Production and Cost in the Long Run

By Thomas, C.R., Maurice, S.C.

Edited by Paul Ducham

PRODUCTION ISOQUANTS

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 8.1 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 6 units of capital and 1 worker or 1 unit of capital and 4 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 8.1 that can produce the same level of output are:

Q = 258: using K = 2, L = 5 or K = 8, L = 2

Q = 400: using K = 9, L = 3 or K = 4, L = 4

Q = 453: using K = 5, L = 4 or K = 3, L = 7

Q = 708: using K = 6, L = 7 or K = 5, L = 9

Q = 753: using K = 10, L = 6 or K = 6, L = 8

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 9 units of capital and 3 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 = 3 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 using 10 units of capital and 75 of labor, or 50 units of capital and 15 of labor, or any other combination of capital and labor on isoquant Q1. Similarly, isoquant Q2shows 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 Q2indicates a higher level of output than isoquant Q1 and Q3, indicates a higher level than Q2.

Marginal Rate of Technical Substitution

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 in order 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

MRTS = -K/L

The minus sign is added to make MRTS a positive number, since K/L, the slope of the isoquant, is negative.

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 K/L 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 5 units (from 15 to 20) in order 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 5 units of labor in order to keep output at 100. The marginal rate of technical substitution in this case is K/L = -(-10)5 = 2, meaning that for every unit of labor added, 2 units of capital can be discharged in order 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 (-K/L) 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 sub- stitution 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. Because Q is constant along an isoquant, Q 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 3 and the marginal product of labor (MPL) is 6.If we add 1 unit of labor, output would increase by 6 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, 2 units of capital must be discharged in order to reduce output by 6 units. In this case the MRTS = -K/L = -(-2)/1 = 2, which is exactly equal to MPL/MPK = 6/3 = 2.

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,

Q = (MPL)(L) + (MPK)(K)

In order to remain on a given isoquant, it is necessary to set Q equal to 0. Then, solving for the marginal rate of technical substitution yields

MRTS = -K/L = MPL/MPK

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.

Fig-9.1

ISOCOST CURVES

Producers must consider relative input prices in order to find the least-cost combination 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. Since the price of capital is $50, the manager can spend all $400 on capital alone and purchase 8 units of capital and 0 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 ( $50 6) are spent on capital and $100 ( $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 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

9P1Fig-9.2Fig-9.3

OPTIMAL INPUT COMBINATIONS

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 minimization problem that a manager can solve by following the rule for CONSTRAINED OPTIMIZATION.

While 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.

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 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:

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 (K''L'') than input combination A, which lies on K'L'. The blowup in Figure 9.4 shows that combination B uses 66L and 90K. Combination B costs $8,040 [= ($40 * 66) + ($60 * 90)]. Thus the manager can decrease the total cost of producing 10,000 units by $360 (= $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 = $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 (in absolute value) is equal to the relative input price ratio, w/r. Thus, at point E, MRTS equals the ratio of input prices. At the cost-minimizing input combination,

MRTS = w/r

To minimize the cost of producing a given level of output, the manager employs the input combination for which MRTS = w/r

The Marginal Product Approch cost Minimization

Finding the optimal levels of two activities A and B in a constrained optimization problem involved equating the marginal benefit per dollar spent on each of the activities (MB/P). A manager compares the marginal benefit per dollar spent on each activity to determine which activity is the “better deal”: that is, which activity gives the higher marginal benefit per dollar spent. At their optimal levels, both activities are equally good deals (MBA/PA = MBB/PB) and the constraint is met.

The tangency condition for cost minimization, MRTS = w/r, is equivalent to the condition of equal marginal benefit per dollar spent. Recall that MRTS = MPL/MPK; thus the cost-minimizing condition can be expressed in terms of marginal products:

MRTS = MPL/MPK = w/r

After a bit of algebraic manipulation, the optimization condition may be expressed as

MPL/w = MPK/r

The marginal benefits of hiring extra units of labor and capital are the marginal products of labor and capital. Dividing each marginal product by its respective input price tells the manager the additional output that will be forthcoming if one more dollar is spent on that input. Thus, at point E in Figure 9.4, the marginal product per dollar spent on labor is equal to the marginal product per dollar spent on capital, and the constraint is met (Q = 10,000 units).

To illustrate how a manager uses information about marginal products and input prices to find the least-cost input combination, we return to point A in Figure 9.4, where MRTS is greater than w/r. Assume that at point A, MPL = 160 and MPK = 80; thus MRTS = 2 (= MPL/MPK = 160/80). Since the slope of the isocost curve is 2/3 (= w/r = 40/60), MRTS is greater than w/r, and

MPL/w = 160/40 = 4 > 1.33 = 80/60 = MPK/r

The firm should substitute labor, which has the higher marginal product per dollar, for capital, which has the lower marginal product per dollar. For example, an additional unit of labor would increase output by 160 units while increasing labor cost by $40. To keep output constant, 2 units of capital must be released, causing output to fall 160 units (the marginal product of each unit of capital released is 80), but the cost of capital would fall by $120, which is $60 for each of the 2 units of capital released. Output remains constant at 10,000 because the higher output from 1 more unit of labor is just offset by the lower output from 2 fewer units of capital. However, because labor cost rises by only $40 while capital cost falls by $120, the total cost of producing 10,000 units of output falls by $80 (= $120 - $40).

This example shows that when MPL/w is greater than MPK/r, the manager can reduce cost by increasing labor usage while decreasing capital usage just enough to keep output constant. Since MPL/w > MPK/r for every input combination along Q1 from point A to point E, the firm should continue to substitute labor for capital until it reaches point E. As more labor is used, MPL falls because of diminishing marginal product. As less capital is used, MPK rises for the same reason. As the manager substitutes labor for capital, MRTS falls until equilibrium is reached.

Now consider point C, where MRTS is less than w/r, and consequently MPL/w is less than MPK/r. The marginal product per dollar spent on the last unit of labor is less than the marginal product per dollar spent on the last unit of capital. In this case, the manager can reduce cost by increasing capital usage and decreasing labor usage in such a way as to keep output constant. To see this, assume that at point C, MPL = 40 and MPK = 240, and thus MRTS = 40/240 = 1/6, which is less than w/r (=2/3). If the manager uses 1 more unit of capital and 6 fewer units of labor, output stays constant while total cost falls by $180. (You should verify this yourself.) The manager can continue moving upward along isoquant Q1, keeping output constant but reducing cost until point E is reached. As capital is increased and labor decreased, MPL rises and MPK falls until, at point E,MPL/w equals MPK/r. We have now derived the following:

Principle To produce a given level of output at the lowest possible cost when two inputs (L and K) are variable and the prices of the inputs are, respectively, w and r, a manager chooses the combination of inputs for which

MRTS = MPL/MPK = w/r

which implies that

MPL/w = MPK/r

The isoquant associated with the desired level of output (the slope of which is the MRTS) is tangent to the isocost curve (the slope of which is w/r) at the optimal combination of inputs. This optimization condition also means that the marginal product per dollar spent on the last unit of each input is the same.

Production of Maximum Output with a Given Level of Cost

As discussed earlier, there may be times when managers can spend only a fixed amount on production and wish to attain the highest level of production consistent with that amount of expenditure. This is a constrained maximization problem, the optimization condition for constrained maximization is the same as that for constrained minimization. In other words, the input combination that maximizes the level of output for a given level of total cost of inputs is that combination for which

MRTS = w/r or MPL/w = MPK/r

This is the same condition that must be satisfied by the input combination that minimizes the total cost of producing a given output level.

