh (\)) 2036 836 sh 384 1000 1393 /Symbol f3 (\() 2152 856 sh (\)) 3030 856 sh 384 /Times-Roman f1 ( ) 1233 292 sh ( ) 3081 292 sh 384 /Times-Roman f1 (,) 1783 292 sh (,) 3631 292 sh 384 /Times-Roman f1 (0) 3763 292 sh end MTsave restore dMATH S S =TC H,S()-TC H,0()S()MC S () FMathType Equation1ELO Equation9q S S =TC H,S()-TC H,0()S()MC S () Equationd,Ole10FmtProgID 5 _892714792E F !Y}B !Y}BOle 6PIC BD7d Z   .  & Timesw@ [wdw0-!TC PSymbolw@ [wdw0-!= Timesw@ [wdw0-!a !PSymbolw@ [wdw0-META 9PICT CGT8CompObjmPObjInfoFHo!- )Timesw@ [wdw0-!cS 2PSymbolw@ [wdw0-!( !) >Timesw@ [wdw0-!H BPSymbolw@ [wdw0-!+ NTimesw@ [wdw0-!bS WPSymbolw@ [wdw0-!= gTimesw@ [wdw0-!aH pPSymbolw@ ![wdw0-!+ Timesw@ [wdw0-!b PSymbolw@ "[wdw0-!- Timesw@ [wdw0-!cH PSymbolw@ #[wdw0-!( !) Timesw@ [wdw0-!S Timesw@ $[wdw0-!, & '8dxpr MTHUGrphbj " currentpoint " + TC)=)a)-) cS ( ()"))H) +) bS)=) aH)+) b) -) cH ( ()%))S),g/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 6080 div 480 3 -1 roll exch div scale currentpoint translate 64 60 translate /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (TC) -12 292 sh (a) 997 292 sh (cS) 1560 292 sh (H) 2079 292 sh (bS) 2749 292 sh (aH) 3547 292 sh (b) 4534 292 sh (cH) 5097 292 sh (S) 5703 292 sh 384 /Symbol f1 (=) 559 292 sh (-) 1271 292 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 857 302 sh (\)) 1934 302 sh 384 /Symbol f1 (+) 2456 292 sh (=) 3234 292 sh (+) 4116 292 sh (-) 4808 292 sh 384 1000 1171 /Symbol f3 (\() 4398 302 sh (\)) 5570 302 sh 384 /Times-Roman f1 (,) 5894 292 sh end MTsave restore dnMATHb TC=a-cS()H+bS=aH+b-cH()S,sh FMathType Equation1ELO EquationOle10NativeIpfOle10FmtProgID r _892714864ASN F !Y}B !Y}BOle sb TC=a-cS()H+bS=aH+b-cH()S, Equationd4 l4    .  & PIC KMtdMETA vPICT LPCompObjPTimesw@D [wdw0-!STimesw@ [wdw0-!HTimesw@D [wdw0-!, Timesw@ [wdw0-!SPSymbolw@D [wdw0-!=Timesw@ [wdw0-!aH 3PSymbolw@D [wdw0-!+ ETimesw@ [wdw0-!bS NPSymbolw@D [wdw0-!- ]Timesw@ [wdw0-!cHS f!H"!a0PSymbolw@D [wdw0-!-8Timesw@ [wdw0-!cSAPSymbolw@D [wdw0-!(+!)MPSymbolw@ [wdw0-!+STimesw@D [wdw0-!S]!bgPSymbolw@ [wdw0-!-pTimesw@D [wdw0-!cHyPSymbolw@ [wdw0-!(c!)- ! & 'dxpr MTHUGrphbj " currentpoint " +S +H),)S (=( 3aH)+) bS)-) cHS("H)a)-) cS (+()"))+) S) b) -) cH (c()$)" !j/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4544 div 992 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 1008 403 moveto 3415 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (S) -3 502 sh (aH) 1587 261 sh (bS) 2449 261 sh (cHS) 3207 261 sh (H) 1049 795 sh (a) 1483 795 sh (cS) 2046 795 sh (S) 2918 795 sh (b) 3249 795 sh (cH) 3812 795 sh 224 ns (H) 185 598 sh (S) 442 598 sh 224 /Times-Roman f1 (,) 367 598 sh 384 /Symbol f1 (=) 690 502 sh (+) 2156 261 sh (-) 2918 261 sh (-) 1757 795 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 1343 805 sh (\)) 2420 805 sh 384 /Symbol f1 (+) 2619 795 sh (-) 3523 795 sh 384 1000 1171 /Symbol f3 (\() 3113 805 sh (\)) 4285 805 sh end MTsave restore dMATH S H,S =aH+bS-cHSHa-cS()+Sb-cH()f  FMathType Equation1ELO Equation S H,S =aH+bS-cHSHa-cS()+Sb-cH() EquationObjInfoOQOle10NativeROle10FmtProgID  _892714976U F !Y}B !Y}BOle PIC TWLMETA PICT VYL    .1  `@&  & MathType-+1  Century Schoolbook- 2 ,S 2 QaH  2 QbS 2 Qc cHS  2 AHa  2 =Sb 2  cHS  Century Schoolbookv- 2 pH 2 p S Century Schoolbookv- 2 p,>Symbol- 2  = 2 Q+ 2 Qn - 2 Q+ 2  - 2  >Century Schoolbookw- 2  2 2 h1 & "System-y y y"!HdPPNTCentury Schoolbook,Century Schoolbook .