Relevant Costs and Contribution Analysis

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Production and Cost Analysis

Learning Objectives

A�er reading this chapter, you should be able to:

Understand why the efficiency of inputs in the produc�on process varies according to the ra�o of fixed to variable inputs u�lized. Examine the rela�onships between produc�on efficiency and average and marginal cost of produc�on. Explain the rela�onships between average and marginal costs and between short-run and long-run costs of produc�on. Dis�nguish between increasing efficiency in produc�on and economies of scale in produc�on. Iden�fy increasing, constant, and decreasing economies of scale and dis�nguish these from economies of scope, purchasing economies, and the learning curve. Discuss the meaning and importance of a series of other cost concepts that are used in managerial decision making.

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Economic analysis can be traced back hundreds of years to a predominantly agricultural world where land and labor were the two main inputs to the produc�on process.

Introduction

As first men�oned in Chapter 1, managerial economics is concerned with maximizing profits while taking into considera�on nonmonetary issues, such as the firm's risk exposure and its impact upon the social and natural environment. Profits are the surplus of revenues over costs, and the nonmonetary issues can be incorporated into the revenue side or the cost side of the profit calcula�on. In the first two chapters, we considered the manager's decision making under condi�ons of risk and uncertainty; in the next pair of chapters we considered the revenue side, and now we turn to the produc�on and costs side of the managerial decision-making problem.

The first and most important dis�nc�on to make in produc�on and cost analysis is between fixed and variable inputs to the produc�on process. Fixed inputs do not vary over the �me period chosen for the analysis, which we shall call the produc�on period—one day, one week, or one month. Fixed inputs include buildings, factories, machines, vehicles, tools, and furniture (o�en collec�vely called "plant and equipment"), and highly skilled employees who cannot be hired or fired at short no�ce. These fixed inputs are durable assets of the business and con�nue to be useful in later periods, but generally cannot be varied in the current produc�on period. Since the amount of fixed resources is constant over the current produc�on period it follows that the cost of those resources is a fixed cost over the produc�on period.

Variable inputs are those that can be varied at short no�ce, that is, the input quan�ty of these can be augmented or reduced during the produc�on period. Examples of variable inputs include raw materials, components, electrical energy, fuel, office supplies, and employees who can be hired, laid-off, or u�lized frac�onally (paid only for hours worked) during the produc�on period. The costs of these variable inputs will depend on how much of each resource is u�lized in the produc�on process during the current produc�on period, and the amounts u�lized will vary according to how much output is to be produced. Smaller volumes of output will require lesser amounts of variable inputs and thus lower costs of variable inputs, while larger volumes of output will require greater amounts of variable inputs and thus higher costs of variable inputs.

Economic analysis of the firm goes back hundreds of years to a predominantly agricultural world where there were two main inputs to the produc�on process, namely land and labor. Tools and other fixed inputs, such as fences, spades, and rakes, were handmade and thus it took �me to make addi�onal units of these available to the produc�on process. As the industrial revolu�on progressed, more and more industrial equipment was made and u�lized, such as plows, harvesters, flour mills, steam engines, and so on, and economists chose a collec�ve noun for all these fixed inputs, calling them capital. So the word capital became used to represent all the fixed inputs that enter the firm's produc�on func�on. Most people remained rela�vely unskilled, and in any case, labor was abundant due to high levels of unemployment and was therefore easily augmented or reduced in the produc�on process. Accordingly, the word labor was used to represent all the variable inputs to the produc�on process.

This conven�on, calling the fixed inputs capital and the variable inputs labor, endures today, despite the fact that most human resources are now fixed during the current produc�on period due to the �me it takes to recruit and train new hires, the constraints imposed by labor unions and legisla�on on firms that want to reduce the number of workers, and the difficulty of finding people to hire with the required skills. Par�cularly, since managerial economists must talk to accountants (about costs) and human resource managers (about wages and salaries), it is be�er that we adopt the terms fixed and variable inputs to avoid confusion in our communica�on with these other managers within the firm. So, whenever we hear an economist say "capital and labor" we will know what they mean, but by using the terms "fixed and variable inputs" we will avoid introducing confusion into our communica�ons with other managers.

Economists also make a dis�nc�on between the short run versus the long run. The short run refers to the period of �me during which the fixed inputs remain fixed. For example, if it would take six months for a firm to change its fixed inputs, then for that firm the short run would be six months. Importantly, the short-run context implies a constraint on the firm's output level—if the plant and equipment are running at full capacity, it will not be possible to increase output beyond the full capacity output level without increasing the size of the firm's plant and equipment. The term plant size refers to the amount of the fixed inputs in the short run, and plant size can be changed only in the long run.

The long run is a hypothe�cal situa�on in which all inputs are variable, and the firm (for planning purposes) can contemplate any plant size and, consequently, any output level for produc�on in a future produc�on period. Thus, managers might install any number and combina�on of machines, vehicles, and other equipment; any number and composi�on of employees; any size of factory and office buildings; any quan�ty of raw material, components, energy usage, and so on. So, in any short-run period, managers of the firm will consider whether the current plant size is appropriate for their future sales projec�ons (i.e., planned output levels) and, if not, will begin to make arrangements to augment (if planning expansion) or reduce (if planning contrac�on) their fixed inputs to enter a new short-run produc�on period with an appropriate range of output levels. For example, suppose a new housing development is planned for the western suburbs of a city. A restaurant that is located near there an�cipates that the demand for restaurant meals will increase as a result of the influx of new residents and decides to expand its plant size (sea�ng and serving capacity) to capitalize on the situa�on. In remodeling and extending the exis�ng restaurant, the managers have an infinite variety of layouts, sea�ng capacity, kitchen sizes, numbers of permanent employees, and other fixed assets that can be considered, and eventually they would choose one par�cular configura�on of these inputs that would then become fixed for the subsequent short-run periods (un�l another expansion or reduc�on in plant size is deemed necessary).

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5.1 Production and Cost Curves

In the short run, the quan�ty of output produced depends on both the quan�ty of the fixed inputs (plant size) and the quan�ty of the variable inputs. The produc�on func�on shows the exact form of the rela�onship between the quan�ty of output and the quan�ty of the inputs. Because the inputs cost money to purchase, hire, or otherwise u�lize, the produc�on func�on is easily translated into the cost func�on, which relates the cost of produc�on to the various output levels that are possible given the current plant size.

The form of the produc�on and cost func�ons is an empirical issue—it needs to be inves�gated from data whether the form of the func�on is linear or curvilinear. The simplest case is a linear produc�on func�on, where output increases linearly as variable inputs are added to the fixed inputs. With a linear produc�on func�on the cost func�on will start with the (lump sum) cost of the fixed inputs and then rise linearly with the output level. Curvilinear produc�on func�ons are more likely, however, because the propor�onality between output and the variable inputs is likely to vary.

The Law of Variable Proportions

The law of variable propor�ons states that total output is likely to increase at an increasing rate at first, and then increase at a decreasing rate as we progressively add more and more variable inputs to the fixed inputs. Total output, conven�onally called total product (TP), thus rises in a "lazy-S" manner (shown in Figure 5.1). The total product (TP) curve shows the total output level that can be produced by a given plant size when augmented by various levels of the variable inputs. For example, in a restaurant with fixed plant size (kitchen and sea�ng capacity) in the short run, as variable inputs (such as food materials, and casual wait staff) are added, output would increase at an increasing rate, at first, as the variable inputs become more efficient in a plant size that is ini�ally "too large" for them. The wait staff would be underu�lized and food materials would be wasted—a roast beef would not be completely eaten and the remainder would need to be thrown out at the end of the evening. A�er the point of inflec�on (where the TP curve changes from concave from below to convex from below), output increases at a diminishing rate as the variable inputs become progressively less efficient in a plant size that is now "too small" for the amount of variable inputs being applied to the produc�on process. In this region of the produc�on func�on, the wait staff would be bumping into each other, kitchen staff might be burning food, and the rapid pace of work is likely to induce mistakes in the ordering, cooking, and serving processes.

The increasing and later decreasing efficiency of the variable inputs to the produc�on func�on can be measured by the average and marginal product values. The average product (AP) curve shows the ra�o of the output level to the variable input level, at any par�cular input level of the variable inputs, or TP/V. As you can see in the lower part of Figure 5.1, average product (AP) rises at first and then declines as more and more units of the variable inputs are added to the produc�on process. In the upper part of Figure 5.1, a ray from the origin (0b) lays just tangent to the TP curve at input level V2. Note that the slope of any ray from the origin that hits the TP curve will represent

the ra�o of total output to the variable inputs, and thus the slope of the ray will indicate the ra�o of TP (rise) to V (run) and thus indicates the AP value. The ray shown in Figure 5.1 indicates the input level (V2) where AP is maximized, since any steeper ray would not touch the TP curve. At lower input levels (such as V1) a ray joining the

origin and a point on the TP curve (0a) is fla�er, and thus AP is lower. At higher input levels (V3), a ray joining the origin and the TP curve (not shown) would also be fla�er

—thus the AP curve rises to a maximum value at input level V2 and falls therea�er.

Figure 5.1: Total, average, and marginal product curves

The marginal product (MP) curve reflects the change in total product for a one-unit change in the variable inputs, that is, ΔTP/ΔV (where ΔV = 1). Since the TP curve is not a straight line, the MP is not constant but varies as addi�onal units of the variable inputs are added. Put another way, the MP curve reflects the slope of the TP curve, rising at first, and then falling as the TP curve gets progressively steeper at first (MP rising) and then progressively less steep (MP falling). For example, in a restaurant, adding addi�onal wait staff might cause the total number of meals to increase as displayed in Table 5.1. As you can see, as wait staff are increased from 1–4, MP increases, but as wait staff are increased from 5–10, MP decreases.

Table 5.1: Total, average, and marginal produc�vity of wait staff in a restaurant Number of wait staff (V) (people) Total product (TP) (meals served) Average product (AP) AP = TP/V Marginal product (MP) MP = ΔTP/ΔV

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Adding addi�onal workers to this produc�on line should be expected to cause average produc�vity to rise at first and later decrease, due to the law of variable propor�ons.

1 2 3 4 5 6 7 8 9 10

3 8

15 23 30 36 40 43 44 44

3 4 5

5.75 6 6

5.71 5.38 4.49 4.4

3 5 7 8 7 6 4 3 1 0

Now consider the shape of the marginal product (MP) curve and its rela�onship with the AP curve. First, since MP equals the slope of the TP curve, it rises from the

beginning to a maximum value when the TP curve is steepest, at input level V1, and subsequently falls. 1 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#ch05txt1) Note

that it must reach zero at input level V3 because the TP becomes flat at that point. Finally, MP must intersect and be equal to the AP curve when the AP is at its maximum,

at input level V2, because the MP is equal to the slope of the TP, and at V2 the AP is equal to the slope of the ray that is just tangent to (i.e., has the same slope as) the TP

curve.2 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#ch05txt2)

Why do we expect these produc�on curves to bend and intersect like this? Because experience has taught us that the produc�ve efficiency of variable inputs will usually vary in this predictable way, and the common observa�on of this phenomenon led to it being called the law of variable propor�ons. Produc�on efficiency is measured by the average and marginal produc�vity of the variable inputs in the context of the plant size to which they are added. At low levels of variable inputs the ra�o of variable to fixed inputs is very high, and, in effect, the variable inputs have "too much" plant size to work with. At higher levels of variable inputs, the ra�o of variable to fixed inputs is rela�vely low, and in effect the variable inputs have "too li�le" plant size to work with.

