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(BQ) Part 2 book Managerial economics has contents: Government regulation of business; decisions under risk and uncertainty, advanced pricing techniques; strategic decision making in oligopoly markets; strategic decision making in oligopoly markets; managerial decisions in competitive markets,...and other contents.
Chapter www.downloadslide.com Production and Cost in the Long Run After reading this chapter, you will be able to: 9.1 Graph a typical production isoquant and discuss the properties of isoquants 9.2 Construct isocost curves for a given level of expenditure on inputs 9.3 Apply optimization theory to find the optimal input combination 9.4 Construct the firm’s expansion path and show how it relates to the firm’s longrun cost structure 9.5 Calculate long-run total, average, and marginal costs from the firm’s expansion path 9.6 Explain how a variety of forces affects long-run costs: scale, scope, learning, and purchasing economies 9.7 Show the relation between long-run and short-run cost curves using long-run and short-run expansion paths N o matter how a firm operates in the short run, its manager can always change things at some point in the future Economists refer to this future period as the “long run.” Managers face a particularly important constraint on the way they can organize production in the short run: The usage of one or more inputs is fixed Generally the most important type of fixed input is the physical capital used in production: machinery, tools, computer hardware, buildings for manufacturing, office space for administrative operations, facilities for storing inventory, and so on In the long run, managers can choose to operate with whatever amounts and kinds of capital resources they wish This is the essential feature of long-run analysis of production and cost In the long run, managers are not stuck with too much or too little capital— or any fixed input for that matter As you will see in this chapter, long-run 311 www.downloadslide.com 312 C H A P T E R 9 Production and Cost in the Long Run flexibility in resource usage usually creates an opportunity for firms to reduce their costs in the long run Since a long-run analysis of production generates the “best-case” scenario for costs, managers cannot make tactical and strategic decisions in a sensible way unless they possess considerable understanding of the long-run cost structure available to their firms, as well as the long-run costs of any rival firms they might face As we mentioned in the previous chapter, firms operate in the short run and plan for the long run The managers in charge of production operations must have accurate information about the short-run cost measures discussed in Chapter 8, while the executives responsible for long-run planning must look beyond the constraints imposed by the firm’s existing short-run configuration of productive inputs to a future situation in which the firm can choose the optimal combination of inputs Recently, U.S auto manufacturers faced historic challenges to their survival, forcing executive management at Ford, Chrysler, and General Motors to examine every possible way of reorganizing production to reduce long-run costs While shortrun costs determined their current levels of profitability—or losses in this case—it was the flexibility of long-run adjustments in the organization of production and structure of costs that offered some promise of a return to profitability and economic survival of American car producers The outcome for U.S carmakers depends on many of the issues you will learn about in this chapter: economies of scale, economies of scope, purchasing economies, and learning economies And, as you will see in later chapters, the responses by rival auto producers—both American and foreign—will depend most importantly on the rivals’ long-run costs of producing cars, SUVs, and trucks Corporate decisions concerning such matters as adding new product lines (e.g., hybrids or electric models), dropping current lines (e.g., Pontiac at GM), allowing some divisions to merge, or even, as a last resort, exiting through bankruptcy all require accurate analyses and forecasts of long-run costs In this chapter, we analyze the situation in which the fixed inputs in the short run become variable inputs in the long run In the long run, we will view all inputs as variable inputs, a situation that is both more complex and more i nteresting than production with only one variable input—labor For clarification and c ompleteness, we should remind you that, unlike fixed inputs, quasi-fixed inputs not become variable inputs in the long run In both the short- and long-run periods, they are indivisible in nature and must be employed in specific lump amounts that not vary with output—unless output is zero, and then none of the quasi-fixed inputs will be employed or paid Because the amount of a quasi-fixed input used in the short run is generally the same amount used in the long run, we not include quasi-fixed inputs as choice variables for long-run production decisions.1 With this distinction in mind, we can say that all inputs are variable