This situation is illustrated in Figure 9.5. The isocost line KL shows all possible combinations of the two inputs that can be purchased for the level of total cost (and input prices) associated with this isocost curve. Suppose the manager chooses point R on the isocost curve and is thus meeting the cost constraint. While 500 units of output are produced using LR units of labor and KR units of capital, the manager could produce more output at no additional cost by using less labor and more capital.

This can be accomplished, for example, by moving up the isocost curve to point S. Point S and point R lie on the same isocost curve and consequently cost the same amount. Point S lies on a higher isoquant, Q2, allowing the manager to produce 1,000 units without spending any more than the given amount on inputs (represented by isocost curve KL). The highest level of output attainable with the given level of cost is 1,700 units (point E), which is produced by using LE labor and KE capital. At point E, the highest attainable isoquant, isoquant Q3, is just tangent to the given isocost, and MRTS = w/r or MPL/w = MPK/r, the same conditions that must be met to minimize the cost of producing a given output level.

To see why MPL/w must equal MPK/r in order to maximize output for a given level of expenditures on inputs, suppose that this optimizing condition does not hold. Specifically, assume that w = $2, r = $3, MPL = 6, and MPK = 12, so that

MPL/w = 6/2 = 3 < 4 = 12/3 = MPK/r

The last unit of labor adds 3 units of output per dollar spent; the last unit of capital adds 4 units of output per dollar. If the firm wants to produce the maximum output possible with a given level of cost, it could spend $1 less on labor, thereby reducing labor by half a unit and hence output by 3 units. It could spend this dollar on capital, thereby increasing output by 4 units. Cost would be unchanged, and total output would rise by 1 unit. And the firm would continue taking dollars out of labor and adding them to capital as long as the inequality holds. But as labor is reduced, its marginal product will increase, and as capital is increased, its marginal product will decline. Eventually the marginal product per dollar spent on each input will be equal. We have established the following:

Principle In the case of two variable inputs, labor and capital, the manager of a firm maximizes output for a given level of cost by using the amounts of labor and capital such that the marginal rate of technical substitution (MRTS) equals the input price ratio (w/r). In terms of a graph, this condition is equivalent to choosing the input combination where the slope of the given isocost curve equals the slope of the highest attainable isoquant. This output-maximizing condition implies that the marginal product per dollar spent on the last unit of each input is the same.

Fig-9.4Fig-9.5

OPTIMIZATION AND COST

Using Figure 9.4 we showed how a manager can choose the optimal (least-cost) combination of inputs to produce a given level of output. We also showed how the total cost of producing that level of output is calculated. When the optimal input combination for each possible output level is determined and total cost is calculated for each one of these input combinations, a total cost curve (or schedule) is generated. In this section, we illustrate how any number of optimizing points can be combined into a single graph and how these points are related to the firm’s cost structure.

An Expansion Path

In Figure 9.4 we illustrated one optimizing point for a firm. This point shows the optimal (least-cost) combination of inputs for a given level of output. However, as you would expect, there exists an optimal combination of inputs for every level of output the firm might choose to produce. And the proportions in which the inputs are used need not be the same for all levels of output. To examine several optimizing points at once, we use the expansion path.

The expansion path shows the cost-minimizing input combination for each level of output with the input price ratio held constant. It therefore shows how input usage changes as output changes. Figure 9.6 illustrates the derivation of an expansion path. Isoquants Q1, Q2, and Q3 show, respectively, the input combinations of labor and capital that are capable of producing 500, 700, and 900 units of output. The price of capital (r) is $20 and the price of labor (w) is $10. Thus any isocost curve would have a slope of 10/20 = 1/2.

The three isocost curves KL, K'L', and K''L'', each of which has a slope of 1/2, represent the minimum costs of producing the three levels of output, 500, 700, and 900, because they are tangent to the respective isoquants. That is, at optimal input combinations A, B, and C, MRTS = w/r = 1/2. In the figure, the expansion path connects these optimal points and all other points so generated.

Note that points A, B, and C are also points indicating the combinations of inputs that can produce the maximum output possible at each level of cost given by isocost curves KL, K'L', and K''L''. The optimizing condition, as emphasized, is the same for cost minimization with an output constraint and output maximization with a cost constraint. For example, to produce 500 units of output at the lowest possible cost, the firm would use 91 units of capital and 118 units of labor. The lowest cost of producing this output is therefore $3,000 (from the vertical intercept, $20 * 150 = $3,000). Likewise, 91 units of capital and 118 units of labor are the input combination that can produce the maximum possible output (500 units) under the cost constraint given by $3,000 (isocost curve KL). Each of the other optimal points along the expansion path also shows an input combination that is the cost-minimizing combination for the given output or the output-maximizing combination for the given cost. At every point along the expansion path,

MRTS = MPL/MPK = w/r

and

MPL/w = MPK/r

Therefore, the expansion path is the curve or locus of points along which the marginal rate of technical substitution is constant and equal to the input price ratio. It is a curve with a special feature: It is the curve or locus along which the firm will expand output when input prices are constant.

Relation The expansion path is the curve along which a firm expands (or contracts) output when input prices remain constant. Each point on the expansion path represents an efficient (least-cost) input combination. Along the expansion path, the marginal rate of technical substitution equals the constant input price ratio. The expansion path indicates how input usage changes when output or cost changes.

The Expansion Path and the Structure of Cost

An important aspect of the expansion path that was implied in this discussion and will be emphasized in the remainder of this chapter is that the expansion path gives the firm its cost structure. The lowest cost of producing any given level of output can be determined from the expansion path. Thus the structure of the relation between output and cost is determined by the expansion path.

Recall from the discussion of Figure 9.6 that the lowest cost of producing 500 units of output is $3,000, which was calculated as the price of capital, $20, times the vertical intercept of the isocost curve, 150. Alternatively, the cost of producing 500 units can be calculated by multiplying the price of labor by the amount of labor used plus the price of capital by the amount of capital used:

wL + rK = ($10 * 118) + ($20 * 91) = $3,000

Using the same method, we calculate the lowest cost of producing 700 and 900 units of output, respectively, as

($10 * 148) + ($20 * 126) = $4,000

and

($10 * 200) + ($20 * 150) = $5,000

Similarly, the sum of the quantities of each input used times the respective input prices gives the minimum cost of producing every level of output along the expansion path. This allows the firm to relate its cost to the level of output used.

I L L U S T R AT I O N 9 . 1

Downsizing or Dumbsizing

Optimal Input Choice Should Guide Restructuring Decisions

One of the most disparaged strategies for cost cutting has been corporate “downsizing” or, synonymously, corporate “restructuring.” Managers downsize a firm by permanently laying off a sizable fraction of their workforce, in many cases, using across-the-board layoffs.

If a firm employs more than the efficient amount of labor, reducing the amount of labor employed can lead to lower costs for producing the same amount of output. Business publications have documented dozens of restructuring plans that have failed to realize the promised cost savings. Apparently, a successful restructuring requires more than “meat-ax,” across-the-board cutting of labor. The Wall Street Journal reported that “despite warnings about downsizing becoming dumbsizing, many companies continue to make flawed decisions—hasty, across-the-board cuts —that come back to haunt them.”