Another version of the law of variable propor�ons is the law of diminishing returns,3

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#ch05txt3) which states that as the firm adds variable inputs to its fixed inputs in its produc�on process, a�er some point the marginal produc�vity of the variable inputs will begin to decline and will progressively fall, poten�ally becoming nega�ve if the firm con�nues to add variable inputs into the produc�on process. Thus, the law of diminishing returns is effec�vely iden�cal to the law of variable propor�ons, but refers explicitly to the range of input levels above V1 in Figure 5.1. We will see that this is the range of input (and output) levels at which the firm

will most likely want to be opera�ng.

Total Variable Costs

The total variable cost (TVC) is the total cost of the variable inputs to the produc�on process. The total variable cost (TVC) curve will necessarily reflect the shape of the TP curve since increasing, and later, diminishing returns to the variable inputs in produc�on will have a corresponding impact on the TVC curve. In Figure 5.2, we show the TP curve on the right-hand side and TVC on the le�-hand side, both rela�ng to the ver�cal axis represen�ng output levels. Note that the le�-hand scale (for TVC) is simply the monetary equivalent of the right-hand scale (for TP). Suppose, for example, that units of the variable inputs cost $100 each, the input quan�ty levels on the right-hand side are simply mul�plied by 100 to find TVC levels on the le�-hand side.

Figure 5.2: Deriva�on of the TVC curve from the TP curve

Note that the shape of the TVC curve is absolutely dependent upon the shape of the TP curve—if there are diminishing returns to the variable inputs, there will simultaneously be increasing average and variable costs of produc�on, as shown in Figure 5.3 where we derive the average variable cost (AVC) and marginal cost (MC) curves from the TVC curve.

Marginal Costs and Average Variable Costs

No�ce that we have rotated the le�-hand side of Figure 5.2 through 90 degrees to show the TVC curve with output (Q) on the horizontal axis in Figure 5.3, and thus have our cost data in the same graphical format as the demand and MR curves, as shown in Chapters 3 and 4. From this TVC curve we can now derive average variable cost and marginal cost curves by observing the shape of the TVC curve. Marginal cost (MC) is a change in TVC for a one-unit change in the output level (i.e., ΔTVC/ΔQ, where ΔQ = 1) and you will note that it is equal to the slope of the TVC curve at any output level. When TVC is at its fla�est (at the point of inflec�on) at output level Q1, MC reaches

its minimum. When the TVC curve goes ver�cal (i.e., there is extra variable cost but no extra output, at Q3) then the value of MC becomes infinite.

Figure 5.3: Deriving the AVC and MC curves from the TVC curve

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Average variable cost (AVC) is the ra�o of total variable cost to output level, or TVC/Q. The slope of a ray from the origin that touches the TVC curve at any output level will give the value of AVC at that output level. You can see that if you were to draw rays from the origin to points on the TVC curve, these rays (not shown) would be progressively fla�er at first, reaching a minimum slope (shown) at output level Q2, and therea�er the rays would be progressively steeper (not shown) and, thus, AVC must

be increasing between output levels Q2 and Q3. It follows that MC = AVC at the output level (Q2) where AVC is minimized since both are equal to the slope of the TVC curve

at that point.4 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#ch05txt4)

Total Costs and Short-Run Average Costs

Total costs (TC) are the sum of total variable costs and total fixed costs. Since total fixed costs (TFC) are constant during the short-run produc�on period, we can simply add a constant ver�cal amount to the TVC curve to find the TC curve. Short-run average costs (SAC) are the total costs divided by the number of units of output, so must be equal to the average variable costs (AVC) plus the average fixed costs. Average fixed costs (AFC) are the total fixed costs divided by the number of units of output. Since TFC is a constant, the AFC must decline from a very high number (equal to TFC when Q = 1) to a very low number as TFC are spread across larger and larger volumes of output. This type of curve is known as a "rectangular hyperbola."

As previously men�oned, we add AVC and AFC to determine short-run average costs (SAC), as shown in Figure 5.4. No�ce that AFC is shown in the lower part of the figure as a monotonically declining line that would progressively approach zero as output levels become very high. We add the AFC curve to the AVC curve by a process of ver�cal addi�on at every output level. Since AFC is declining monotonically, and we add this increasing smaller ver�cal distance to AVC, the ver�cal distance between AVC and SAC must also become smaller as output levels rise. Accordingly, the SAC con�nues to fall a�er the output level where AVC was minimized but, at some point, the rise in AVC exceeds the fall in AFC, and so the summa�on of these two (i.e., SAC) must also begin to rise. Note that the MC curve must pass through the minimum point of SAC

because the TC curve is changing only due to changes in the TVC curve.5 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#ch05txt5)

Figure 5.4: Finding the SAC curve by adding the AFC curve to the AVC curve

In Figure 5.4, the ver�cal addi�on of the AVC and the AFC curves to find the SAC curve is illustrated at output levels Q1 and Q4. You can see that the value of AFC at output

level Q1 is equal to the ver�cal distance between the AVC and the SAC curves at that output level, and similarly, the AFC value at output level Q4 is equal to the ver�cal

distance between the AVC and the SAC curves at that output level.6 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#ch05txt6)

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A Numerical Example: Short-Run Cost Curves for The Robust Coffee Place

It will help put all these short-run produc�on and cost concepts into focus if we take a look at a prac�cal example. Suppose that entrepreneur Eddie opens a new coffee shop called The Robust Coffee Place and collects data on his variable produc�on inputs and their costs over the first 10 days of opera�on. His variable costs include the cost of coffee beans, cream, sugar, napkins and coffee s�rrers, and casual labor for coffee-making and cleaning the shop. His fixed costs include his rental of the premises, his lease payments on the coffee machine and other equipment, his purchases of coffee cups, utensils, and cleaning materials, and his own salary. During the first 10 days, his output levels (coffees sold) varied up and down according to customer traffic at the �me of the day and the day of the week. Eddie arranges his data in ascending order of coffees sold to see how costs of produc�on vary with the output level, as shown in Table 5.2.

Table 5.2: Produc�on and variable cost data for The Robust Coffee Place Coffees sold (Q) (1)

Coffee bean cost (2)

Cream and sugar cost (3)

Napkins and s�rrers cost (4)

Casual labor cost (5)

TVC (6)

AVC = TVC/Q (7)

MC = ΔTVC/ΔQ (8)

80 40.00 24.00 8.00 80.00 152.00 1.90

102 45.90 28.56 9.18 85.00 168.64 1.65 0.76

112 48.16 30.24 9.52 90.00 177.92 1.59 0.93

118 53.10 33.04 9.68 95.00 190.82 1.62 2.15

124 62.00 37.20 10.54 98.00 207.74 1.68 2.82

132 72.60 43.56 11.88 106.00 234.04 1.77 3.29

140 81.20 49.00 14.00 120.00 264.20 1.89 3.77

144 86.40 53.28 15.84 124.00 279.52 1.94 3.83

150 97.50 60.00 18.00 140.00 315.50 2.10 6.00

In columns 6, 7, and 8 of Table 5.2, we can calculate the TVC, AVC, and MC values for Eddie's coffee produc�on process. The TVC is simply the sum of the variable cost categories applicable to this business, which are shown in columns 2–5, and it goes up with the numbers of coffees sold, as expected. The average variable costs are equal to TVC divided by the output level (column 1) for each data observa�on. You can see that AVC falls at first, then rises, as expected. No�ce that AVC falls to a minimum value of about $1.59 per cup somewhere around the output rate of 112 coffees per day, and then rises to more than $2 per cup at high output rates, being pulled up by the MC value. Marginal costs are es�mated over each of the discrete ranges of outputs given by the day-to-day varia�ons in the number of coffees sold. We calculate MC as the change in TVC (ΔTVC) divided by the change in quan�ty of number of coffees produced (ΔQ). As you can see, the MC value rises and con�nues to rise as output levels rise, indica�ng diminishing marginal produc�vity of the variable inputs as they are applied to the fixed inputs of the coffee shop.

You might be thinking that this data indicates that Eddie should try to keep his AVC down near its minimum level by restric�ng coffee sales to around 110–120 coffees per day. But, before we conclude anything like that, we need to consider Eddie's fixed costs as well and add them to the variable costs already considered. In Table 5.3, we show TFC, AFC, and SAC data in addi�on to the cost data repeated from Table 5.2.

Table 5.3: Short-run total costs and average costs for The Robust Coffee Place Output (Q) (1) TVC ($) (2) TFC ($) (3) TC ($) (4) AVC ($) (5) AFC ($) (6) SAC ($) (7) MC ($) (8)

80 152.00 200.00 352.00 1.90 2.50 4.40

102 168.64 200.00 368.64 1.65 1.96 3.61 0.76

112 177.92 200.00 377.92 1.59 1.79 3.37 0.93

118 190.82 200.00 390.82 1.62 1.69 3.31 2.15

124 207.74 200.00 407.74 1.68 1.61 3.29 2.82

132 234.04 200.00 434.04 1.77 1.52 3.29 3.29

140 264.20 200.00 464.20 1.89 1.43 3.32 3.77

144 279.52 200.00 479.52 1.94 1.39 3.33 3.83

150 315.50 200.00 515.50 2.10 1.33 3.44 6.00

As shown in Table 5.3, we assume that Eddie's fixed costs are $200 per day. Total costs (TC) in column 4 are the summa�on of the TVC and the TFC values in columns 2 and 3. The AFC value (column 6) is the TFC divided by the output level (column 1). No�ce that the AFC values decline quickly as output levels increase because the constant level of TFC is spread over more and more units of output. The SAC values (column 7) may be found either by dividing the TC values by the output levels, or by adding together the AVC and AFC values at each output level. Now, no�ce what is happening to the SAC values; although SAC is rela�vely high at low output levels due to the heavy burden of AFC, it falls rela�vely quickly to a minimum of $3.29 in the output range of about 124–132 coffees a day, a�er which it starts to rise slowly. Note that the SAC value is rela�vely stable in the range $3.29 to $3.33 over a quite wide range of output levels from about 115 to 144 coffees per day. Also note that the SAC keeps falling a�er the AVC has started to rise, because the rise in AVC is outweighed by the fall in AFC as output levels con�nue to rise. Even though the MC is above the AVC, it is below the SAC value un�l it intersects the SAC curve at $3.29 per cup.

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Variable costs for a coffee shop include the cost of coffee beans, cream, sugar, napkins, mugs, and employees.