in the long run An exception to this rule occurs when, as output increases, the fixed lump amount of input eventually becomes fully utilized and constrains further increases in output Then, the firm must add another lump of quasi-fixed input in the long run to allow further expansion of output This exception is not particularly important because it does not change the principles set forth in this chapter or other chapters in this textbook Thus, we will continue to assume that when a quasi-fixed input is required, only one lump of the input is needed for all positive levels of output www.downloadslide.com C H A P T E R 9 Production and Cost in the Long Run 313 9.1 PRODUCTION ISOQUANTS isoquant A curve showing all possible combinations of inputs p hysically capable of p roducing a given fixed level of output An important tool of analysis when two inputs are variable is the production isoquant or simply isoquant An isoquant is a curve showing all possible combinations of the inputs physically capable of producing a given (fixed) level of output Each point on an isoquant is technically efficient; that is, for each combination on the isoquant, the maximum possible output is that associated with the given isoquant The concept of an isoquant implies that it is possible to substitute some amount of one input for some of the other, say, labor for capital, while keeping output constant Therefore, if the two inputs are continuously divisible, as we will assume, there are an infinite number of input combinations capable of producing each level of output To understand the concept of an isoquant, return for a moment to Table 8.2 in the preceding chapter This table shows the maximum output that can be produced by combining different levels of labor and capital Now note that several levels of output in this table can be produced in two ways For example, 108 units of output can be produced using either units of capital and worker or unit of capital and workers Thus, these two combinations of labor and capital are two points on the isoquant associated with 108 units of output And if we assumed that labor and capital were continuously divisible, there would be many more combinations on this isoquant Other input combinations in Table 8.2 that can produce the same level of output are Q 258: using K 2, L 5 or K 8, L Q 400: using K 9, L 3 or K 4, L Q 453: using K 5, L 4 or K 3, L Q 708: using K 6, L 7 or K 5, L Q 753: using K 10, L 6 or K 6, L Each pair of combinations of K and L is two of the many combinations associated with each specific level of output Each demonstrates that it is possible to increase capital and decrease labor (or increase labor and decrease capital) while keeping the level of output constant For example, if the firm is producing 400 units of output with units of capital and units of labor, it can increase labor by 1, decrease capital by 5, and keep output at 400 Or if it is producing 453 units of output with K and L 7, it can increase K by 2, decrease L by 3, and keep output at 453 Thus an isoquant shows how one input can be substituted for another while keeping the level of output constant Characteristics of Isoquants We now set forth the typically assumed characteristics of isoquants when labor, capital, and output are continuously divisible Figure 9.1 illustrates three such isoquants Isoquant Q1 shows all the combinations of capital and labor that yield 100 units of output As shown, the firm can produce 100 units of output by u sing 10 units of capital and 75 of labor, or 50 units of capital and 15 of labor, or any other www.downloadslide.com 314 C H A P T E R 9 Production and Cost in the Long Run F I G U R E 9.1 A Typical Isoquant Map Units of capital (K ) 50 40 20 Q2 = 200 T 10 Q3 = 300 A 15 20 40 Q1 = 100 75 Units of labor (L) isoquant map A graph showing a group of isoquants combination of capital and labor on isoquant Q1 Similarly, isoquant Q2 shows the various combinations of capital and labor that can be used to produce 200 units of output And isoquant Q3 shows all combinations that can produce 300 units of output Each capital–labor combination can be on only one isoquant That is, isoquants cannot intersect Isoquants Q1, Q2, and Q3 are only three of an infinite number of isoquants that could be drawn A group of isoquants is called an isoquant map In an isoquant map, all isoquants lying above and to the right of a given isoquant indicate higher levels of output Thus in Figure 9.1 isoquant Q2 indicates a higher level of output than isoquant Q1, and Q3 indicates a higher level than Q2 Marginal Rate of Technical Substitution marginal rate of technical substitution (MRTS) The rate at which one input is substituted for another along an isoquant DK 2 _ DL As depicted in Figure 9.1, isoquants slope downward over the relevant range of production This negative slope indicates that if the firm decreases the amount of capital employed, more labor must be added to keep the rate of output constant Or if labor use is decreased, capital usage must be increased to keep output constant Thus the two inputs can be substituted for one another to maintain a constant level of output The rate at