The reason that across-the-board cuts in labor do not generally deliver the desired lower costs can be seen by applying the efficiency rule for choosing inputs that we have developed in this chapter. In order to either minimize the total cost of producing a given level of output or to maximize the output for a given level of cost, managers must base employment decisions on the marginal product per dollar spent on labor, MP/w. Across-theboard downsizing, when no consideration is given to productivity or wages, cannot lead to an efficient reduction in the amount of labor employed by the firm.Workers with the lowest MP/w ratios must be cut first if the manager is to realize the greatest possible cost savings.

Consider this example: Amanager is ordered to cut the firm’s labor force by as many workers as it takes to lower its total labor costs by $10,000 per month. The manager wishes to meet the lower level of labor costs with as little loss of output as possible. The manager examines the employment performance of six workers: workers A and B are senior employees, and workers C, D, E, and F are junior employees. The accompanying table shows the productivity and wages paid monthly to each of these six workers. The senior workers (A and B) are paid more per month than the junior workers (C, D, E, and F), but the senior workers are more productive than the junior workers. Per dollar spent on wages, each senior worker contributes 0.50 unit of output per month, while each dollar spent on junior workers contributes 0.40 unit per month. Consequently, the senior workers provide the firm with more “bang per buck,” even though their wages are higher. The manager, taking an across-the-board approach to cutting workers, could choose to lay off $5,000 worth of labor in each category: lay off worker A and workers C and D. This across-the-board strategy saves the required $10,000, but output falls by 4,500 units per month (= 2,500 + 2 * 1,000). Alternatively, the manager could rank the workers according to the marginal product per dollar spent on each worker. Then, the manager could start by sequentially laying off the workers with the smallest marginal product per dollar spent. This alternative approach would lead the manager to lay off four junior workers. Laying off workers C, D, E, and F saves the required $10,000 but reduces output by 4,000 units per month (= 4 * 1,000). Sequentially laying off the workers that give the least bang for the buck results in a smaller reduction in output while achieving the required labor savings of $10,000.

This illustration shows that restructuring decisions should be made on the basis of the production theory presented in this chapter. Input employment decisions cannot be made efficiently without using

9T1

information about both the productivity of an input and the price of the input. Across-the-board approaches to restructuring cannot, in general, lead to efficient reorganizations because these approaches do not consider information about worker productivity per dollar spent when making the layoff decision. Reducing the amount of labor employed is not “dumbsizing” if a firm is employing more than the efficient amount of labor. Dumbsizing occurs only when a manager lays off the wrong workers or too many workers.

Fig-9.6

LONG-RUN COSTS

Now that we have demonstrated how a manager can find the cost-minimizing input combination when more than one input is variable, we can derive the cost curves facing a manager in the long run. The structure of long-run cost curves is determined by the structure of long-run production, as reflected in the expansion path.

Derivation of Cost Schedules from a Production Function

We begin our discussion with a situation in which the price of labor (w) is $5 per unit and the price of capital (r) is $10 per unit. Figure 9.7 shows a portion of the firm’s expansion path. Isoquants Q1, Q2, and Q3 are associated, respectively, with 100, 200, and 300 units of output.

For the given set of input prices, the isocost curve with intercepts of 12 units of capital and 24 units of labor, which clearly has a slope of -5/10 (=-w/r), shows the least-cost method of producing 100 units of output: Use 10 units of labor and 7 units of capital. If the firm wants to produce 100 units, it spends $50 ($5 * 10) on labor and $70 ($10 * 7) on capital, giving it a total cost of $120.

9P2

The first row of Table 9.1 gives the level of output (100), the least-cost combination of labor and capital that can produce that output, and the long-run total, average, and marginal costs when output is 100 units.

Returning to Figure 9.7, you can see that the least-cost method of producing 200 units of output is to use 12 units of labor and 8 units of capital. Thus producing 200 units of output costs $140 (= $5 * 12 + $10 * 8). The average cost is $0.70 (= $140/200) and, since producing the additional 100 units increases total cost from $120 to $140, the marginal cost is $0.20 (= $20/100). These figures are shown in the second row of Table 9.1, and they give additional points on the firm’s longrun total, average, and marginal cost curves.

Figure 9.7 shows that the firm will use 20 units of labor and 10 units of capital to produce 300 units of output. Using the same method as before, we calculate total, average, and marginal costs, which are given in row 3 of Table 9.1.

Figure 9.7 shows only three of the possible cost-minimizing choices. But, if we were to go on, we could obtain additional least-cost combinations, and in the same way, we could calculate the total, average, and marginal costs of these other outputs. This information is shown in the last four rows of Table 9.1 for output levels from 400 through 700.

Thus, at the given set of input prices and with the given technology, column 4 shows the long-run total cost schedule, column 5 the long-run average cost schedule, and column 6 the long-run marginal cost schedule. The corresponding long-run total cost curve is given in Figure 9.8, Panel A. This curve shows the least cost at which each quantity of output in Table 9.1 can be produced when no input is fixed. Its shape depends exclusively on the production function and the input prices.

This curve reflects three of the commonly assumed characteristics of long-run total cost. First, because there are no fixed costs, LTC is 0 when output is 0. Second, cost and output are directly related; that is, LTC has a positive slope. It costs more to produce more, which is to say that resources are scarce or that one never gets something for nothing. Third, LTC first increases at a decreasing rate, then increases at an increasing rate. This implies that marginal cost first decreases, then increases.

Turn now to the long-run average and marginal cost curves derived fromTable 9.1 and shown in Panel B of Figure 9.8. These curves reflect the characteristics of typical LAC and LMC curves. They have essentially the same shape as they do in the short run—but, as we shall show below, for different reasons. Long-run average cost first decreases, reaches aminimum(at 300 units of output), then increases. Long-run marginal cost first declines, reaches its minimum at a lower output than that associated with minimum LAC (between 100 and 200 units), and then increases thereafter.

In Figure 9.8, marginal cost crosses the average cost curve at approximately the minimum of average cost. As we will show next, when output and cost are allowed to vary continuously, LMC crosses LAC at exactly the minimum point on the latter. (It is only approximate in Figure 9.8 because output varies discretely by 100 units in the table.)

The reasoning is the same as that given for short-run average and marginal cost curves. When marginal cost is less than average cost, each additional unit produced adds less than average cost to total cost, so average cost must decrease. When marginal cost is greater than average cost, each additional unit of the good produced adds more than average cost to total cost, so average cost must be increasing over this range of output. Thus marginal cost must be equal to average cost when average cost is at its minimum.

Figure 9.9 shows long-run marginal and average cost curves that reflect the typically assumed characteristics when output and cost can vary continuously.

Relations As illustrated in Figure 9.9, (1) long-run average cost, defined as

L AC = LTC/Q

first declines, reaches a minimum (here at Q2 units of output), and then increases. (2) When LAC is at its minimum, long-run marginal cost, defined as

L MC = LTC/Q

equals LAC. (3) LMC first declines, reaches a minimum (here at Q1, less than Q2), and then increases. LMC lies below LAC over the range in which LAC declines; it lies above LAC when LAC is rising.

Fig-9.7

Fig-9.8Fig-9.9

FORCES AFFECTING LONG-RUN COSTS

As they plan for the future, business owners and managers make every effort to avoid undertaking operations or making strategic plans that will result in losses or negative profits. When managers foresee market conditions that will not generate enough total revenue to cover long-run total costs, they will plan to cease production in the long run and exit the industry by moving the firm’s resources to their best alternative use. Similarly, decisions to add new product lines or enter new geographic markets will not be undertaken unless managers are reasonably sure that long-run costs can be paid from revenues generated by entering those new markets. Because the long-run viability of a firm—as well as the number of product lines and geographic markets a firm chooses—depends crucially on the likelihood of covering long-run costs, managers need to understand the various economic forces that can affect long-run costs. We will now examine several important forces that affect the long-run cost structure of firms. While some of these factors cannot be directly controlled by managers, the ability to predict costs in the long run requires an understanding of all forces, internal and external, that affect a firm’s long-run costs. Managers who can best forecast future costs are likely to make the most profitable decisions.