So, is Eddie making any money in his new coffee business? That depends, of course, on what he earns per coffee sold, that is, what price he is charging. Although we will see in Chapter 7 exactly what price he should charge to maximize profits, for now let's assume that he is charging a compe��ve price of $4 per cup. Since this price exceeds average cost for all output levels, he has made a profit every day so far, but no�ce that when the output levels are at or near 150 cups a day, his marginal cost exceeds $4 so he is losing money on the last few coffees! Although he should not turn away the last few customers (since they might turn into repeat and regular customers), he should start thinking about increasing his plant size (e.g., install a larger or faster coffee

machine) to produce higher output levels at lower average and marginal cost levels.7 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#ch05txt7)

1. To reconcile Figure 5.1 (where symbols represent the variable input levels) and Table 5.1 (where numbers represent the input levels) note that V1 in Figure 5.1 (where MP is maximized) is shown as 4 units

of the variable inputs in Table 5.1; V2 in Figure 5.1 (where AP is maximized and also MP = AP) is shown as 6 units of the variable inputs; V3 in Figure 5.1 (where MP = 0) is shown as 10 units of the variable

inputs. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#return1) ]

2. You may note that the TP curve takes the shape of a cubic func�on, TP = a + bV + cV2 + dV3, where the parameter a represents the intercept on the ver�cal axis (zero in this case), b and c will take posi�ve

values and d will have a smaller nega�ve value, which ul�mately causes the V3 term to outweigh the V and V2 terms and cause the TP to reach a maximum and therea�er decline. Algebraically, AP = TP/V

= a/V + b + cV + dV2 and MP = δTP/δV = b + 2cV + 3dV2. You can see that both AP and MP are quadra�c equa�ons and thus have an inverted-U shape. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#return1) ]

3. Note that these laws are not legal laws, but instead are empirical laws, that is, they are frequently observed in prac�ce and have been validated by data collec�on and es�ma�on of the produc�on and cost func�ons. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#return3) ]

4. There is a general rela�onship between the average and the marginal value of any sta�s�c. If the marginal observa�on is below the average, the average must be falling, since the marginal (last) observa�on will pull the average down, and conversely if the marginal is above the average, the average must be rising, since the marginal observa�on will push the average up. You already know this in the context of your grade point average (GPA) or your baseball ba�ng percentage, for example. It follows that the marginal value must equal the average value at the minimum value of the average. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#return4) ]

5. We saw earlier that the MC is equal to the slope of the TVC curve. Because the TC curve is simply shi�ed ver�cally by the addi�on of the TFC curve, the TVC and the TC curves must have the same slope at any par�cular output level. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#return5) ]

6. Because the TVC curve reflects the shape of the TP curve, it is no surprise that the TVC curve is a cubic func�on of the output level, namely TVC = e + fQ + gQ2 + hQ3, and thus AVC = TVC/Q = f + 2gQ

+3hQ2 and MC = δTVC/δQ = f + 2gQ + 3hQ2. Thus, AVC and MC are both quadra�c expressions and have an inverted-U shape when graphed. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#return6) ]

7. This example assumes that all costs of opera�on are included. If Eddie was staying open un�l midnight on some days and only sells a few coffees a�er 9 p.m., and was not accoun�ng for the opportunity cost of his own �me, he should revise his decision. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.1#return7) ]

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When communica�on among mangers and other employees becomes less efficient in large plants, diseconomies of plant size may occur leading to reduced efficiency.

5.2 The Long-Run Cost Curves

As men�oned earlier, the long run is a hypothe�cal situa�on in which all inputs and costs are variable, so managers can choose whichever plant size they think is appropriate for their demand situa�on. Thus, managers can visualize many different SAC curves (and their underlying AVC, AFC, and MC curves) all at the same �me, and will then select one set of short-run curves, build the chosen plant size, and proceed ahead into a new short-run produc�on period. The long-run average cost (LAC) curve shows the least cost of produc�on for each output level when all inputs are variable. It is composed of a small segment of many different SAC curves. In Figure 5.5, we show the long-run average costs (LAC) curve as the envelope curve of a series of SAC curves, where each of the SAC curves sits on the LAC and is tangent to the LAC for a

small distance.8 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.2#ch05txt8)

Figure 5.5: The LAC curve is an envelope curve for the SAC curves

Note that this does not mean the LAC joins the minimum point of each SAC curve. On the downward sloping (le�-hand side) sec�on of the LAC curve, the part of the SAC that contributes to the LAC is found to the le� of the minimum point on the SAC curve. You can see in Figure 5.5 that at output level Q1 it is SAC2, rather than the minimum

point on SAC1,, that contributes to the LAC curve since the average cost of Q1 is minimized at cʹ dollars rather than at c dollars. Oppositely, on the upward-sloping right-hand

side of the LAC curve, the part of the SAC that contributes to the LAC is to the right of the minimum point on each SAC curve.

In Figure 5.5, we also show the long-run marginal cost (LMC) curve, which is the locus of least-cost short-run marginal cost levels when all inputs to the produc�on process are variable. The LMC curve has the expected rela�onship with LAC, lying below LAC when the la�er is falling, lying above LAC when the la�er is rising, and intersec�ng LAC when the la�er is at its minimum level. At the minimum point on the LAC curve, note that LAC = LMC and also that SAC = SMC so that all four values are equal at output level Q2. (Note that we are using SMC to refer to short-run marginal cost to dis�nguish it from LMC.)

Because the fixed inputs are infinitely variable in the long run, there would be an infinite number of SAC curves, which allows the LAC curve to be a smooth curve composed of infinitely small sec�ons of each of the possible SAC curves. We have shown only five SAC curves in Figure 5.5, but those five curves are enough to demonstrate that we expect the LAC curve to be U-shaped. Said another way, we expect the produc�on efficiency of the firm to improve at first and later reduce as we progressively increase the size of the firm's plant.

Economies and Diseconomies of Scale

The increase in produc�on efficiency as plant size is increased is known as economies of scale and is characterized by successive SAC curves lying below and to the right of the preceding SAC curve (such that the LAC curve is downward sloping). Economies of scale are also known as economies of plant size. In Figure 5.5, economies of plant size are evident up un�l output level Q2. Diseconomies of plant size (or scale) are evident when successively larger plant sizes cause the SAC curves to lie above and to the right

of the preceding SAC curve, and thus the LAC curve is upward sloping and LAC values are rising.

What causes economies of plant size? They occur because the ra�o of fixed to variable inputs is becoming progressively more efficient. This increased efficiency occurs because some of the fixed resources with unused capacity (such as management �me, or factory, office, and storage space) become more fully u�lized without cos�ng any more money, and because larger plant sizes allow more and more workers to specialize on those parts of the work where they are most produc�ve, rather than be "jacks of all trades, masters of none." Economies of scale may also arise due to purchasing economies (e.g., buying in bulk), as we shall see later. Diseconomies of plant size may occur when plants become very large and there are many people working in the same workplace. Communica�on among coworkers, and between bosses and other employees, becomes less and less efficient, and employee morale might break down leading to reduced personal efficiency of individual workers.

The SAC curve that nestles in the bo�om of the U-shaped LAC curve is known as the op�mum plant size, or the op�mal scale of plant. The op�mum scale of plant is the plant size that allows the product to be produced at the least cost per unit when all inputs to the produc�on process can be varied.

Constant Returns to Scale

It is possible that the firm might experience constant returns to plant size, where the LAC of produc�on remains constant at a par�cular level as the scale of plant is increased, as shown in Figure 5.6. In such cases, constant returns to scale are usually preceded by economies of scale and are later followed by diseconomies of scale, but there is a range of plant sizes over which LAC neither falls nor rises. In Figure 5.6, a selec�on of SAC curves lie on a horizontal sec�on of the LAC curve between output levels Q1 and Q2. Note the LMC curve is the dashed line that joins the le�-hand sec�on of the LAC when constant returns to plant size begin, and is then equal to LAC while

the la�er is constant, and finally LMC rises above LAC, pulling that curve upwards a�er diseconomies of scale set in.

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Figure 5.6: Constant returns to plant size

In this case, there are mul�ple op�mal plant sizes, as each SAC curve that is tangent to the flat bo�om of the LAC curve can produce the product at the minimum average cost of produc�on. Constant returns to plant size are preferable to diseconomies of plant size, since the firm can keep its costs per unit of output at a stable level in the long run and, thus, avoid price increases and the loss of market share that might otherwise have been necessary. Constant returns might occur when the forces that deliver economies of scale are just balanced by the forces that deliver diseconomies of scale over a range of output levels. They might occur because managerial decisions have been taken to improve the produc�vity of resources that were seen to be approaching the point of diminishing marginal produc�vity, such as decisions to supply workers with more efficient computers, tools, machines, vehicles, and so on. In the case of The Robust Coffee Place, if demand was sufficient, Eddie could poten�ally gain constant returns to plant size by doubling his plant size by leasing the shop next door and installing similar coffee-making equipment and personnel there to replicate his exis�ng opera�on.

8. An envelope curve forms the outer boundary of a set of observa�ons. The LAC curve shows the minimum SAC for every output level, presuming that the size of plant can be varied infinitesimally to minimize the SAC for each output level. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec5.2#return8) ]

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5.3 Other Cost Concepts

It is important to understand a variety of other cost concepts that are important for managerial decision making. As indicated in Chapter 1, economic and accoun�ng concepts of costs and profits differ due to the use of different cost and revenue conven�ons. Thus, it is important here to clarify several cost concepts and to make it perfectly clear what we mean by costs of produc�on in managerial economics.

Economic Costs Versus Historical Costs

In business situa�ons, there are �mes when the actual cost of an input is not the economic cost of the input. The economic cost of an item is defined as its value in the best alterna�ve u�liza�on of that item. The best alterna�ve use of the item is also known as the best alterna�ve opportunity, and so economic costs are also known as opportunity costs. For example, suppose a machine had been purchased last month for $200,000 but, due to currency exchange rate fluctua�ons, it would now cost $250,000 to buy and import the same machine. Rather than use the machine in its produc�on process, the firm could alterna�vely sell it to another firm for $250,000, assuming it was s�ll in as-new condi�on. This means that the economic cost (opportunity cost) of the machine is $250,000 not the ini�al purchase cost. The ini�al purchase cost is known as the historical cost, or what it actually cost in the past period.

Another example is the inventory of copper tubing that a plumbing firm holds, which cost, let's say, $100,000 to buy last year. Now, let's suppose that the price of copper had recently risen significantly and would now cost $150,000 to buy the same amount of copper tubing. Again, the historical cost would be $100,000 whereas the economic cost would be $150,000 since it would cost that much to replace the copper tubing, or, alterna�vely, it could be sold to another plumber for that amount. Thus, the value of the inventory of copper tubing held by the firm would need to be revised upwards to reflect the change in copper prices. Oppositely, if the market value of another inventory item, say, finished goods, were to fall significantly due to the design becoming outmoded, the value of those inventories would need to be revised downwards to reflect their economic value to the firm.

It is important to understand that the cost concept underlying the cost curves in our analysis is economic (or opportunity) costs. To rely on historical costs would poten�ally undervalue (or overvalue) the true economic costs of producing any par�cular output level.

Sunk Costs, Unavoidable Costs, and Incremental Costs

Costs that have been incurred previously and cannot be retrieved are called sunk costs. Most fixed costs will be sunk costs unless the asset can be sold to someone else who wants to buy it, and usually the price that someone else will pay for used equipment or other asset is less than the historical cost of those items. The price the firm can obtain by selling these used items is known as the salvage value, and, as men�oned, unless the value is apprecia�ng (e.g., a collector's item), the salvage value is usually less than the historical cost. To account for the declining value of a fixed asset, accountants apply a deprecia�on charge that reduces the depreciated value of the asset to reflect its reduced salvage value.