which one input is substituted for another along an isoquant is called the marginal rate of technical substitution (MRTS) and is defined as DK MRTS 2 _ DL The minus sign is added to make MRTS a positive number because DK/DL, the slope of the isoquant, is negative www.downloadslide.com C H A P T E R 9 Production and Cost in the Long Run 315 Over the relevant range of production, the marginal rate of technical substitution diminishes As more and more labor is substituted for capital while holding output constant, the absolute value of DK/DL decreases This can be seen in Figure 9.1 If capital is reduced from 50 to 40 (a decrease of 10 units), labor must be increased by units (from 15 to 20) to keep the level of output at 100 units That is, when capital is plentiful relative to labor, the firm can discharge 10 units of capital but must substitute only units of labor to keep output at 100 The marginal rate of technical substitution in this case is 2DK/DL 2(210)/5 2, meaning that for every unit of labor added, units of capital can be discharged to keep the level of output constant However, consider a combination where capital is more scarce and labor more plentiful For example, if capital is decreased from 20 to 10 (again a decrease of 10 units), labor must be increased by 35 units (from 40 to 75) to keep output at 100 units In this case the MRTS is 10/35, indicating that for each unit of labor added, capital can be reduced by slightly more than one-quarter of a unit As capital decreases and labor increases along an isoquant, the amount of capital that can be discharged for each unit of labor added declines This relation is seen in Figure 9.1 As the change in labor and the change in capital become extremely small around a point on an isoquant, the absolute value of the slope of a tangent to the isoquant at that point is the MRTS (2DK/DL) in the neighborhood of that point In Figure 9.1, the absolute value of the slope of tangent T to isoquant Q1 at point A shows the marginal rate of technical substitution at that point Thus the slope of the isoquant reflects the rate at which labor can be substituted for capital As you can see, the isoquant becomes less and less steep with movements downward along the isoquant, and thus MRTS declines along an isoquant Relation of MRTS to Marginal Products For very small movements along an isoquant, the marginal rate of technical substitution equals the ratio of the marginal products of the two inputs We will now demonstrate why this comes about The level of output, Q, depends on the use of the two inputs, L and K Since Q is constant along an isoquant, DQ must equal zero for any change in L and K that would remain on a given isoquant Suppose that, at a point on the isoquant, the marginal product of capital (MPK) is and the marginal product of labor (MPL) is If we add unit of labor, output would increase by units To keep Q at the original level, capital must decrease just enough to offset the 6-unit increase in output generated by the increase in labor Because the marginal product of capital is 3, units of capital must be discharged to reduce output by units In this case the MRTS 2DK/DL 2(22)/1 2, which is exactly equal to MPL/MPK 6/3 In more general terms, we can say that when L and K are allowed to vary slightly, the change in Q resulting from the change in the two inputs is the marginal product of L times the amount of change in L plus the marginal product of K times its change Put in equation form DQ (MPL)(DL) (MPK)(DK) www.downloadslide.com 316 C H A P T E R 9 Production and Cost in the Long Run To remain on a given isoquant, it is necessary to set DQ equal to Then, solving for the marginal rate of technical substitution yields DK MPL MRTS 2 _ 5 _ DL MPK Now try Technical Problem Using this relation, the reason for diminishing MRTS is easily explained As additional units of labor are substituted for capital, the marginal product of labor diminishes Two forces are working to diminish labor’s marginal product: (1) Less capital causes a downward shift of the marginal product of labor curve, and (2) more units of the variable input (labor) cause a downward movement along the marginal product curve Thus, as labor is substituted for capital, the marginal product of labor must decline For analogous reasons the marginal product of capital increases as less capital and more labor are used The same two forces are present in this case: a movement along a marginal product curve and a shift in the location of the curve In this situation, however, both forces work to increase the marginal product of capital Thus, as labor is substituted for capital, the marginal product of capital increases Combining these two conditions, as labor is substituted for capital, MPL decreases and MPK increases, so MPL/MPK will decrease 9.2 ISOCOST CURVES isocost curve Line that shows the various combinations of inputs that may be purchased for a given level of expenditure at given input prices Producers must consider relative input prices to find the least-cost c ombination of inputs to produce a given level of output An extremely useful tool for analyzing the cost of purchasing inputs is an