Economies and Diseconomies of Scale

The shape of a firm’s long-run average cost curve (LAC) determines the range and strength of economies and diseconomies of scale. Economies of scale occur when long-run average cost falls as output increases. In Figure 9.10, economies of scale exist over the range of output from zero up to Q2 units of output. Diseconomies of scale occur when long-run average cost rises as output increases. As you can see in the figure, diseconomies of scale set in beyond Q2 units of output.

The strength of scale economies or diseconomies can been seen, respectively, as the reduction in unit cost over the range of scale economies or the increase in LAC above its minimum value LAC2 beyond Q2. Average cost falls when marginal cost is less than average cost. As you can see in the figure, over the output range from 0 to Q2, LAC is falling because LMC is less than LAC. Beyond Q2, LMC is greater than LAC, and LAC is rising.

Reasons for Scale Economies and Diseconomies

Before we begin discussing reasons for economies and diseconomies of scale, we need to remind you of two things that cannot be reasons for rising or falling unit costs as quantity increases along the LAC curve: changes in technology and changes in input prices. Recall that both technology and input prices are held constant when deriving expansion paths and long-run cost curves. Consequently, as a firm moves along its LAC curve to larger scales of operation, any economies and diseconomies of scale the firm experiences must be caused by factors other than changing technology or changing input prices. When technology or input prices do change, as we will show you later in this section, the entire LAC curve shifts upward or downward, perhaps even changing shape in ways that will alter the range and strength of existing scale economies and diseconomies.

Probably the most fundamental reason for economies of scale is that larger-scale firms have greater opportunities for specialization and division of labor. As an example, consider Precision Brakes, a small-scale automobile brake repair shop servicing only a few customers each day and employing just one mechanic. The single mechanic at Precision Brakes must perform every step in each brake repair: moving the car onto a hydraulic lift in a service bay, removing the wheels, removing the worn brake pads and shoes, installing the new parts, replacing the wheels, moving the car off the lift and out of the service bay, and perhaps even processing and collecting a payment from the customer. As the number of customers grows larger at Precision Brakes, the repair shop may wish to increase its scale of operation by hiring more mechanics and adding more service bays. At this larger scale of operation, some mechanics can specialize in lifting the car and removing worn out parts, while others can concentrate on installing the new parts and moving cars off the lifts and out of the service bays. And, a customer service manager would probably process each customer’s work order and collect payments. As you can see from this rather straightforward example, large-scale production affords the opportunity for dividing a production process into a number of specialized tasks. Division of labor allows workers to focus on single tasks, which increases worker productivity in each task and brings about very substantial reductions in unit costs.

A second cause of falling unit costs arises when a firm employs one or more quasi-fixed inputs. Quasi-fixed inputs must be used in fixed amounts in both the short run and long run. As output expands, quasi-fixed costs are spread over more units of output causing long-run average cost to fall. The larger the contribution of quasi-fixed costs to overall total costs, the stronger will be the downward pressure on LAC as output increases. For example, a natural gas pipeline company experiences particularly strong economies of scale because the quasi-fixed cost of its pipelines and compressor pumps accounts for a very large portion of the total costs of transporting natural gas through pipelines. In contrast, a trucking company can expect to experience only modest scale economies from spreading the quasi-fixed cost of tractor-trailer rigs over more transportation miles, because the variable fuel costs account for the largest portion of trucking costs.

A variety of technological factors constitute a third force contributing to economies of scale. First, when several different machines are required in a production process and each machine produces at a different rate of output, the operation may have to be quite sizable to permit proper meshing of equipment. Suppose only two types of machines are required: one that produces the product and one that packages it. If the first machine can produce 30,000 units per day and the second can package 45,000 units per day, output will have to be 90,000 units per day in order to fully utilize the capacity of each type of machine: three machines making the good and two machines packaging it. Failure to utilize the full capacity of each machine drives up unit production costs because the firm is paying for some amount of machine capacity it does not need or use.

Another technological factor creating scale economies concerns the costs of capital equipment: The expense of purchasing and installing larger machines is usually proportionately less than for smaller machines. For example, a printing press that can run 200,000 papers per day does not cost 10 times as much as one that can run 20,000 per day—nor does it require 10 times as much building space, 10 times as many people to operate it, and so forth. Again, expanding size or scale of operation tends to reduce unit costs of production.

A final technological matter might be the most important technological factor of all: As the scale of operation expands, there is usually a qualitative change in production process and type of capital equipment employed. For a simple example, consider ditch digging. The smallest scale of operation is one worker and one shovel. But as the scale expands, the firm does not simply continue to add workers and shovels. Beyond a certain point, shovels and most workers are replaced by a modern ditch-digging machine. Furthermore, expansion of scale also permits the introduction of various types of automation devices, all of which tend to reduce the unit cost of production.

You may wonder why the long-run average cost curve would ever rise. After all possible economies of scale have been realized, why doesn’t the LAC curve become horizontal, never turning up at all? The rising portion of LAC is generally attributed to limitations to efficient management and organization of the firm. As the scale of a plant expands beyond a certain point, top management must necessarily delegate responsibility and authority to lower- Echelon employees. Contact with the daily routine of operation tends to be lost, and efficiency of operation declines. Furthermore, managing any business entails controlling and coordinating a wide variety of activities: production, distribution, finance, marketing, and so on. To perform these functions efficiently, a manager must have accurate information, as well as efficient monitoring and control systems. Even though information technology continues to improve in dramatic ways, pushing higher the scale at which diseconomies set in, the cost of monitoring and controlling large-scale businesses eventually leads to rising unit costs.

As an organizational plan for avoiding diseconomies, large-scale businesses sometimes divide operations into two or more separate management divisions so that each of the smaller divisions can avoid some or all of the diseconomies of scale. Unfortunately, division managers frequently compete with each other for allocation of scarce corporate resources—such as workers, travel budget, capital outlays, office space, and R & D expenditures. The time and energy spent by division managers trying to influence corporate allocation of resources is costly for division managers, as well as for top-level corporate managers who must evaluate the competing claims of division chiefs for more resources. Overall corporate efficiency is sacrificed when lobbying by division managers results in a misallocation of resources among divisions. Scale diseconomies, then, remain a fact of life for very large-scale enterprises.

Constant Costs: Absence of Economies and Diseconomies of Scale

In some cases, firms may experience neither economies nor diseconomies of scale, and instead face constant costs. When a firm experiences constant costs in the long run, its LAC curve is flat and equal to its LMC curve at all output levels. Figure 9.11 illustrates a firm with constant costs of $20 per unit: Average and marginal costs are both equal to $20 for all output levels. As you can see by the flat LAC curve, firms facing constant costs experience neither economies nor diseconomies of scale.