Similar to sunk costs are unavoidable costs—these are costs that the firm is contractually commi�ed to pay regardless of output levels. Lease costs of a fixed asset, for example, are set by contractual agreement and must be paid monthly in most cases. Salaries of top management are also unavoidable costs for the firm in most cases, although in dire circumstances even these might be avoidable or at least postponed.

Incremental costs are those costs that will be incurred in the future because of a decision to be made. Thus, variable costs are incremental costs that follow the decision to produce output at a par�cular level, whereas fixed costs are not incremental. In making forward-looking decisions about output levels in the current or future periods, we will see that only incremental costs are relevant and that sunk costs and unavoidable costs are not relevant since they will occur anyway. And, con�nuing the discussion above, the incremental costs that we do consider must be valued at their economic (or opportunity) cost rather than at their historical costs.

Economies of Scope

Economies of scope are reduc�ons in average costs that occur because the cost of an input can be spread across more than one product line. Our analysis has been concerned with the produc�on and cost curves for a single product line, but, in most firms, there will be mul�ple product lines. Economies of scope arise when fixed or variable inputs to the produc�on process have underu�lized capacity and this underu�lized capacity may be used in another produc�on process. For fixed inputs, there might be unused or underu�lized space in buildings, �me of managers, or �me of machines and equipment that could be u�lized in the produc�on process for another product line. Thus, Toyota, which has at least 45 different vehicle types and models, gains economies of scope when it introduces a new model because the extra product line does not require an en�rely new set of buildings, managers, and machines but can instead u�lize the spare space and �me of exis�ng fixed inputs. Accordingly, some frac�on of the fixed costs will be allocated to the new model and thereby reduce the fixed cost alloca�on to the previously exis�ng models.

Economies of scope are also possible for variable inputs that are incompletely u�lized in the produc�on of a par�cular product. Consider a metal-working firm that has a produc�on process in its factory (fixed inputs) that processes mild steel (variable input) into steel gates and fences for sale to firms in the construc�on industry. The off-cuts of steel that are generally too small to be used in fences and gates might be called waste products and simply disposed of; alterna�vely, these can be used to produce one or more other product lines, such as small brackets. Because products of saleable value (brackets) can be made from the waste product of the metal-working opera�on, accountants will want to assign some of the cost of the steel to the bracket product line and thus, reduce the cost of the steel to the gates and fences product line.

Increasing the scope of the firm's opera�on to u�lize the underu�lized space and �me of fixed input, or the unused �me or capability of variable inputs, serves to reduce the fixed or variable costs associated with the other product lines produced by the firm. Economies of scope, therefore, cause the SAC and the LAC curves to shi� ver�cally downwards at every output level due to a downward shi� of the AFC or the AVC curves.

Purchasing Economies

Purchasing economies are the reduc�on in average costs that are due to purchasing inputs in larger volumes, where the firm receives discounts for buying in bulk. For example, a glass manufacturing company might pay $30 per ton of sand for orders of 10 tons or less, but the price of sand might fall to $27 per ton (i.e., 10% discount) for order quan��es between 10 and 50 tons, and fall again to $24 per ton (i.e., 20% discount) for order quan��es greater than 50 tons. Because larger glass manufacturing firms would use more sand per period in their produc�on process, they will gain a cost advantage over smaller firms whose lesser rate of produc�on makes it economically inefficient to take delivery of huge amounts of sand and have their factory space (and their funds) �ed up in a large inventory of sand. Along with the reduced cost per ton

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Purchasing economies are the reduc�on in average costs due to purchasing inputs in larger volumes, where the firm receives discounts for buying in bulk.

of sand, the delivery cost per ton will generally be smaller for larger delivery volumes, so this is another purchasing economy accruing to larger firms.

As indicated earlier, purchasing economies are one of the main causes of economies of scale—firms with larger plant sizes gain purchasing economies on all kinds of fixed and variable inputs that cause the LAC curve to slope downward as plant size is increased, up to the point of the op�mum scale of plant. While very large firms might s�ll be gaining purchasing economies as they expand to larger and larger plant sizes, the rise of other inefficiencies might offset the gains from purchasing economies such that overall the firm eventually experiences diseconomies of plant size.

Learning Curves

Learning curves, also known as experience curves, show the decline in average cost per unit of output as the firm's experience producing that product accumulates. As the firm's experience in producing a par�cular product increases, the managers and other employees discover ways to cut �me and materials cost in the produc�on of that product. The famous economist Kenneth Arrow called this "learning by doing," but it might equally be called learning by making mistakes (and not repea�ng the mistakes!) (Arrow, 1962, 1970). Empirical studies indicate that average costs tend to decline by a rela�vely stable percentage each �me that cumula�ve output doubles. Figure 5.7 displays a learning curve where the average cost declines by 20% each �me cumula�ve output doubles. This can be verified using simple arithme�c: The average cost falls to 80% of the preceding level each �me cumula�ve output doubles.

Figure 5.7: The learning curve

Note that the learning curve relates the average cost of produc�on to the cumula�ve volume of produc�on and therefore refers to SAC values across several different produc�on periods, poten�ally involving several different plant sizes, differing input prices, and so on. Thus, a learning curve shows the decline in average costs where nothing (necessarily) stays the same. Indeed, the learning curve accounts for all of the cost reduc�ons that take place over �me, including those due to economies of scale, economies of scope, and purchasing economies, as well as those that are due to changes in technology causing the inputs to be more produc�ve than before.

Ques�on: How does the learning curve impact the SAC and LAC curves? Answer: It causes them to sink downward gradually from one period to the next. Early in the life of a firm, for a new business venture for example, the downward shi� of the LAC and SAC curves would be quite significant as that firm doubles then doubles again its cumula�ve output level. But as the firm gets older and accumulates greater and greater cumula�ve output (i.e., greater produc�on experience) the rela�vely large cumula�ve output figures need to double again to see the same percentage decline in average costs. So, as the firm matures and has behind it a rela�vely long history of produc�on, the period-to-period downward shi� of the SAC curve will become negligible. Note that the rate of learning varies across produc�on processes, being rela�vely low for simplis�c produc�on methods and being higher for more complex produc�on methods, but it usually falls within the range of 5–20%.

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Summary

In this chapter, we have laid the groundwork for our analysis of produc�on and costs and have encountered a variety of cost concepts that are useful for decision making by managerial economists. We first introduced the concepts of fixed and variable inputs in the produc�on process, no�ng that while there are fixed inputs in the short run, in the long run all inputs are theore�cally variable as the firm is free to choose any size plant that it wants. In the short run, the produc�vity of the variable inputs will increase at first and later decrease in accordance with the law of variable propor�ons, the la�er part of which is also known as the law of diminishing marginal produc�vity. Essen�ally, the marginal product of the variable inputs (MP) might increase at first but a�er some point begins to decline and con�nues to decline as more and more variable inputs are added to the fixed inputs.

The total variable cost (TVC) curve mirrors the shape of the total product (TP) curve because it is essen�ally the monetary version of the TP curve. At first, when average product (AP) is rising, average variable cost (AVC) is falling, and later, when AP is falling, AVC is rising. Similarly, when marginal product (MP) is ini�ally rising, marginal cost (MC) is falling, and later when MP is falling, MC is rising. The MC curve lies below the AVC and SAC curves when they are falling, and lies above those two curves when they are rising. Inevitably, the MC curve cuts the minimum point on both the AVC and the SAC curves.

Total costs (TC) are the sum of total variable costs (TVC) and total fixed costs (TFC). TFC are constant at a par�cular level rela�ng to the plant size throughout the current produc�on period. To find TC we simply add the constant TFC amount to the varying TVC amount at each output level. Short-run average costs (SAC) are the sum of AVC and average fixed costs (AFC) at each output level, or alterna�vely are equal to TC/Q at each output level.

In the long run the firm may choose any size of plant and thus can regard the long-run average cost (LAC) curve as a smorgasbord of available plant sizes and choose the plant size that best suits its output plans. The long-run marginal cost (LMC) curve cuts the LAC curve from below at its minimum point when the LAC curve is U-shaped, that is, exhibi�ng both economies and diseconomies of plant size (or scale). In some cases the LAC might be found to have a flat bo�om, and thus exhibit more than one op�mal size of plant, in which case the LMC curve is coextensive with the LAC curve for the dura�on of plant sizes that exhibit such constant returns to scale.

Finally, we considered a series of cost concepts that are important for managerial decision making. In economic analysis we always consider the economic costs of resources or assets, which is equal to the opportunity costs of those items. Historical costs might overstate or understate the economic costs of an input to the produc�on process or an asset held by the firm if the opportunity cost has changed since the �me the item was purchased. Sunk costs are costs that have been paid and cannot be retrieved, while unavoidable costs may not yet have been paid but must be paid regardless of the output level chosen. The fixed costs of produc�on are usually either sunk costs or unavoidable costs. Incremental costs are those that are incurred because of a decision, so variable costs are always incremental costs.

The concept of economies of scope was introduced to illustrate that adding addi�onal product lines can cause the short-run cost curves to shi� ver�cally downwards as the cost of underu�lized fixed or variable inputs is transferred to the new product lines. Purchasing economies refer to the decline in average costs that is due to the firm's ability to gain discounts by buying in bulk as the firm becomes larger. This led to a discussion of the learning curve, whereby the average cost of produc�on declines over �me as the firm (and its employees) learns from cumula�ve produc�on experience and also benefits from economies of scale, scope, and purchasing.

Ques�ons for Review and Discussion

Click on each ques�on to reveal the answer.

1. Suppose your produc�on process has three inputs—machinery, highly skilled labor, and raw materials. If you wanted a new (larger or smaller) machine, it would take six months to be fabricated, delivered, and installed. Your highly skilled workers are all under contract for another eight months. New skilled workers take three months to acquire, because of the lengthy process of adver�sing, interviewing, signing them up, and familiarizing them with your opera�ons. Raw materials must be ordered four weeks in advance. (a) How long is your short run? (b) When can you make your long run decision to change plant size? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

(a) The short run con�nues un�l you can change the input quan��es of all inputs that need to be changed to increase or decrease output levels. If the firm wants to increase produc�on, the current skilled labor will presumably be retained and new skilled labor will require three months to hire and train, but a new machine will require a six-month delay. So the firm is s�ll in the short run for another six months for output increases. For a reduc�on in plant size it will take 8 months to reduce the input of skilled labor, such that in this case the short run is 8 months. (In prac�ce, the firm would buy out the contract of the skilled labor and reduce output levels sooner, depending on whether the machine could be run at less than its full capacity rate.) (b) You can make the decision to change plant size as soon as you like—it just takes �me to implement the change.

2. How does the law of diminishing returns differ from the law of variable propor�ons? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

The law of diminishing returns (LDR) is the la�er part of the law of variable propor�ons (LVP). The LVP says that the marginal product (MP) of the variable inputs will increase at first and later decrease progressively, while the LDR says that a�er some point the MP of the variable inputs will decrease progressively.

3. Why is the point of inflec�on on the total product curve the point where diminishing returns begin? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

The point of inflec�on is where the total product (TP) curve changes from "convex from below" to "concave from below" and thus reflects the point where the rate of change of TP (i.e., the marginal product) stops increasing and starts decreasing.