+S( $aH)bS)cHS("Ha)Sb)cHSdPPNTCentury Schoolbook (H)SdPPNTCentury Schoolbook(,dPPNTSymbol, Symbol (=( 5+)-(2+)-(k>dPPNTCentury Schoolbook(O2(r1dPPNT"System FMathType Equation Equation Equation S H,S =aH+bS-cHSHa+Sb-2cHS>1CompObjPObjInfoXZOle10Native_892715052Jm_ F !Y}B%#Y}BOle PIC \^dMETA PICT ]aqd Od  O     .  & Timesw@ z[wdw0-!STimesw@ [wdw0-!HPSymbolw@ {[wdw0-!=Timesw@ [wdw0-!aH PSymbolw@ |[wdw0-!+ 1Timesw@ [wdw0-!bS :PSymbolw@ }[wdw0-!- ITimesw@ [wdw0-!cHS RPSymbolw@ ~[wdw0-!( !) gPSymbolw@ [wdw0-!- mTimesw@ [wdw0-!bS v!H8!aEPSymbolw@ [wdw0-!-NTimesw@ [wdw0-!cSWPSymbolw@ [wdw0-!(A!)c- & 'q dxpr MTHU Grphbj " currentpoint " +S +H (=( aH)+) bS)-) cHS ( ()L))-) bS(8H) a) -) cS (A()")"g/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4256 div 1024 3 -1 roll exch div scale curren    !"#$%&'()*+,-./012345789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWY\]acdefghijklmnopqrstuvwxyz{}~tpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 809 434 moveto 3322 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (S) -3 533 sh (aH) 961 292 sh (bS) 1823 292 sh (cHS) 2581 292 sh (bS) 3722 292 sh (H) 1736 826 sh (a) 2170 826 sh (cS) 2733 826 sh 224 ns (H) 185 629 sh 384 /Symbol f1 (=) 491 533 sh (+) 1530 292 sh (-) 2292 292 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 821 302 sh (\)) 3232 302 sh 384 /Symbol f1 (-) 3430 292 sh (-) 2444 826 sh 384 1000 1171 /Symbol f3 (\() 2030 836 sh (\)) 3107 836 sh end MTsave restore d~MATHr S H =aH+bS-cHS()-bSHa-cS() FMathType Equation1ELO Equationr S H =aH+bS-cHS()-bSHa-cS()CompObj PObjInfo`b Ole10NativecvOle10FmtProgID  EquationdO O <    .  & Timesw@ [wdw0-!STimesw@n [wdw0_892715128h F%#Y}B%#Y}BOle PIC egdMETA -!HPSymbolw@ [wdw0-!=Timesw@n [wdw0-!aH PSymbolw@ [wdw0-!- 1Timesw@n [wdw0-!cHS :PSymbolw@ [wdw0-!( !) OTimesw@n [wdw0-!H !a.PSymbolw@ [wdw0-!-6Timesw@n [wdw0-!cS?PSymbolw@ [wdw0-!()!)K-RPSymbolw@n [wdw0-!=VTimesw@ [wdw0-!a ePSymbolw@n [wdw0-!- nTimesw@ [wdw0-!cS wPSymbolw@n [wdw0-!( a!) Timesw@ [wdw0-!aePSymbolw@n [wdw0-!-nTimesw@ [wdw0-!cSwPSymbolw@n [wdw0-!(a!)`PSymbolw@ [wdw0-!=Timesw@n [wdw0-!1 & 'Z dxpr MTHU Grphbj " currentpoint " PICT fj6ZCompObjXPObjInfoikZOle10Nativel[+S +H (=( aH)-) cHS ( ()4)( H)a)-) cS ()()")"7(V=( ea) -) cS ( a()!)(ea) -) cS (a()!)"