isocost curve An isocost curve shows all combinations of inputs that may be purchased for a given level of total expenditure at given input prices As you will see in the next section, isocost curves play a key role in finding the combination of inputs that produces a given output level at the lowest possible total cost Characteristics of Isocost Curves Suppose a manager must pay $25 for each unit of labor services and $50 for each unit of capital services employed The manager wishes to know what combinations of labor and capital can be purchased for $400 total expenditure on inputs Figure 9.2 shows the isocost curve for $400 when the price of labor is $25 and the price of capital is $50 Each combination of inputs on this isocost curve costs $400 to purchase Point A on the isocost curve shows how much capital could be purchased if no labor is employed Because the price of capital is $50, the manager can spend all $400 on capital alone and purchase units of capital and units of labor Similarly, point D on the isocost curve gives the maximum amount of labor— 16 units—that can be purchased if labor costs $25 per unit and $400 are spent on labor alone Points B and C also represent input combinations that cost $400 At point B, for example, $300 (5 $50 6) are spent on capital and $100 (5 $25 4) are spent on labor, which represents a total cost of $400 If we continue to denote the quantities of capital and labor by K and L, and denote their respective prices by r and w, total cost, C, is C wL rK Total cost is www.downloadslide.com C H A P T E R 9 Production and Cost in the Long Run 317 F I G U R E 9.2 An Isocost Curve (w $25 and r $50) Capital (K ) 10 A B K=8– C L D 10 12 14 16 18 20 Labor (L) simply the sum of the cost of L units of labor at w dollars per unit and of K units of capital at r dollars per unit: C wL rK In this example, the total cost function is 400 25L 50K Solving this equation for 400 _ 25 K, you can see the combinations of K and L that can be chosen: K 50 2 50 L 5 1 L More generally, if a fixed amount C is to be spent, the firm can choose 8 2 among the combinations given by C w L K r r If C is the total amount to be spent on inputs, the most capital that can be purchased (if no labor is purchased) is C /r units of capital, and the most labor that can be purchased (if no capital is purchased) is C /w units of labor The slope of the isocost curve is equal to the negative of the relative input price ratio, 2w/r This ratio is important because it tells the manager how much capital must be given up if one more unit of labor is purchased In the example just given and illustrated in Figure 9.2, 2w/r 2$25/$50 21/2 If the manager wishes to purchase more unit of labor at $25, 1/2 unit of capital, which costs $50, must be given up to keep the total cost of the input combination constant If the price of labor happens to rise to $50 per unit, r remaining constant, the slope of the isocost curve is 2$50/$50 21, which means the manager must give up unit of capital for each additional unit of labor purchased to keep total cost constant Shifts in Isocost Curves If the constant level of total cost associated with a particular isocost curve changes, the isocost curve shifts parallel Figure 9.3 shows how the isocost curve shifts www.downloadslide.com 318 C H A P T E R 9 Production and Cost in the Long Run F I G U R E 9.3 Shift in an Isocost Curve 12 Capital (K ) 10 K = 10 – L K=8– 2 L 10 12 14 16 18 20 Labor (L) when the total expenditure on resources (C ) increases from $400 to $500 The isocost curve shifts out parallel, and the equation for the new isocost curve is K 10 1 L The slope is still 21/2 because 2w/r does not change The K-intercept is now 10, indicating that a maximum of 10 units of capital can be purchased if no labor is purchased and $500 are spent In general, an increase in cost, holding input prices constant, leads to a parallel upward shift in the isocost curve A decrease in cost, holding input prices constant, leads to a parallel downward shift in the isocost curve An infinite number of isocost curves exist, one for each level of total cost Relation At constant input prices, w and r for labor and capital, a given expenditure on inputs (C ) will purchase any combination of labor and capital given by the following equation, called an isocost curve: C w K r 2 r L Now try Technical Problem 9.3 FINDING THE OPTIMAL COMBINATION OF INPUTS We have shown that any given level of output can be produced by many combinations of inputs—as illustrated by isoquants When a manager wishes to produce a given level of output at the lowest possible total cost, the manager chooses the combination on the desired isoquant that costs the least This is a constrained m inimization problem that a manager can solve by following the rule for constrained optimization set forth in Chapter www.downloadslide.com C H A P T E R 9 Production and Cost in the Long Run 319 Although managers whose goal is profit maximization are generally and primarily concerned with searching for the least-cost combination of inputs to produce a given (profit-maximizing) output, managers of nonprofit organizations may face an alternative