4. Why is marginal product maximized at the same variable input level at which marginal cost is minimized? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

Marginal product (MP) is maximized at the same output level where marginal cost (MC) is minimized because the la�er is the monetary equivalent of the former's (physical) value. MP is some value of the output units (e.g., 2.5 units) due to the increment of one unit of the variable factors (that costs, let's say $10). So the MC = ΔTC/ΔQ = 10/2.5 = $4. If the produc�vity of the variable factors therea�er declines—i.e., produces less than 2.5 units of output per unit of input—yet s�ll costs $10 per unit, then the MC must therea�er increase.

5. Why does the marginal cost (MC) curve cut the minimum point of the average variable cost (AVC) curve? Why does it also cut the minimum point of the short-run average cost (SAC) curve?

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(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

If MC is less than AVC, the AVC must be falling, since the lesser cost of the marginal unit of output will pull down the average variable cost per unit of output. Conversely, if MC is above AVC, the la�er must be rising, since the higher cost of the marginal unit of output will push up the average variable cost. Since MC falls at first and later rises when diminishing returns set in, it follows that the MC must fall and later rise as the AVC is s�ll falling, then cross the AVC at its minimum point and then con�nue to rise, pulling up the AVC.

6. Why is the long-run average cost (LAC) curve not a line joining the minimum points on the possible SAC curves? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

The LAC is an envelope curve (or lower boundary) of the mul�tude of possible SAC curves, composed of tangency points with the lowest SAC value at each output level. When there are economies of plant size the LAC will be falling as output levels rise, and thus the tangency point with LAC of any SAC curve must occur on a downward-sloping sec�on of those SAC curves (and not the minimum point on the SAC curves). Similarly when the LAC is rising due to diseconomies of plant size, the envelope curve will be tangent on the upward-sloping sec�on of the SAC curves (and not the minimum points of the SAC curves).

7. Define economies of scale, diseconomies of scale, and constant returns to scale. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

Economies of scale are indicated by declining average costs of produc�on on the LAC curve as plant size (i.e., the fixed inputs) increase. Diseconomies of scale are indicated by rising average costs of produc�on on the LAC curve as plant size is increased. In between these stages, there may be constant returns to scale that are indicated by constant average costs of produc�on (horizontal LAC curve) as plant size is increased.

8. Explain the difference between economics costs and historical costs. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

Economic cost is the current value of the item or resource, measured by its opportunity cost, which is its value in its best alterna�ve usage. That value would be equal to what it would cost to replace it. Historical cost is the actual purchase cost in the current or previous period. Economic and historical costs may be the same, or one might be higher or lower than the other, depending on the forces that are influencing supply and demand of the item or resource in ques�on in the present period.

9. Dis�nguish between sunk costs and unavoidable costs. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

Sunk costs are payments that have been paid in the past and cannot be retrieved. Unavoidable costs are costs that are to be paid in the future and cannot be avoided. In neither case should these costs enter a decision to be made in the present period since neither are incremental to that decision.

10. Explain the learning curve in terms of economies of plant size, economies of scope, purchasing economies and changes in technology. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

The learning curve traces the firm's unit cost of produc�on over �me and typically takes a hyperbolic shape, with unit cost decreasing at a decreasing rate. It is the net result of all influences on unit costs, including economies of scale (or plant size), economies of scope (or breadth of product line), purchasing economies (due to buying materials in bulk), and improvements in technology that make the produc�on process more efficient.

Decision Problems

1. Donald's Oyster & Pearl Company operates a pearl-diving business in the North Pacific Ocean. Donald owns a large trawler boat and hires local divers from the nearby islands and pays them on the basis of the weight of oysters recovered. He sells the pearls and the oyster meat separately. Over the past month he has been out pearling eight �mes in the same general area, each �me taking all the divers who showed up looking for work. The details of the number of divers and the weight of oysters recovered are listed in the following table:

Trip number Divers employed Oysters recovered (kilograms)

1 2 3 4 5 6 7 8

6 17 9 5

12 3

14 15

38 76 56 32 74 15 80 78

a. Over what ranges do there appear to be increasing, constant and/or diminishing returns to the variable factor? b. What number of divers appears to be most efficient in terms of output per diver? c. What number of divers appears to minimize the marginal cost of oysters and pearls?

2. The Peaches and Plums Cosme�cs Company (PPCC) produces face cream products mostly for women. Management has established the following rela�onships between units of the variable inputs and units of output. One variable input unit comprises one (unskilled) person working 40 hours per week, the electric power to run the mixing machines, the ingredients in the required propor�ons, and the small glass jars and other packaging materials needed for a week's produc�on. The variable costs are $1,200 per unit of the variable inputs. Overhead (i.e., fixed) costs are $120,000 per week.

Variable input (units) Output (jars)

100 200 300 400 500 600

6,500 14,300 20,200 24,400 27,800 30,000

a. Derive a table showing the firm's AVC and SAC values for the output levels shown. b. Graph the AVC and SAC curves and sketch in your best es�mate of the marginal cost curve. c. At what output do you think diminishing returns first start? Explain with reference to the graph and reconcile this with the input–output data.

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3. Suppose that one of the very expensive ingredients in Peaches and Plums cosme�c cream (refer to problem 2 for details) was suddenly found in abundance, such that cost per unit of the variable inputs fell to $500 each.

a. What is the impact of this discovery on the firm's SAC, AVC, and MC curves? b. Does it change the point where diminishing returns set in? Why or why not? c. Now suppose that a�er the cost reduc�on for the expensive ingredient, worker morale improves such that worker produc�vity goes up by 10% at each input level.

What is the impact of this on the firm's AVC, SAC, and MC curves? 4. Panache Shirts Limited is a small manufacturing business that makes men's shirts in dis�nc�ve designs that appeal to a limited segment of the market. Sales have been

growing steadily and since beginning the business with one produc�on center, Mr. Panache has first doubled, then tripled and finally quadrupled the number of produc�on centers by leasing larger spaces within a warehouse building. Each produc�on center consists of a cu�ng machine, three sewing machines, and three skilled operators. Variable inputs include co�on and silk cloth, sewing threads, bu�ons, casual labor, and packaging materials. Throughout the expansion, Mr. Panache has personally supervised all of the operators and has handled all other aspects of the business, including the marke�ng and financial aspects. He has kept records of the daily produc�on of shirts from each produc�on center, as follows:

Produc�on center Shirts per day (average)

1 2 3 4

61.8 127.2 182.4 228.9

Each produc�on center costs $12,000 per month in fixed and variable costs. Mr. Panache draws $9,000 a month from the business for his salary, and the remaining fixed costs are $3,000 a month. Assume there are 20 working days in each month.

a. Has Panache Shirts Limited increased the rate of output or the scale of output? Explain. b. Are there economies or diseconomies of scale apparent in the data? Explain. c. Indulge in some specula�on as to the causes of the economies and diseconomies, if any.

5. Greenfield Farms Bakery (GFB) is currently producing below full capacity with a rela�vely stable demand of 15,000 loaves of bread per week for its regular wholesale customers. At this �me, the business is reasonably profitable—average cost per loaf is $1.48 and the delivered price is $1.70 per loaf to the various food stores that retail the bread to the end-user customers. The CEO of GFB, Ms. Brianna Puddington, has been nego�a�ng to supply bread to another large food chain that currently buys its bread from a rival bakery. This would involve a minimum fixed order of 10,000 loaves per week with the requirement that GFB must also supply any addi�onal demand by this food chain up to another 20,000 loaves per week (i.e., 30,000 loaves total). Ms. Puddington considers the probability distribu�on of the extra demand from this wholesaler to be as follows:

Quan�ty demanded 10,000 15,000 20,000 25,000 30,000

Probability 0.4 0.3 0.15 0.1 0.05

While GFB's present produc�on facili�es could supply the addi�onal 30,000 loaves per week, it would prove to be very expensive to run the plant at such a high produc�on rate. Alterna�vely, Ms. Puddington is considering the purchases of a new con�nuous-process mixing and baking machine that would cost $41,600 (installed and ready to start). This machine could always be sold at its market value, which is expected to decline linearly at one sixth of its value per year, and would have no scrap value at the end of its 6-year life. Alterna�vely, the funds could be invested at 18% per annum at similar risk. The new plant would not require any other changes in overhead costs, which are currently $7,200 per week including management and skilled labor salaries. This current level of fixed cost includes no allowance for deprecia�on since the present plant has been completely depreciated. Variable costs for the present and the proposed plant are as follows (the missing data for each plant indicates that these output levels are not possible in that plant):

Output per week Present plant TVC per week Proposed plant TVC per week

10,000 15,000 20,000 25,000 30,000 35,000 40,000 45,000 50,000

12,600 15,000 18,400 25,500 37,200 53,900 80,800

153,000 —

— — —

30,000 30,600 30,800 31,200 45,900 71,000

a. Calculate the AVC and SAC values, plot the curves for each plant size, and comment on the differences. b. Calculate the expected value of weekly profits under three alterna�ve scenarios:

i. Present plant with no addi�onal sales ii. Present plant with new sales contract

iii. Proposed plant with new sales contract

c. Advise Ms. Puddington whether to install the proposed new plant, and explain your decision.

Key Terms

Click on each key term to see the defini�on.

average fixed costs (AFC) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Total fixed cost (TFC) divided by the number of output units (Q). Since TFC is a constant, AFC is a constant divided by an increasing number (as Q increases) and so the AFC curve takes the shape of a rectangular hyperbola.

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average product (AP) curve (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Shows the ra�o of the output level (or Total Product, TP) to the variable input level (V), or TP/V, at any par�cular input level of the variable inputs.

average variable cost (AVC) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The value of total variable cost (TVC) divided by the quan�ty of output units (Q).

capital (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The economic value of the firm's investment in fixed inputs (such as land, buildings, equipment, vehicles, and skilled human resources) that enter the firm's produc�on func�on, referred to as fixed inputs in this book, since capital has other meanings, o�en associated with financial issues.

constant returns to plant size (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A situa�on where the long-run average cost (LAC) of produc�on remains constant at a par�cular level as the scale of the plant is progressively increased.

constant returns to scale (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Another expression that means the same as constant returns to plant size, since plant size is o�en referred to as scale.

cost func�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A mathema�cal expression showing the firm's total costs as a func�on of the fixed and variable costs necessary to produce a range of output levels.

curvilinear produc�on func�ons (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Reflect a state where the propor�onality between output and the variable inputs varies, such that there is either increasing returns to the variable inputs, or diminishing returns to the variable inputs, or both in sequence.

diseconomies of plant size (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Occur when successively larger plant sizes cause the next short-run average cost (SAC) curve to lie above and to the right of the preceding SAC curve, and thus the long-run average cost (LAC) curve is upward sloping.

economies of scale (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Occurs when the short-run average cost (SAC) curves associated with successively larger plant sizes lie on a downward-sloping sec�on of the long-run average cost (LAC) curve, and thus allow reduced average costs per unit of output as produc�on volume increases in the long run.

economies of scope (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Reduc�ons in the average cost of produc�on per unit of output (SAC or LAC) that are due to (a) the spreading of overhead (fixed) costs over a broader product line, thus reducing the average fixed costs (AFC) of an exis�ng product; and/or (b) u�lizing material off-cuts, idle �me of variable labor, and other underu�lized variable inputs to produce another product thus reducing the average variable cost (AVC) of exis�ng products offered.

fixed inputs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Inputs to the produc�on process that do not vary over the �me period chosen for the analysis (the produc�on period). These include land, buildings, equipment, and skilled personnel that take �me to build, acquire, assemble, or recruit.

historical cost (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The actual monetary cost that was incurred to purchase a fixed input or an item in inventory in a past period. The present period cost of that asset or item (its opportunity cost) might be quite different from its historical cost due to infla�on, current shortage or abundance of supply, obsolescence, or collectors' value recognized.

incremental costs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Costs that will be incurred in the current or future produc�on period because of a decision to be made. In making forward-looking decisions about output levels in the current or future periods, only incremental costs are relevant.

labor (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Historically used by economists to refer to the variable inputs to the produc�on func�on, but these days only unskilled human resources can be varied from day to day according to produc�on requirements. Conversely, skilled labor takes �me to recruit and train and is thus treated as a fixed input to the produc�on process.