`&(=) 1/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4896 div 1024 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 809 434 moveto 1798 0 rlineto stroke 3033 434 moveto 1227 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (S) -3 533 sh (aH) 961 292 sh (cHS) 1818 292 sh (H) 974 826 sh (a) 1408 826 sh (cS) 1971 826 sh (a) 3185 292 sh (cS) 3748 292 sh (a) 3185 826 sh (cS) 3748 826 sh 224 ns (H) 185 629 sh 384 /Symbol f1 (=) 491 533 sh (-) 1529 292 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 821 302 sh (\)) 2469 302 sh 384 /Symbol f1 (-) 1682 826 sh 384 1000 1171 /Symbol f3 (\() 1268 836 sh (\)) 2345 836 sh 384 /Symbol f1 (=) 2715 533 sh (-) 3459 292 sh 384 1000 1171 /Symbol f3 (\() 3045 302 sh (\)) 4122 302 sh 384 /Symbol f1 (-) 3459 826 sh 384 1000 1171 /Symbol f3 (\() 3045 836 sh (\)) 4122 836 sh 384 /Symbol f1 (=) 4368 533 sh 384 /Times-Roman f1 (1) 4644 533 sh end MTsave restore dMATH! S H =aH-cHS()Ha-cS()=a-cS()a-cS()=129 FMathType Equation1ELO Equation S H =aH-cHS()Ha-cS()=a-cS()a-cS()=1 EquationOle10FmtProgID ^ _892715257d~q F%#Y}B%#Y}BOle _PIC np`ddO O 0    .  & Timesw@ [wdw0-!STimesw@v [wdw0-!SPSymbolw@ [wdw0-!=META bhPICT os|CompObj\ObjInfortTimesw@v [wdw0-!aH PSymbolw@ [wdw0-!+ /Timesw@v [wdw0-!bS 8PSymbolw@ [wdw0-!- GTimesw@v [wdw0-!cHS PPSymbolw@ [wdw0-!( !) dPSymbolw@v [wdw0-!- kTimesw@ [wdw0-!aH t!S7!bBPSymbolw@v [wdw0-!-JTimesw@ [wdw0-!cHSPSymbolw@v [wdw0-!(=!)b-PSymbolw@ [wdw0-!=Timesw@v [wdw0-!1!. & ' dxpr MTHU Grphbj " currentpoint " +S +S (=( aH)+) bS)-) cHS ( ()K))-) aH(7S) b)-) cH (=()%)"j(=) 1)./MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 4928 div 1024 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 744 434 moveto 3425 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (S) -3 533 sh (aH) 896 292 sh (bS) 1758 292 sh (cHS) 2516 292 sh (aH) 3661 292 sh (S) 1718 826 sh (b) 2049 826 sh (cH) 2612 826 sh 224 ns (S) 178 629 sh 384 /Symbol f1 (=) 426 533 sh (+) 1465 292 sh (-) 2227 292 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 756 302 sh (\)) 3167 302 sh 384 /Symbol f1 (-) 3365 292 sh (-) 2323 826 sh 384 1000 1171 /Symbol f3 (\() 1913 836 sh (\)) 3085 836 sh 384 /Symbol f1 (=) 4277 533 sh 384 /Times-Roman f1 (1) 4553 533 sh 384 /Times-Roman f1 (.) 4745 533 sh end MTsave restore dMATH{ S S =aH+bS-cHS()-aHSb-cH()=1. FMathType Equation1ELO Equation9q{ S S =aH+bS-cHS()-aHSb-cH()=1. EquationL~itOle10NativeuOle10FmtProgID  _892715380x F%#Y}B%#Y}BOle PIC wzLPICT qCompObjy|\ObjInfoq dxpr MTHU Grphbj " currentpoint " +S +c (=( TC) ) H) ,)0 ( +()))+) TC) )0),)S ( d()))-) TC) )H) ,)S ( ())(QTC) )H) ,)S (c()) /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5920 div 1024 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 725 434 moveto 5088 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (S) -3 533 sh (TC) 745 292 sh (H) 1474 292 sh (TC) 2586 292 sh (S) 3623 292 sh (TC) 4314 292 sh (H) 5043 292 sh (S) 5471 292 sh (TC) 2530 826 sh (H) 3259 826 sh (S) 3687 826 sh 224 ns (c) 173 629 sh 384 /Symbol f1 (=) 407 533 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 1320 302 sh (\)) 2097 302 sh 384 /Symbol f1 (+) 2296 292 sh 384 1000 1171 /Symbol f3 (\() 3161 302 sh (\)) 3827 302 sh 384 /Symbol f1 (-) 4025 292 sh 384 1000 1171 /Symbol f3 (\() 4889 302 sh (\)) 5675 302 sh (\() 3105 836 sh (\)) 3891 836 sh 384 /Times-Roman