situation In a nonprofit situation, a manager may have a budget or fixed amount of money available for production and wish to maximize the amount of output that can be produced As we have shown using isocost curves, there are many different input combinations that can be purchased for a given (or fixed) amount of expenditure on inputs When a manager wishes to maximize output for a given level of total cost, the manager must choose the input combination on the isocost curve that lies on the highest isoquant This is a constrained maximization problem, and the rule for solving it was set forth in Chapter Whether the manager is searching for the input combination that minimizes cost for a given level of production or maximizes total production for a given level of expenditure on resources, the optimal combination of inputs to employ is found by using the same rule We first illustrate the fundamental principles of cost minimization with an output constraint; then we will turn to the case of output maximization given a cost constraint Production of a Given Output at Minimum Cost The principle of minimizing the total cost of producing a given level of output is illustrated in Figure 9.4 The manager wants to produce 10,000 units of output F I G U R E 9.4 Optimal Input Combination to Minimize Cost for a Given Output 140 134 Capital (K ) 120 100 KЈ K' 100 K'' KЉ Kٞ A B 90 A L' L'' B 60 66 E 60 C 40 Q1 = 10,000 Lٞ 60 90 150 Labor (L) 180 LЉ 201 LЈ 210 www.downloadslide.com 320 C H A P T E R 9 Production and Cost in the Long Run at the lowest possible total cost All combinations of labor and capital capable of producing this level of output are shown by isoquant Q1 The price of labor (w) is $40 per unit, and the price of capital (r) is $60 per unit Consider the combination of inputs 60L and 100K, represented by point A on isoquant Q1 At point A, 10,000 units can be produced at a total cost of $8,400, where the total cost is calculated by adding the total expenditure on labor and the total expenditure on capital:2 C wL rK ($40 60) ($60 100) $8,400 The manager can lower the total cost of producing 10,000 units by moving down along the isoquant and purchasing input combination B, because this combination of labor and capital lies on a lower isocost curve (K0L0) than input combination A, which lies on K9L9 The blowup in Figure 9.4 shows that combination B uses 66L and 90K Combination B costs $8,040 [5 ($40 66) ($60 90)] Thus the manager can decrease the total cost of producing 10,000 units by $360 (5 $8,400 $8,040) by moving from input combination A to input combination B on isoquant Q1 Since the manager’s objective is to choose the combination of labor and capital on the 10,000-unit isoquant that can be purchased at the lowest possible cost, the manager will continue to move downward along the isoquant until the lowest possible isocost curve is reached Examining Figure 9.4 reveals that the lowest cost of producing 10,000 units of output is attained at point E by using 90 units of labor and 60 units of capital on isocost curve K'''L''', which shows all input combinations that can be purchased for $7,200 Note that at this cost-minimizing input combination C wL rK ($40 90) ($60 60) $7,200 No input combination on an isocost curve below the one going through point E is capable of producing 10,000 units of output The total cost associated with input combination E is the lowest possible total cost for producing 10,000 units when w 5 $40 and r $60 Suppose the manager chooses to produce using 40 units of capital and 150 units of labor—point C on the isoquant The manager could now increase capital and reduce labor along isoquant Q1, keeping output constant and moving to lower and lower isocost curves, and hence lower costs, until point E is reached Regardless of whether a manager starts with too much capital and too little labor (such as point A) or too little capital and too much labor (such as point C), the manager can move to the optimal input combination by moving along the isoquant to lower and lower isocost curves until input combination E is reached At point E, the isoquant is tangent to the isocost curve Recall that the slope (in absolute value) of the isoquant is the MRTS, and the slope of the isocost curve Alternatively, you can calculate the cost associated with an isocost curve as the maximum amount of labor that could be hired at $40 per unit if no capital is used For K9L9, 210 units of labor could be hired (if K 0) for a cost of $8,400 Or 140 units of capital can be hired at $60 (if L 0) for a cost of $8,400 ... intercepts of 12 units of capital and 24 units of labor, which clearly has a slope of 25 /10 (5 −w/r), shows the least-cost method of producing 100 units of output: Use 10 units of labor and 7 units of. .. (w $5, r $10) Long-run average cost (LAC ) Long-run marginal cost (LMC ) 100 20 0 300 400 500 600 700 10 12 20 30 40 52 60 7 8 10 15 22 30 42 $ 120 140 20 0 300 420 560 720 $1 .20 0.70 0.67 0.75... method of producing 20 0 units of output is to use 12 units of labor and units of capital Thus producing T A B L E 9.1 Derivation of a Long-Run Cost Schedule (1) (2) (3) Least-cost combination of