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law of diminishing returns (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A phenomenon in produc�on situa�ons reflec�ng decreasing produc�vity of the variable inputs that causes the marginal product (MP) of the variable inputs to decline progressively, a�er some point, as more and more variable inputs are added to the fixed inputs in a produc�on process.

law of variable propor�ons (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A rule that states that the firm's total output is likely to increase at an increasing rate at first, and then increase at a decreasing rate, as we progressively add more and more variable inputs to the fixed inputs. The law of diminishing returns refers to the la�er part of this law.

learning curves (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The downward-sloping curvilinear trajectory of the firm's average costs of output as produc�on experience increases over mul�ple produc�on periods (they are also known as experience curves). Learning curves incorporate all the reasons for changes in average cost per unit of output, including economies of scale, economies of scope, purchasing economies, changes in technology and changes in employee efficiency.

linear produc�on func�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A situa�on where output increases linearly as variable inputs are added to the fixed inputs. With a linear produc�on func�on the total cost (TC) curve would start with the (lump sum) cost of the fixed inputs and then rise linearly with the output level.

long run (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A hypothe�cal situa�on in which a firm can contemplate the cost implica�ons of various plant sizes, that is, the long-run average (LAC) curve. The firm can choose the plant size (a specific SAC curve) that best serves its produc�on plans. A�er this plant-size decision is made, it takes �me to purchase the fixed inputs and set up the new plant, a�er which point the firm will transi�on from the old short-run situa�on to the new short-run situa�on.

long-run average cost (LAC) curve (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The curve that shows the least cost of produc�on for each output level assuming that all inputs are variable (in the long run). It is composed of a small segment of many different SAC curves.

long-run marginal cost (LMC) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The curve that is marginal to the LAC curve. It lies below LAC when the la�er is falling, lies above LAC when the la�er is rising, and intersects LAC when the la�er is at its minimum level. It is the locus of short-run marginal cost (SMC) levels at the output levels where the SAC curves rela�ng to each SMC are tangent to the LAC curve, and necessarily requires that all inputs to the produc�on process are variable.

marginal cost (MC) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The change in total cost for a one-unit change in the output level, or ΔTC/ΔQ. Since ΔTC only happens (in the short run) due to ΔTVC, short-run marginal cost (SMC) is also equal to ΔTVC/ΔQ.

marginal product (MP) curve (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A curve that reflects the change in total product for a one-unit change in the variable input level. Due to the law of variable propor�ons, the MP is typically not constant but increases at first and later diminishes as addi�onal units of the variable inputs are added to the fixed inputs.

op�mal scale of plant (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

See op�mum plant size.

op�mum plant size (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The SAC curve that sits in the bo�om of the U-shaped LAC curve. The op�mum size (or scale) of plant is the plant size that allows the product to be produced at the least cost per unit when all inputs to the produc�on process can be varied. With constant returns to scale (horizontal sec�on of LAC curve) there will be several op�mum plant sizes.

plant size (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A term that refers to the amount of the fixed inputs involved in the produc�on process in the short run—plant size can be changed only in the long run.

produc�on efficiency (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A condi�on of business opera�ons when output can only be increased by also increasing the costs of produc�on, that is, there is no slack in the system.

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produc�on func�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A formula that shows the dependence of total product (TP) or output (Q) on the quan��es of fixed and variable inputs to the produc�on process.

purchasing economies (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The reduc�on in average costs per unit of a variable input that is due to purchasing that input in larger volumes, where the firm receives discounts for buying in bulk.

short run (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The period of �me in which the firm is unable to change the size of its plant (and thus its maximum output capacity) due to the �me it takes to purchase and assemble addi�onal fixed inputs to the produc�on process.

short-run average costs (SAC) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The (short run) total costs divided by the number of units of output. SAC is equal to the average variable costs (AVC) plus the average fixed costs (AFC).

sunk costs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Costs that have been incurred previously and cannot now be retrieved, such as the historical costs of assets purchased. In most cases only the salvage value of these assets (if any) can be retrieved.

total costs (TC) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The sum of all indirect (fixed) and direct (variable) costs of produc�on that are involved in a business firm.

total fixed costs (TFC) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The cost of all fixed inputs to the produc�on process. Since fixed inputs cannot be varied in the short run, TFC will remain constant in the short run unless the prices of any of the fixed inputs changes—such as managers' salaries and lease costs per month.

total product (TP) curve (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A curve that shows the total output level that can be produced by a given plant size when augmented by various levels of the variable inputs.

total variable cost (TVC) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A value that increases as more units are produced because higher levels of output require higher levels of the variable inputs to be added to the firm's fixed inputs.

unavoidable costs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Costs that the firm is contractually commi�ed to pay regardless of output levels, such as management salaries, rental of factory and office space, lease payments for equipment, and so on.

variable inputs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Inputs to the produc�on process that can be varied at short no�ce, such as raw materials, components, and unskilled labor, because the firm can readily purchase these inputs in the markets for these resources.

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Managerial Cost Analysis and Estimation

Learning Objectives

A�er reading this chapter, you should be able to:

Dis�nguish between incremental costs and nonincremental costs for the purposes of managerial decision making. Iden�fy incremental revenues as dis�nct from nonincremental revenues. Conduct contribu�on analysis in a range of decision-making scenarios, including Project A versus Project B decisions, make-or-buy decisions, and take-it-or-leave-it decisions. Es�mate unknown cost values for par�cular output levels using known cost data from other output levels using extrapola�on, interpola�on, and gradient analysis techniques. Use regression analysis to find which form of the cost func�on (linear, quadra�c, or cubic) provides the line of best fit to the data and thus the most reliable cost es�mates for managerial decision making.

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Introduction

In this chapter, we con�nue with the general topic of cost analysis for managers and will introduce several cost es�ma�on and forecas�ng techniques that allow managers to make profit-maximizing decisions. As we know, profit is the difference between revenues and costs, and if we can minimize costs rela�ng to any par�cular decision that is expected to generate revenues, this will allow the decision to be profit maximizing (or loss minimizing). In this chapter, we con�nue our focus on cost minimiza�on (for any given output and quality level) and will turn our a�en�on to the profit-maximizing issues in the following chapter.

In the first half of this chapter, we will focus on "contribu�on analysis" whereby decisions are evaluated based on the financial contribu�on they make to the firm’s overheads and profits a�er covering the variable costs associated with those decisions. Contribu�on analysis allows managers to make choices among compe�ng alterna�ves —their decision choices—such that the alterna�ve chosen is the one that contributes most to the firm’s overheads and profits. Thus, contribu�on analysis is concerned with accurately es�ma�ng the incremental costs and incremental revenues of each decision alterna�ve such that the best (profit-maximizing) op�on can be chosen.

In the second half of this chapter, we will explore the es�ma�on of cost curves. We begin with some simple techniques and con�nue on to apply the regression analysis technique that was introduced in Chapter 4. Es�ma�ng cost values for the current period involves collec�ng at least one data point on total variable costs (TVC) or average variable costs (AVC) or marginal costs (MC) from the current or past periods and methodically projec�ng that/those values forward or backward to predict the TVC, AVC, and MC levels for any par�cular output level in the current produc�on period.

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Fixed and variable costs are incremental to decisions made by managers. Incremental costs represent the change in total costs resul�ng from a par�cular decision.

Future costs of repairs and maintenance must be taken into considera�on when purchasing equipment.

6.1 Contribution Analysis

The contribu�on of a decision is defined as the excess of incremental revenues over incremental costs, and it is called the contribu�on because it contributes to the firm’s fixed and unavoidable costs, and also to profits if total revenues are more than total costs. Contribu�on analysis is a form of cost–benefit analysis where the costs are confined to incremental costs and the benefits are confined to incremental revenues. Incremental costs, you will recall from the preceding chapter, are the costs that are consequen�al to a decision made, while incremental revenues are the revenues that are consequen�al to the decision made. As an example, if the firm can sell any amount of its product at a price $10 per unit, and average variable cost (AVC) is constant at $6 per unit, then the contribu�on made towards fixed costs and profit is $4 per unit. This difference between incremental costs (which equal AVC in this case), and incremental revenue (which equals price in this case), is known as the contribu�on margin when it applies to the sale of a single unit of the firm’s output.

Incremental Costs and Revenues

In more complex cases, AVC will not be constant, as we saw in the preceding chapter, so incremental cost generally is not equal to AVC. Similarly, incremental revenue is generally not equal to price, as we shall see in the next chapter. Moreover, decisions made by managers usually cause both fixed and variable costs to change, since the implementa�on of a new decision (e.g., to expand produc�on) may cause not only extra direct labor and direct material costs but also require the purchase of addi�onal capital assets and the employment of salaried managers and other workers who must be treated as fixed costs since their salaries must be paid regardless of output levels. Thus, incremental costs are defined as the change in total costs that result from a par�cular decision.

But isn’t that the defini�on of marginal costs? No, as defined in Chapter 5, the marginal cost (MC) is the change in total costs (TC) for a one-unit change in the output level (Q). Incremental cost, on the other hand, is the change in TC that results from a decision that may or may not involve a change in the output level. For example, the decision might be to purchase a new machine or introduce a new produc�on method that will allow produc�on of the same output level at a lower AVC level.

Incremental costs must be accurately iden�fied for good decision making. All costs that change must be included, and costs that do not change must not be included. For example, capital assets that have been idle with no alterna�ve use do not have an incremental cost and can be regarded as costless for the decision at hand. On the other hand, if capital assets are currently being used to produce an alterna�ve product, and the decision at hand would require them to be used elsewhere, we have to include the foregone contribu�on as an opportunity cost of the decision to be made and to treat the opportunity cost as an incremental cost of that decision. For example, a trucking firm u�lizing an idle truck to complete a special delivery would include driver, fuel, and toll costs, but would not include any cost for the use of the truck in the calcula�on of incremental costs. However, if the truck would have been used to carry groceries to earn $200 during that �me, that foregone revenue would be the opportunity cost of using the truck to make the special delivery and should be included as an incremental cost of the decision to use the truck that way instead.