Instances of truly constant costs at all output levels are not common in practice. However, businesses frequently treat their costs as if they are constant even when their costs actually follow the more typical U-shape pattern shown in Figure 9.9. The primary reason for assuming constant costs, when costs are in fact U-shaped, is to simplify cost (and profit) computations in spreadsheets. This simplifying assumption might not adversely affect managerial decision making if marginal and average costs are very nearly equal. However, serious decision errors can occur when LAC rises or falls by even modest amounts as quantity rises. In most instances in this textbook, we will assume a representative LAC, such as that illustrated earlier in Figure 9.9. Nonetheless, you should be familiar with this special case because many businesses treat their costs as constant.

Minimum Efficient Scale (MES)

In many situations, a relatively modest scale of operation may enable a firm to capture all available economies of scale, and diseconomies may not arise until output is very large. Figure 9.12 illustrates such a situation by flattening LAC between points m and d to create a range of output over which LAC is constant. Once a firm reaches the scale of operation at point m on LAC, it will achieve the lowest possible unit costs in the long run, LACmin. The minimum level of output (i.e., scale of operation) that achieves all available economies of scale is called minimum efficient scale (MES), which is output level QMES in Figure 9.12. After a firm reaches minimum efficient scale, it will enjoy the lowest possible unit costs for all output levels up to the point where diseconomies set in at QDIS in the figure.

Firms can face a variety of shapes of LAC curves, and the differences in shape can influence long-run managerial decision making. In businesses where economies of scale are negligible, diseconomies may soon become of paramount importance, as LAC turns up at a relatively small volume of output. PanelA of Figure 9.13 shows a long-run average cost curve for a firm of this type. Panel B illustrates a situation in which the range and strength of the available scale economies are both substantial. Firms that must have low unit costs to profitably enter or even just to survive in this market will need to operate at a large scale when they face the LAC in Panel B. In many real-world situations, Panel C typifies the longrun cost structure: MES is reached at a low level of production and then costs remain constant for a wide range of output until eventually diseconomies of scale take over.

Before leaving this discussion of scale economies, we wish to dispel a commonly held notion that all firms should plan to operate at minimum efficient scale in the long run. As you will see in Part IV of this book, the long run profitmaximizing output or scale of operation can occur in a region of falling, constant, or rising long-run average cost, depending on the shape of LAC and the intensity of MARKET COMPETITION. Decision makers should ignore average cost and focus instead on marginal cost when trying to reach the optimal level of any activity. For now, we will simply state that profit-maximizing firms do not always operate at minimum efficient scale in the long run. We will postpone a more detailed statement until Part IV, where we will examine profit-maximization in various market structures.

Economies of Scope in Multiproduct Firms

Many firms produce a number of different products. Typically, multiproduct firms employ some resources that contribute to the production of two or more goods or services: Citrus orchards produce both oranges and grapefruit, oil wells pump both crude oil and natural gas, automotive plants produce both cars and trucks, commercial banks provide a variety of financial services, and hospitals perform a wide array of surgical operations and medical procedures. Economies of scope are said to exist whenever it is less costly for a multiproduct firm to produce two or more products together than for separate single-product firms to produce identical amounts of each product. Economists believe the prevalence of scope economies may be the best explanation for why we observe so many multiproduct firms across most Industries and in most countries.

Multiproduct Cost Functions and Scope Economies

Thus far, our analysis of production and costs has focused exclusively on singleproduct firms. We are now going to examine long-run total cost when a firm produces two or more goods or services. Although we will limit our discussion here to just two goods, the analysis applies to any number of products.

A multiproduct total cost function is derived from a multiproduct expansion path. To construct a multiproduct expansion path for two goods X and Y, production engineers must work with a more complicated production function—one that gives technically efficient input combinations for various pairs of output quantities (X, Y ). For a given set of input prices, engineers can find the economically efficient input combination that will produce a particular output combination (X, Y) at the lowest total cost. In practice, production engineers use reasonably complicated computer algorithms to repeatedly search for and identify the efficient combinations of inputs for a range of output pairs the manager may wish to produce. This process, which you will never undertake as a manager, typically results in a spreadsheet or table of input and output values that can be rather easily used to construct a multiproduct total cost function: LTC(X, Y). A multiproduct total cost function— whether expressed as an equation or as a spreadsheet—gives the lowest total cost for a multiproduct firm to produce X units of one good and Y units of some other good.

While deriving multiproduct cost functions is something you will never actually do, the concept of multiproduct cost functions nonetheless proves quite useful in defining scope economies and explaining why multiproduct efficiencies arise. Economies of scope exist when

LTC(X, Y) < LTC(X, 0) + LTC(0, Y)

where LTC(X, 0) and LTC(0, Y) are the total costs when single-product firms specialize in production of X and Y, respectively. As you can see from this mathematical expression, a multiproduct firm experiencing scope economies can produce goods X and Y together at a lower total cost than two single-product firms, one firm specializing in good X and the other in good Y.

Consider Precision Brakes and Mufflers—formerly our single-product firm known as Precision Brakes—that now operates as a multiservice firm repairing brakes and replacing mufflers. Precision Brakes and Mufflers can perform 4 brake jobs (B) and replace 8 mufflers (M) a day for a total cost of $1,400:

LTC(B, M) = LTC(4, 8) + $1,400

A single-service firm specializing in muffler replacement can install 8 replacement mufflers daily at a total cost of $1,000: LTC(0, 8) = $1,000. A different singleservice firm specializing in brake repair can perform 4 brake jobs daily for a total cost of $600: LTC(4, 0) = $600. In this example, a multiproduct firm can perform 4 brake jobs and replace 8 mufflers at lower total cost than two separate firms producing the same level of outputs:

LTC(4, 8) < LTC(0, 8) + LTC(4, 0)

$1,400 < $1,000 + $600

$1,400 < $1,600

Thus, Precision Brakes and Mufflers experiences economies of scope for this combination and muffler repair services.

An important consequence of scope economies for managerial decision making concerns the incremental or marginal cost of adding new product or service lines:

Firms that already produce good X can add production of good Y at lower cost than a specialized, single-product firm can produce Y. You can quickly confirm the validity of this statement by subtracting LTC(X, 0) from both sides of the original mathematical expression for economies of scope:

LTC(X, Y) - LTC(X, 0) < LTC(0, Y)

The left side of this expression shows the marginal cost of adding Y units at a firm already producing good X, which, in the presence of scope economies, costs less than having a single-product firm produce Y units. To illustrate this point, suppose Precision Brakes, the single-product firm specializing in brake jobs, is performing 4 brake jobs daily. If Precision Brakes wishes to become a multiservice company by adding 8 muffler repairs daily, the marginal or incremental cost to do so is $800:

LTC(4, 8) - LTC(4, 0) = $1,400 - $600

= $800

Recall that a single-product firm specializing in muffler repair incurs a total cost of $1,000 to perform 8 muffler repairs: LTC(0, 8) = $1,000, which is more costly than letting a multiproduct firm add 8 muffler repairs a day to its service mix.

As you can see from this example, the existence of economies of scope confers a Cost Advantage to multiproduct firms compared to single-product producers of the same goods. In product markets where scope economies are strong, managers should expect that new firms entering a market are likely to be multiproduct firms, and existing single-product firms are likely to be targets for acquisition by multiproduct firms.

Reasons for Economies of Scope

Economists have identified two situations that give rise to economies of scope. In the first of these situations, economies of scope arise because multiple goods are produced together as joint products. Goods are joint products if employing resources to produce one good causes one or more other goods to be produced as by-products at little or no additional cost. Frequently, but not always, the joint products come in fixed proportions. One of the classic examples is that of beef carcasses and the leather products produced with hides that are by-products of beef production. Other examples of joint products include wool and mutton, chickens and fertilizer, lumber and saw dust, and crude oil and natural gas. Joint products always lead to economies of scope. However, occurrences of scope economies are much more common than cases of joint products.