f1 ( ) 1214 292 sh ( ) 3055 292 sh ( ) 4783 292 sh ( ) 2999 826 sh 384 /Times-Roman f1 (,) 1764 292 sh (,) 3485 292 sh (,) 5333 292 sh (,) 3549 826 sh 384 /Times-Roman f1 (0) 1896 292 sh (0) 3297 292 sh end MTsave restore dMATH S c =TC H,0()+TC 0,S()-TC H,S()TC H,S()6  FMathType Equation1ELO Equation9q S c =TC H,0()+TC 0,S()-TC H,SOle10Native{}Ole10FmtProgID  _892715465v F%#Y}B%#Y}BOle ()TC H,S() EquationLiD  dxpr MTHU Grphbj " currentpoint " +S +c (=( aH)+) bSPIC LPICT CompObj\ObjInfo)-) aH)+) bS)-) cHS ( K()L)(CTC) )H) ,)S (T()) /MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 5024 div 1024 3 -1 roll exch div scale currentpoint translate 64 43 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 725 434 moveto 4188 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (S) -3 533 sh (aH) 752 292 sh (bS) 1614 292 sh (aH) 2504 292 sh (bS) 3366 292 sh (cHS) 4124 292 sh (TC) 2080 826 sh (H) 2809 826 sh (S) 3237 826 sh 224 ns (c) 173 629 sh 384 /Symbol f1 (=) 407 533 sh (+) 1321 292 sh (-) 2083 292 sh (+) 3073 292 sh (-) 3835 292 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 2364 302 sh (\)) 4775 302 sh (\() 2655 836 sh (\)) 3441 836 sh 384 /Times-Roman f1 ( ) 2549 826 sh 384 /Times-Roman f1 (,) 3099 826 sh end MTsave restore dMATH~ S c =aH+bS-aH+bS-cHS()TC H,S() FMathType Equation1ELO Equation9q~ S c =aH+bS-aH+bS-cHS()TC H,S() EquationOle10NativeOle10FmtProgID  _892715551 F%#Y}B%#Y}BOle d 4Xl 4 K  ^ .  & Timesw@ [wdw0-!STimesw@ [wdw0-!cPSymbolw@PIC dMETA PICT CompObj\     "%&'()*,-./1 [wdw0-!=Timesw@ [wdw0-!cHS &!TCTimesw@ [wdw0-! 'Timesw@ [wdw0-!H0Timesw@ [wdw0-!,9Timesw@ [wdw0-!S=PSymbolw@ [wdw0-!(+!)C-  GPSymbolw@ [wdw0-!>KTimesw@ [wdw0-!0T!.Z & '^dxpr MTHU^Grphbj ^"^ currentpoint " +S +c (=( &cHS(TC) ) H) ,)S (+())" /(K>) 0).}/MTsave save def 40 dict begin currentpoint 3 -1 roll sub neg 3 1 roll sub 3008 div 992 3 -1 roll exch div scale currentpoint translate 64 42 translate /thick 0 def /th { dup setlinewidth /thick exch def } def 16 th 725 403 moveto 1519 0 rlineto stroke /cat { dup length 2 index length add string dup dup 5 -1 roll exch copy length 4 -1 roll putinterval } def /ff { dup FontDirectory exch known not { dup dup length string cvs (|______) exch cat dup FontDirectory exch known {exch} if pop } if findfont } def /fs 0 def /cf 0 def /sf {exch dup /fs exch def dup neg matrix scale makefont setfont} def /f1 {ff dup /cf exch def sf} def /ns {cf sf} def /sh {moveto show} def 384 /Times-Italic f1 (S) -3 502 sh (cHS) 1162 261 sh (TC) 745 795 sh (H) 1474 795 sh (S) 1902 795 sh 224 ns (c) 173 598 sh 384 /Symbol f1 (=) 407 502 sh /f3 {ff 3 -1 roll .001 mul 3 -1 roll .001 mul matrix scale makefont dup /cf exch def sf} def 384 1000 1171 /Symbol f3 (\() 1320 805 sh (\)) 2106 805 sh 384 /Symbol f1 (>) 2347 502 sh 384 /Times-Roman f1 ( ) 1214 795 sh 384 /Times-Roman f1 (,) 1764 795 sh (.) 2842 502 sh 384 /Times-Roman f1 (0) 2650 502 sh end MTsave restore dbMATHVW S c =cHSTC H,S()>0.  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