Incremental costs are o�en called relevant costs since they are the costs that are relevant to the decision that is to be made, as dis�nct from the irrelevant costs that will be incurred regardless of the decision to be made. Irrelevant costs are either sunk costs (fixed costs incurred in the past) or unavoidable costs (costs that must be incurred in the present or future period) as discussed in Chapter 5.

Incremental Cost Categories

There are three main categories of relevant or incremental costs. The first is present-period explicit costs. These are actual outlays of cash to pay for the variable and fixed inputs that are required to implement the decision that is made. Of course, incremental costs will not include unavoidable costs that must be paid in the present period regardless of the decision to be made.

The second category of incremental costs is opportunity costs. Raw materials or components or finished goods taken from inventory do not have a present-period explicit cost but could presumably be sold to another producer or an end user at a fair market value for the item, and that fair market value is the opportunity cost and should be accounted for as an incremental cost. Alterna�vely, if an item in inventory has li�le or no market value (i.e., "dead stock") and would not be replaced in inventory then it has no opportunity cost and, thus, its use does not involve an incremental cost. The fact that there was previously a historic cost of purchasing or manufacturing that item is an irrelevant sunk cost for the purposes of the present decision.

The third category is future costs. Many decisions will have implica�ons for future costs, such as repairs and maintenance to equipment, vehicles, or other capital assets that will be necessitated as a result of their u�liza�on for the decision to be made. Of course, as we saw in Chapters 1 and 2, future costs must be evaluated in present-value terms (if known for certain) or in expected present-value terms (if there is uncertainty surrounding the actual cost to be incurred in the future).

In Table 6.1, we can see a summary of the various costs that are relevant or irrelevant for managerial decision making. Note that by relevant and irrelevant we mean with respect to the decision to be made. If a cost is a consequence of the decision to be made, it is a relevant or incremental cost. Some current period and future costs are irrelevant because the firm is commi�ed to them and they are, thus, unavoidable costs. No prior expenditures (sunk costs) are incremental costs, unless they have an opportunity cost of being involved in the present decision to be made.

Table 6.1: Summary of cost concepts for decision making Relevant costs = incremental costs Irrelevant costs = nonincremental costs

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Present-period explicit costs Variable costs Direct labor costs Direct materials Variable overheads Fixed costs New equipment needed New salaried personnel needed Opportunity costs Contribu�on foregone on the best alterna�ve use of the resources involved Future-period incremental costs EPV of probable costs to follow in the future as a consequence of this decision

Unavoidable costs Managers’ salaries Payments on debt Rental and lease costs Salaries for ongoing workers All other payments that must be made regardless of the decision at hand Sunk costs Previously paid for purchases of assets including land, buildings, plant and equipment, and deprecia�on expenses based on these

All prepaid and nonrecoverable expenses

Incremental Revenues

Similarly on the revenue side, there are some revenues that are relevant to the decision to be made and some that are irrelevant. Incremental revenues are those that will be received as a result of the decision, so these are the relevant revenues. Irrelevant revenues (for the purpose of the decision to be made) are those that would be received or lost regardless of the decision to be made.

So, any decision to be made may cause some costs to be incurred (incremental costs) and may also cause some revenues to be earned (incremental revenues). Note that some decisions impact only incremental costs (such as replacing broken equipment) while other decisions may impact only incremental revenues (such as selling an unwanted asset). Most decisions have both incremental costs and incremental revenues to consider. We shall proceed to work through some realis�c business examples using contribu�on analysis and will examine three main types of contribu�on analysis with these sugges�ve names: Project A versus Project B, make-or-buy decisions, and take-it-or-leave-it decisions.

Project A Versus Project B Decisions

Managers o�en have to choose between two or more projects that are both poten�ally profitable because they do not have the produc�ve capacity or the funding to handle both projects at the same �me. A profit-maximizing firm would want to undertake the most profitable project first, and defer the less profitable project to a later period when it would be compared with other poten�al projects that were available for implementa�on at that �me. The appropriate method for choosing between compe�ng projects is contribu�on analysis—profits will be maximized by choosing the project that contributes the most towards overheads and profits.

Suppose a firm is considering implementa�on of either Project A or Project B as detailed in Table 6.2. Project A promises sales of 10,000 units at $2 each, with materials, labor, variable overhead, and allocated overhead costs as shown, and so apparently makes a profit of $2,000. Project B promises sales revenue of $18,000 with $14,000 of direct and allocated costs, and, thus, apparently makes a profit of $4,000. It might seem that Project B is superior to Project A, because it seems to make higher profits. But, what do we know about relevant and irrelevant costs?

Table 6.2: Income statements for Project A and Project B Project A Project B

Revenues Costs

$20,000 Revenues Costs

$18,000

Materials Direct labor Variable overhead Allocated overhead Total costs

$2,000 6,000 4,000 6,000 $18,000

Materials Direct labor Variable overhead Allocated overhead Total costs

$5,000 3,000 3,000 3,000 $14,000

Profit $2,000 Profit $4,000

When contribu�on analysis is applied to the above choice situa�on the result may be surprising. In Table 6.3, we show only the incremental costs and revenues and see that the contribu�on of Project A actually exceeds that of Project B. For each project we include only the materials, direct labor, and variable overhead costs, presuming that these costs would not be incurred unless the project is undertaken. We exclude allocated overhead charges since these relate to the sunk costs of previously purchased capital assets or the salaries of management and other workers who must be paid whether or not the project is undertaken. Thus, we see that Project A promises the larger contribu�on to overheads and profit and should be chosen for implementa�on by the profit-maximizing firm.

Table 6.3: Contribu�on analysis for Project A and Project B decision Project A Project B

Incremental revenues Incremental costs

$20,000 Incremental revenues Incremental costs

$18,000

Materials Direct labor Variable overhead Incremental costs

$2,000 6,000 4,000

$12,000

Materials Direct labor Variable overhead Incremental costs

$5,000 3,000 3,000

$11,000

Contribu�on $8,000 Contribu�on $7,000

The danger of using an arbitrary rule to allocate fixed overhead costs is illustrated in this example. In Table 6.2, the overhead charge is set equal to the cost of direct labor for both projects, implying that the manager who produced the data in Table 6.2 used an alloca�on rule of "100% of direct labor costs." Simple rules like that almost certainly do not correctly reflect the relevant costs of undertaking any project. Contribu�on analysis allows an incisive look at the actual changes in the costs and revenues that would follow the decision to choose one project over the other.

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Companies are frequently faced with the decision to make or buy. This model can also be applied to a firm deciding between hiring an external cleaning crew or doing cleaning and maintenance work itself.

Also, note that in this simple example we implicitly assumed there were no opportunity costs or opportunity revenues1

(h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec6.1#footernote1) and no future costs or revenues associated with either project. In this case, there is only $1,000 difference in the contribu�ons of the two projects, so the decision to implement Project A is very sensi�ve to the assump�on of zero opportunity and future costs and revenues. This calls for sensi�vity analysis, an analysis of the impact of inaccurate data on the desirability of the decision made. A decision is called sensi�ve to the assump�ons on which it is based if a rela�vely small change in those assump�ons would cause a different decision to be made. It is useful to express the degree of sensi�vity in terms of the percentage varia�on in costs that it would take to reverse the decision; in this case, the change in costs of Project A that would reverse the decision is the propor�on (or percentage) by which the incremental costs of Project A could increase without reducing its contribu�on to that of the next-best alterna�ve (Project B). In this case it is $1,000/$12,000 = 8.33%, assuming the costs of Project B are calculated accurately. In prac�ce, decision makers must be careful to assess whether there are any opportunity or future implica�ons of their decision. If they cannot easily es�mate these addi�onal costs they should consider, if such addi�onal costs are thought to exist, whether they are likely to be more than 8.33% (in this case) higher than the ini�al es�mate. Where the sensi�vity percentage is rela�vely high, for example 40%, we say the decision is rela�vely insensi�ve to the accuracy of the cost es�mates of incremental costs, unless the costs involved are likely to be highly vola�le (e.g., the pump price of gasoline).

Make-or-Buy Decisions

The next category of decision that involves incremental costs and revenues is the make-or-buy decision. Such decisions are required when the firm could either manufacture the product in-house (i.e., make) or outsource the manufacture from another firm (i.e., buy). Similarly, the firm might consider doing its own cleaning and maintenance work (using current employees and purchasing the necessary equipment and supplies) and compare the incremental cost of this with the alterna�ve solu�on of having an outside firm supply the maintenance and cleaning work.

In Table 6.4, we consider the make-or-buy problem facing Wilson Tools. Wilson Tools manufactures high-quality power tools such as drills, jigsaws, and sanders. All these tools require the same roller-bearing unit, which the company manufactures in its own bearing department. Table 6.4 shows the costs data for the past month for the bearing department.

Table 6.4: Wilson Tool Company—Bearing department costs, July 2012 Cost category Total Per unit

Direct materials Direct labor Allocated overhead Total costs

$38,640 $126,390 $252,780 $417,810

$0.56 $1.81 $3.63 $6.00

Now suppose that Wilson Tools has an opportunity to expand the sales of its power tools by an addi�onal 7,500 units a month by supplying its tools to a chain of hardware stores in another state. Wilson could produce the addi�onal 7,500 bearings in its bearing department, but this addi�onal output would congest opera�ons somewhat, so management is considering having the addi�onal roller-bearing units supplied by a specialist bearing manufacturer. It is es�mated that it will require an addi�onal 15% in direct labor costs and an addi�onal 12% in total materials costs to make the bearings in-house. No addi�onal capital expenditures will be required as all machines have excess capacity currently. A specialist bearing manufacturer has been asked to submit a quote to produce and supply the 7,500 bearings per month and has studied the specifica�ons and submi�ed a proposal to provide the bearings at a total cost of $30,000 per month, or $4 each. So, should Wilson make or buy the addi�onal bearing units?

To answer this we must find the incremental cost of producing the bearings in-house, to compare this with the incremental cost of buying them from outside, which is $30,000 per month. The es�mated incremental cost of direct labor is 15% of $126,390, or $18,959 per month. The es�mated incremental cost of direct materials is 12% of $38,640, or $4,637 per month. Wilson expects no change in overhead costs, so the incremental cost of producing the bearings in-house totals $23,596 per month, or about $3.15 per unit. The decision to make rather than to buy the bearings would, thus, appear to save Wilson about $6,404 per month, or about $0.85 per unit.

Variability of Overheads

The above analysis assumes no variability at all in overhead costs as a result of the decision to make the bearings in-house. It is likely, however, that some costs that are treated as fixed overheads, such as electricity expenses, office and administra�on expenses, and equipment maintenance expenses, might actually increase as a result of this decision to make the extra 7,500 bearings in-house each month. But the changes in these overhead costs are likely to be hard to measure. Rather than undertake expensive search costs or make arbitrary assump�ons, the manager should first apply sensi�vity analysis to the assump�on that overhead costs will remain unchanged—that is, the manager should ask by how much could the overhead costs actually change without causing the decision (to make the bearings) to be the wrong decision. In this case, the percentage varia�on in overhead costs would need to be $6,404/$252,780, which equates to about 2.5%. This is a very slim margin for error, so if the manager thinks that the overhead expenses are indeed likely to change by that amount or more, the decision should be reversed and the extra bearings should be bought from the outside supplier.