A second cause for economies of scope, one more commonplace than joint products, arises when common or shared inputs contribute to the production of two or more goods or services. When a common input is purchased to make good X, as long as the common input is not completely used up in producing good X, then it is also available at little or no extra cost to make good Y. Economies of scope arise because the marginal cost of adding good Y by a firm already producing good X— and thus able to use common inputs at very low cost—will be less costly than producing good Y by a single-product firm incurring the full cost of using common inputs. In other words, the cost of the common inputs gets spread over multiple products or services, creating economies of scope.

The common or shared resources that lead to economies of scope may be the inputs used in the manufacture of the product, or in some cases they may involve only the administrative, marketing, and distribution resources of the firm. In our example of Precision Brakes and Mufflers, the hydraulic lift used to raise cars— once it has been purchased and installed for muffler repair—can be used at almost zero marginal cost to lift cars for brake repair. As you might expect, the larger the share of total cost attributable to common inputs, the greater will be the costsavings from economies of scope. We will now summarize this discussion of economies of scope with the following relations:

Relations When economies of scope exist: (1) The total cost of producing goods X and Y by a multiproduct firm is less than the sum of the costs for specialized, single-product firms to produce these goods: LTC(X, Y) < LTC(X, 0) + LTC(0, Y), and (2) Firms already producing good X can add production of good Y at lower cost than a single-product firm can produce Y: LTC(X, Y) - LTC(X, 0) < LTC(0, Y). Economies of scope arise when firms produce joint products or when firms employ common inputs in production.

Purchasing Economies of Scale

As we stressed previously in the discussion of economies of scale, changing input prices cannot be the cause of scale economies or diseconomies because, quite simply, input prices remain constant along any particular LAC curve. So what does happen to a firm’s long-run costs when input prices change? As it turns out, the answer depends on the cause of the input price change. In many instances, managers of individual firms have no control over input prices, as happens when input prices are set by the forces of demand and supply in resource markets. A decrease in the world price of crude oil, for example, causes a petroleum refiner’s long-run average cost curve to shift downward at every level of output of refined product. In other cases, managers as a group may influence input prices by expanding an entire industry’s production level, which, in turn, significantly increases the demand and prices for some inputs.

Sometimes, however, a purchasing manager for an individual firm may obtain lower input prices as the firm expands its production level. Purchasing economies of scale arise when large-scale purchasing of raw materials—or any other input, for that matter—enables large buyers to obtain lower input prices through quantity discounts. At the threshold level of output where a firm buys enough of an input to qualify for quantity discounting, the firm’s LAC curve shifts downward. Purchasing economies are common for advertising media, some raw materials, and energy supplies.

Figure 9.14 on page 354 illustrates how purchasing economies can affect a firm’s long-run average costs. In this example, the purchasing manager gets a quantity discount on one or more inputs once the firm’s output level reaches a threshold of QT units at point A on the original LAC curve. At QT units and beyond, the firm’s LAC will be lower at every output level, as indicated by LAC' in the figure. Sometimes input suppliers might offer progressively steeper discounts at several higher output levels. As you would expect, this creates multiple downward shifting points along the LAC curve.

Learning or Experience Economies

For many years economists and production engineers have known that certain industries tend to benefit from declining unit costs as the firms gain experience producing certain kinds of manufactured goods (airframes, ships, and computer chips) and even some services (heart surgery and dental procedures). Apparently, workers, managers, engineers, and even input suppliers in these industries “learn by doing” or “learn through experience.” As total cumulative output increases, learning or experience economies cause long-run average cost to fall at every output level.

Notice that learning economies differ substantially fromeconomies of scale.With scale economies, unit costs falls as a firm increases output, moving rightward and downward along its LAC curve.With learning or experience economies, the entire LAC curve shifts downward at every output as a firm’s accumulated output grows. The reasons for learning economies also differ fromthe reasons for scale economies.

The classic explanation for learning economies focuses on labor’s ability to learn how to accomplish tasks more efficiently by repeating them many times; that is, learning by doing. However, engineers and managers can also play important roles in making costs fall as cumulative output rises. As experience with production grows, design engineers usually discover ways to make it cheaper to manufacture a product by making changes in specifications for components and relaxing tolerances on fit and finish without reducing product quality.With experience, managers and production engineers will discover new ways to improve factory layout to speed the flow of materials through all stages of production and to reduce input usage and waste. Unfortunately, the gains from learning and experience eventually run out, and then the LAC curve no longer falls with growing cumulative output.

In Figure 9.15, learning by doing increases worker productivity in Panel A, which causes unit costs to fall at every output level in Panel B. In Panel A, average productivity of labor begins at a value of 10 units of output per worker at the time a firm starts producing the good. As output accumulates over time from 0 to 8,000 total units, worker productivity rises from 10 units per worker (point s) to its greatest level at 20 units of output per worker (point l) where no further productivity gains can be obtained through experience alone. Notice that the length of time it takes to accumulate 8,000 units in no way affects the amount by which AP rises. In Panel A, to keep things simple, we are showing only the effect of learning on labor productivity. (As labor learns better how to use machines, capital productivity also increases, further contributing to the downward shift of LAC in Panel B.)

As a strategic matter, the ability of early entrants in an industry to use learning economies to gain a cost advantage over potential new entrants will depend on how much time start-up firms take going from point s to point l. As we will explain later, faster learning is not necessarily better when entry deterrence is the manager’s primary goal. For now, you can ignore the speed at which a firm gains experience. Generally, it is difficult to predict where the new minimum efficient scale (MES) will lie once the Learning Process is completed at point l in the figure. In Panel B, we show MES increasing from 500 to 700 units, but MES could rise, fall, or stay the same.

As a manager you will almost certainly rely on production engineers to estimate and predict the impact of experience on LAC and MES. A manager’s responsibility is to use this information, which improves your forecasts of future costs, to make the most profitable decisions concerning pricing and output levels in the current period and to plan long-run entry and exit in future periods—topics we will cover in the next two parts of this book.

In this section, we examined a variety of forces affecting the firm’s long-run cost structure. While scale, scope, purchasing, and learning economies can all lead to lower total and average costs of supplying goods and services, we must warn you that managers should not increase production levels solely for the purpose of chasing any one of these cost economies. As you will learn in Part IV of this book, where we show you how to make profit-maximizing output and pricing decisions, the optimal positions for businesses don’t always require taking full advantage of any scale or scope economies available to the firm. Furthermore, it may not be profitable to expand production to the point where economies arise in purchasing inputs or at a rate that rapidly exploits potential productivity gains from learning by doing. However, as you can now understand, estimating and forecasting longrun cost of production will not be accurate if they overlook these important forces affecting the long-run structure of costs. All of these forces provide firms with an opportunity to reduce costs in the long run in ways that simply are not available in the short run when scale and scope are fixed.

I L L U S T R AT I O N 9 . 2

Declining Minimum Efficient Scale (MES) Changes the Shape of Semiconductor Manufacturing

Even those who know relatively little about computer technology have heard of Moore’s Law, which has correctly predicted since 1958 that the number of transistors placed on integrated circuits will double every two years. This exponential growth is expected to continue for another 10 to 15 years. Recently, transistor size has shrunk from130 nanometers (one nanometer = 1 billionth of a meter) to 90 nanometers, and Intel Corp. is on the verge of bringing online 65-nanometer production technology for its semiconductor chips. The implication of Moore’s Law for consumers has been, of course, a tremendous and rapid increase in raw computing power coupled with higher speed, and reduced power consumption.

Unfortunately for the many semiconductor manufacturers— companies like Intel, Samsung, Texas Instruments, Advanced Micro Devices, and Motorola, to name just a few—Moore’s Law causes multibillion dollar semiconductor fabrication plants to become outdated and virtually useless in as little as five years. When a $5 billion dollar fabrication plant gets amortized over a useful lifespan of only five years, the daily cost of the capital investment is about $3 million per day. The only profitable way to operate a semiconductor plant, then, is to produce and sell a very large number of chips in order to take advantage of the sizable scale economies available to the industry. As you know from our discussion of economies of scale, semiconductor manufacturers must push production quantities at least to the point of minimum efficient scale, or MES, to avoid operating at a cost disadvantage.

As technology has continually reduced the size of transistors, the long-run average cost curve has progressively shifted downward and to the right, as shown in the accompanying figure. While falling LAC is certainly desirable, chip manufacturers have also experienced rising MES with each cycle of shrinking. As you can see in the figure, MES increases from point a with 250-nanometer technology to point d with the now widespread 90-nanometer technology. Every chip plant—or “fab,” as they are called—must churn out ever larger quantities of chips in order to reach MES and remain financially viable semiconductor suppliers. Predictably, this expansion of output drives down chip prices and makes it increasingly difficult for fabs to earn a profit making computer chips.

Recently, a team of engineering consultants succeeded in changing the structure of long-run average cost for chipmakers by implementing the lean manufacturing philosophy and rules developed by Toyota Motor Corp. for making its cars. According to the consultants, applying the Toyota Production System (TPS) to chip manufacturing “lowered cycle time in the (plant) by 67 percent, . . . reduced costs by 12 percent, . . . increased the number of products produced by 50 percent, and increased production capacity by 10 percent, all without additional investment.” (p. 25) As a result of applying TPS to chip making, the long-run average cost curve is now lower at all quantities, and it has a range of constant costs beginning at a significantly lower production rate. As shown by LACTPS in the figure, LAC is lower and MES is smaller (MES falls from Q’ to QMES). The consultants predict the following effects on competition in chip manufacturing caused by reshaping long-run average costs to look like LACTPS:

The new economics of semiconductor manufacturing now make it possible to produce chips profitably in much smaller volumes. This effect may not be very important for the fabs that make huge numbers of high-performance chips, but then again, that segment will take up a declining share of the total market. This isn’t because demand for those chips will shrink. Rather, demand will grow even faster for products that require chips with rapid time-to-market and lower costs . . . (p. 28)

We agree with the technology geeks: The new shape of LAC will enhance competition by keeping more semiconductor manufacturers, both large and “small,” in the game.

LAC

I L L U S T R AT I O N 9 . 3

Scale and Scope Economies in the Real World

Government policymakers, academic economists, and industry analysts all wish to know which industries are subject to economies of scale and economies of scope. In this Illustration, we will briefly summarize some of the empirical estimates of scale and scope economies for two service industries: commercial banking and life insurance.

Commercial Banking

When state legislatures began allowing interstate banking during the 1980s, one of the most controversial outcomes of interstate banking was the widespread consolidation that took place through Mergers and Acquisitions of local banks by large out-of-state banks. According to Robert Goudreau and Larry Wall, one of the primary incentives for interstate expansion is a desire by banks to exploit economies of scale and scope.a To the extent that significant economies of scale exist in banking, large banks will have a cost advantage over small banks. If there are economies of scope in banking, then banks offering more banking services will have lower costs than banks providing a smaller number of services. Thomas Gilligan, Michael Smirlock, and William Marshall examined 714 commercial banks to determine the extent of economies of scale and scope in commercial banking.b They concluded that economies of scale in banking are exhausted at relatively low output levels. The long-run average cost curve (LAC) for commercial banks is shaped like LAC in Panel C of Figure 9.13, with minimum efficient scale (MES) occurring at a relatively small scale of operation. Based on these results, small banks do not necessarily suffer a cost disadvantage as they compete with large banks.

Economies of scope also appear to be present for banks producing the traditional set of bank products (i.e., various types of loans and deposits). Given their empirical evidence that economies of scale do not extend over a wide range of output, Gilligan, Smirlock, and Marshall argued that public policymakers should not encourage bank mergers on the basis of cost savings. They also pointed out that Government Regulation restricting the types of loans and deposits that a bank may offer can lead to higher costs, given their evidence of economies of scope in banking.

Life Insurance

Life insurance companies offer three main types of services: life insurance policies, financial annuities, and accident and health (A & H) policies. Don Segal used data for approximately 120 insurance companies in the U.S. over the period 1995–1998 to estimate a multiproduct cost function for the three main lines of services offered by multiproduct insurance agencies. He notes “economies of scale and scope may affect managerial decisions regarding the scale and mix of output” (p. 169). According to his findings, insurance companies experience substantial scale economies, as expected, because insurance policies rely on the statistical law of large numbers to pool risks of policyholders. The larger the pool of policyholders, the less risky, and hence less costly, it will be to insure risk. He finds LAC is still falling—but much less sharply—for the largest scale firms, which indicates that MES has not been reached by the largest insurance companies in the U.S.

Unfortunately, as Segal points out, managers cannot assume a causal relation holds between firm size and unit costs—a common statistical shortcoming in most empirical studies of scale economies. The problem is this: Either (1) large size causes lower unit costs through scale economies or (2) those firms in the sample that are more efficiently managed and enjoy lower costs of operation will grow faster and end up larger in size than their less efficient rivals. In the second scenario, low costs are correlated with large size even in the absence of scale economies. So, managers of insurance companies—and everyone else for that matter— need to be cautious when interpreting statistical evidence of scale economies.

As for scope economies, the evidence more clearly points to economies of scope: “a joint production of all three lines of business by one firm would be cheaper than the overall cost of producing these products separately” (p. 184). Common inputs for supplying life insurance, annuities, and A&H policies include both the labor and capital inputs, as long as these inputs are not subject to “complete congestion” (i.e., completely exhausted or used up) in the production of any one service line. As you would expect, the actuaries, insurance agents, and managerial and clerical staff who work to supply life insurance policies can also work to provide annuities and A&H policies as well. Both physical capital—office space and equipment—and financial capital—monetary assets held in reserve to pay policy claims—can serve as common inputs for all three lines of insurance services. Segal’s multiproduct cost function predicts a significant cost advantage for large, multiservice insurance companies in the U.S.

Fig-9.10Fig-9.11Fig-9.12Fig-9.14

SHORT-RUN AND LONG-RUN COSTS

Now that you understandhowlong-runproduction decisions determine the structure of long-run costs, we can demonstrate more clearly the important relations between short-run and long-run costs. The long run or planning horizon is the collection of all possible short-run situations, one for every amount of fixed input that may be chosen in the long-run planning period. In the first part of this section we will show you how to construct a firm’s long-run planning horizon—in the form of its long-run average cost curve (LAC)— from the short-run average total cost (ATC) curves associated with each possible level of capital the firm might choose. Then, in the next part of this section, we will explain how managers can exploit the flexibility of input choice available in longrun decision making to alter the structure of short-run costs in order to reduce production costs (and increase profit).

Long-Run Average Cost as the Planning Horizon

To keep matters simple, we will continue to discuss a firm that employs only two inputs, labor and capital, and capital is the plant size that becomes fixed in the short run (labor is the variable input in the short run). Since the long run is the set of all possible short-run situations, you can think of the long run as a catalog, and each page of the catalog shows a set of short-run cost curves for one of the possible plant sizes. For example, suppose a manager can choose from only three plant sizes, say plants with 10, 30, and 60 units of capital. In this case, the firm’s long-run planning horizon is a catalog with three pages: page 1 shows the shortrun cost curves when 10 units of capital are employed, page 2 shows the short-run cost curves when 30 units of capital are employed, and page 3 the cost curves for 60 units of capital.

The long-run planning horizon can be constructed by overlaying the cost curves from the three pages of the catalog to form a “group shot” showing all three short-run cost structures in one figure. Figure 9.16 shows the three short-run average total cost (ATC) curves for the three plant sizes that make up the planning horizon in this example: ATCK=10, ATCK=30, and K1.Note that we have omitted the associated AVC and SMC curves to keep the figure as simple as possible.

When the firm wishes to produce any output from 0 to 4,000 units, the manager will choose the small plant size with the cost structure given by ATCK=10because the average cost, and hence the total cost, of producing each output over this range is lower in a plant with 10 units of capital than in a plant with either 30 units or 60 units of capital. For example, when 3,000 units are produced in the plant with 10 units of capital, average cost is $0.50 and total cost is $1,500, which is better than spending $2,250 (= $0.75 * 3,000) to produce 3,000 units in themedium plant with 30 units of capital. (Note that if the ATC curve for the large plant in Figure 9.16 is extended leftward to 3,000 units of production, the average and total cost of producing 3,000 units in a plant with 60 units of capital is higher than both of the other two plant sizes.) When the firm wishes to produce output levels between 4,000 and 7,500 units, the manager would choose the medium plant size (30 units of capital) because K2 lies below both of the other two ATC curves for all outputs over this range. Following this same reasoning, the manager would choose the large plant size (60 units of capital) with the cost structure shown by K1 for any output greater than 7,500 units of production. In this example, the planning horizon, which is precisely the firm’s long-run average cost (LAC) curve, is formed by the light-colored, solid portions of the three ATC curves shown in Figure 9.16.

Firms can generally choose from many more than three plant sizes. When a very large number of plant sizes can be chosen, the LAC curve smoothes out and typically takes a U-shape as shown by the dark-colored LAC curve in Figure 9.16. The set of all tangency points, such as r, m, and e in Figure 9.16, form a lower envelope of average costs. For this reason, long-run average cost is called an “envelope” curve.

While we chose to present the firm’s planning horizon as the envelope of shortrun average cost curves, the same relation holds between the short-run and long-run total or marginal cost curves: Long-run cost curves are always comprised of all possible short-run curves (i.e., they are the envelope curves of their short-run counterparts). Nowthat we have established the relation between short- and long-run costs, we can demonstrate why short-run costs are generally higher than long-run costs.

Restructuring Short-Run Costs

In the long run, a manager can choose any input combination to produce the desired output level. The optimal amount of labor and capital for any specific output level is the combination that minimizes the long-run total cost of producing that amount of output. When the firm builds the optimal plant size and employs the optimal amount of labor, the total (and average) cost of producing the intended or planned output will be the same in both the long run and the short run. In other words, long-run and shortrun costs are identical when the firm produces the output in the short run for which the fixed plant size (capital input) is optimal. However, if demand or cost conditions change and the manager decides to increase or decrease output in the short run, then the current plant size is no longer optimal. Now the manager will wish to restructure its short-run costs by adjusting plant size to the level that is optimal for the new output level, as soon as the next opportunity for a long-run adjustment arises.

We can demonstrate the gains from restructuring short-run costs by returning to the situation presented in Figure 9.4, which is shown again in Figure 9.17. Recall that the manager wishes to minimize the total cost of producing 10,000 units when the price of labor (w) is $40 per unit and the price of capital (r) is $60 per unit. As explained previously, the manager finds the optimal (cost-minimizing) input combination at point E: L* = 90 and K* = 60. As you also know from our previous discussion, point E lies on the expansion path, which we will now refer to as the “long-run” expansion path in this discussion.

We can most easily demonstrate the gains from adjusting plant size (or capital levels) by employing the concept of a short-run expansion path. A short-run expansion path gives the cost-minimizing (or output-maximizing) input combination for each level of output when capital is fixed at K3units in the short run. To avoid any confusion in terminology, we must emphasize that the term “expansion path” always refers to a long-run expansion path, while an expansion path for the short run, to distinguish it fromits long-run counterpart, is always called a short-run expansion path.

Suppose the manager wishes to produce 10,000 units. From the planning horizon in Figure 9.16, the manager determines that a plant size of 60 units of capital is the optimal plant to build for short-run production. As explained previously, once the manager builds the production facility with 60 units of capital, the firm operates with the short-run cost structure given by K1. This cost structure corresponds to the firm’s short-run expansion path in Figure 9.17, which is a horizontal line at 60 units of capital passing through point E on the long-run expansion path. As long as the firm produces 10,000 units in the short run, all of the firm’s inputs are optimally adjusted and its long- and short-run costs are identical: Total cost is $7,200 (= $40 * 90 + $60 * 60) and average cost is $0.72 (= $7,200/10,000). In general, when the firm is producing the output level in the short run using the longrun optimal plant size, ATC and LAC are tangent at that output level. For example, when the firm produces 10,000 units in the short run using 60 units of capital, K1 is tangent to LAC at point e.

If the manager decides to increase or decrease output in the short run, short-run production costs will then exceed long-run production costs because input levels will not be at the optimal levels given by the long-run expansion path. For example, if the manager increases output to 12,000 units in the short run, the manager must employ the input combination at point S on the short-run expansion path in Figure 9.17. The short-run total cost of producing 12,000 units is $9,600 (= $40 * 150 + $60 * 60) and average total cost is $0.80 (= $9,600 / 12,000) at point s in Figure 9.17. Of course, the manager realizes that point F is a less costly input combination for producing 12,000 units, since input combination F lies on a lower isocost line than S. In fact, with input combination F, the total cost of producing 12,000 units is $9,000 (= $40 * 120 + $60 * 70), and average cost is $0.75 (= $9,000 / 12,000), as shown at point f in Figure 9.16. Short-run costs exceed longrun costs for output levels below 10,000 units as well, because a plant size of 60 units of capital is too big (i.e., larger than the optimal plant size) for every output below point E on the long-run expansion path.

At the next opportunity to adjust plant size, the manager will increase plant size to 70 units, as long as the firm plans to continue producing 12,000 units. Increasing capital to 70 units causes the short-run expansion path to shift upward as shown by the broken horizontal line in Figure 9.17. By restructuring short-run production, the manager reduces the short-run total costs of producing 12,000 units by $600 (= $9,600 - $9,000). As you will see in Part IV, firms can increase their profits— sometimes even convert losses to profits—by adjusting their fixed inputs to create a lower cost structure for short-run production operations. We can now summarize this discussion with the following principle.

Principle Since managers have the greatest flexibility to choose inputs in the long run, costs are lower in the long run than in the short run for all output levels except the output level for which the fixed input is at its optimal level. Thus the firm’s short-run costs can generally be reduced by adjusting the fixed inputs to their optimal long-run levels when the long-run opportunity to adjust fixed inputs arises.

Fig-9.16Fig-9.17