Other Considera�ons

Quality. A number of other considera�ons should also enter the make-or-buy decision. First, the manager should be concerned about the quality of the bearings supplied by the outside manufacturer. A make-or-buy decision that ignores the compara�ve quality of the bearings and simply focuses on incremental costs might be a very bad decision if the outside supplier’s bearings are of poorer quality. Low-quality bearings in the power tools could lead to product failure, increased warranty claims, unhappy customers, and a reduc�on in the market reputa�on enjoyed by Wilson Tools. On the other hand, if the quality of the specialist manufacturer’s bearing was significantly be�er, Wilson should consider paying extra to get bearings of higher quality than can be produced in-house and, perhaps, begin making and selling a premium line of power tools.

Longer-term supplier rela�ons. Wilson Tools may be nearing its full capacity output level and, if demand con�nues to grow, may need to expand its plant in the near future. An alterna�ve strategy would be to establish a rela�onship with a specialist supplier of bearings to allow it to meet demand for its power tools in the future. The "buy"

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alterna�ve gives it the opportunity to both test out a poten�al longer term supplier and to start building a mutually beneficial long-term supply rela�onship or work towards a poten�al strategic alliance that might be desirable in the future.

Labor rela�ons. If Wilson management chooses the "make" decision, the extra workers required will lead to greater crowding of the factory floor, cafeteria, toilets, and the parking lot, and may, as a consequence, reduce job sa�sfac�on. As a consequence, worker produc�vity (marginal and average product) may fall resul�ng in a rise in average variable and marginal costs per unit of output, thus, poten�ally nullifying the cost advantage of the "make" op�on.

Conversely, if Wilson takes the "buy" decision and contracts with the outside firm for ongoing supply of bearings, workers in the bearing department (in par�cular) may fear that they might lose their jobs if demand for power tools later falls. This might seriously hurt management-labor workplace rela�ons.

Of course, these addi�onal considera�ons may be difficult (or costly) to quan�fy. Thus, the manager will need to exercise judgment and, perhaps, make a calculated gamble to decide whether to make or buy, par�cularly when the incremental costs of both op�ons are rela�vely close together.

Take-It-or-Leave-It Decisions

In other situa�ons, the manager might be faced with an offer that is non-nego�able and must decide whether to accept or decline that offer. For example, a prospec�ve buyer might offer a fixed sum of money for a par�cular capital asset, such as land, buildings, or piece of equipment owned by the firm, or for a par�cular quan�ty of the firm’s output. Or, the purchasing agent for a chain of discount stores might approach your firm and ask for a special deal on a bulk purchase (e.g., 10,000 units) of your firm’s output. Or, a poten�al customer might say, "I can get this (e.g., car) for $X from another supplier. If you can beat that price you have a deal." The manager’s task is to evaluate the contribu�on of the offer and compare it with the status quo—if the deal offers a posi�ve contribu�on to overheads and profits it will be profit-maximizing to take the offer since it will contribute addi�onal funds towards the firm’s profit (or reduce the firm’s loss if revenues are insufficient to fully cover its overhead costs).

Let us work through an example to demonstrate how take-it-or-leave-it analysis works. Suppose Idaho Instruments Ltd. makes hand-held and dashboard-mounted GPS (global posi�oning satellite) devices that allow pedestrians and drivers to navigate unfamiliar streets or highways and to find their way to specific des�na�ons. Normally, the company manufactures these devices and sells them to a distributor at an agreed distributor price. That distributor, in turn, sells the product to retailers at the wholesale

price, and the retail stores then sell the product to end-user customers at the retail price.2 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec6.1#footernote2) But, yesterday, the purchasing agent of a large retail store (that does not currently stock the Idaho GPS device) has come directly to the manufacturer and says she wants to "cut out the middle man" and buy 20,000 units of Idaho Instrument’s X1 model for $40 each, which is $10 less per unit than the distributor price. Idaho’s sales manager knows that the present produc�on level of the X1 model is nearly at full capacity at 160,000 units, but he could supply the addi�onal 20,000 units by foregoing produc�on and sale of 5,000 units of the more sophis�cated and more expensive X2 model. Per�nent data rela�ng to these two models is shown in Table 6.5. Because of the automated produc�on process, the per unit variable costs (AVC) of both units is constant at those levels over a wide range of output levels. The sales manager is reluctant to sell the X1 model for $40 per unit when he normally gets $50, par�cularly since he will have to sacrifice 5,000 units of sales of the more expensive X2 model. He also thinks that about 20% of the X1 units that would go to the new retailer customer (i.e., 4,000 units) will simply replace sales to customers who would have purchased the device through other stores that already stock Idaho’s GPS devices. He has tried to nego�ate for a be�er price but the purchasing agent is adamant and insists that $40 is her only offer. Should Idaho Instruments take it or leave it?

Table 6.5: Per-unit cost-price-profit data for Idaho Instruments GPS devices

Cost item Model X1 $

Model X2 $

Direct materials Direct labor Variable overhead Average variable cost (AVC) Allocated overhead Short-run average cost (SAC) Profit margin (20%) Price to distributor

6.88 9.67 4.29 20.83 20.83 41.67 8.33 50.00

7.79 12.58 4.63 25.00 25.00 50.00 10.00 60.00

Because AVC is expected to be constant over a wide range of output levels for both products, we can calculate the incremental costs on the basis of the AVC data shown in Table 6.5, where AVC is equal to the sum of the direct materials, direct labor, and variable overheads. Note that Idaho Instruments appears to have a very simple rule for the alloca�on of overhead costs: It simply adds 100% of the AVC for each product to arrive at the short-run average cost (SAC). Subsequently, the company marks up its SAC by 20% to find the normal price to their distributor, and the profit margin is the difference between the SAC and the price it normally receives from its distributor. Note also that we can deduce from Table 6.5 that the contribu�on per unit to overheads and profit (i.e., the contribu�on margin) for the X1 model is the sum of the allocated overhead charge ($20.83) plus the profit margin ($8.33) equals $29.17 (rounded), and for the X2 model it is $25 + $10 = $35.

So, to produce 20,000 more units of model X1 will cause an incremental cost of 20,000 �mes the AVC of $20.83, or $416,667 in total. But this is not the total incremental costs of taking the deal, because there are also opportunity costs associated with sacrificing the produc�on and sale of 5,000 units of the X2 model and 4,000 units of the X1 model. To calculate that opportunity cost we note that if 5,000 units of X2 are not produced, the firm will not spend 5,000 �mes the $25 AVC of the X2 model, that is $125,000, but neither will the firm receive the foregone sales revenue of 5,000 �mes the $60 price, that is $300,000. Thus, taking the deal will also cause a net reduc�on of $175,000 (i.e., $300,000 minus $125,000) in the contribu�on of the X2 model towards the overheads and profits of the company. Alterna�vely, and more quickly, we could mul�ply X2’s contribu�on margin ($35) by 5,000 units to find the $175,000 figure for the contribu�on foregone if the sales manager decides to take the deal. Similarly, the foregone contribu�on from the 4,000 units of the X1 model that is expected to be lost is 4,000 �mes its contribu�on margin of $29.17, or $116,667 in total. In Table 6.6, we have assembled the data to allow the sales manager to make a decision.

Table 6.6: Contribu�on analysis of the take-it-or-leave-it offer

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When a business is able to en�ce new customers to its store by selling products at discount prices while s�ll realizing a profit, customers are likely to come back to buy more products in the future.

Incremental revenue Sale of 20,000 units of X1 at $40 each

Incremental costs Variable costs of 20,000 units of X1 at $20.83 each Foregone contribu�on of 5,000 units of X2 at $35.00 each Foregone contribu�on of 4,000 units of X1 at $29.17 each Total incremental costs Contribu�on to overheads and profit

$800,000

$416,667 $175,000 $116,667 $708,333 $91,667

You can see that there will be a net contribu�on of $91,667 to the overheads and profit of Idaho Instruments if the sales manager takes the deal—despite the fact that ini�ally the take-it-or-leave-it offer looked like a bad deal, being $10 less than the normal distributor’s price and requiring the sacrifice of 5,000 units of sales of the more expensive X2 model, plus the probable loss of contribu�on from 4,000 units of the X1 model. But this demonstrates the beauty of contribu�on analysis: It cuts through arbitrary overhead cost alloca�ons and pricing rules to focus only on what costs and revenues actually change as a result of making a par�cular decision.

Other Considera�ons

The preceding analysis is subject to some simplifying assump�ons of course, and the sales manager must consider these before ge�ng back to the purchasing manager. The first issue is the sales manager’s assump�on that 4,000 units of the X1 model that will be placed directly into this retail store will simply replace or cannibalize exis�ng sales, that is, will be sold via this new retailer instead of through Idaho’s regular distribu�on channel and by the exis�ng retailers of Idaho’s GPS devices. The sales manager must carefully consider the extent to which sales via this retailer will be at the expense of sales via its normal distribu�on channels. Applying sensi�vity analysis to his assump�on he could make a few simple calcula�ons and find that the cri�cal ra�o of sales that replace normal sales in this example is about 35%. For example, if more than 35% of sales (7,000 units) via this new channel replace sales via the normal channel, the deal will give almost exactly the same result as con�nuing to supply the market

through the regular distribu�on channels, and so, the decision should be reversed.3 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec6.1#footernote3) The sales manager must find out where the extra units will be retailed. Will this be in a new geographic area where the firm currently has li�le or no sales? Or, perhaps the new retail stores (that currently do not carry Idaho’s product line) do carry a rival manufacturer’s GPS devices, and Idaho’s X1 model would then be accessible to poten�al new customers who would poten�ally buy it instead of the rival’s product. So, once again, faced by the need to get data that is hard or expensive to get, the sales manager must exercise his judgment and hypothesize about the propor�on of this 20,000-unit deal that will simply cannibalize exis�ng sales and make the decision accordingly.

The second main issue is the rela�onship with the distributor and the exis�ng retailers. Undoubtedly, the distributor and the exis�ng retailers will become aware that a new store has entered the market to sell Idaho’s X1 GPS devices. Having bought them at 20% below the normal wholesale price, the new outlet is likely to set a somewhat lower retail price on this product, and perhaps also promote them as a special, and will consequently a�ract customers away from the exis�ng marke�ng channel. Losing sales (and share of the market) will possibly cause the distributor and exis�ng retailers to become unhappy with what they might feel is unfair compe��on, and they may take their business elsewhere in the future. This would cause a future reduc�on in demand through the regular marke�ng channels and poten�al loss of future income. If this is likely to occur, the sales manager must include it as another opportunity cost of taking the deal.

A third issue is the prospect of future (or repeat) business with the new retail store. If indeed this store is able to reach mostly new customers, and, especially if it sells at lower (discount store) prices, it is likely to come back to buy more product from Idaho in future periods, perhaps also to expand purchases to include the X2 device and other items in Idaho’s product line. In the above analysis, we have treated the deal as a "once-off" deal, which is the most conserva�ve assump�on to make. But if this new customer were to repeat or increase this purchase periodically in the future, the expected present value of such future contribu�ons must be considered by the sales manager before making the decision. Of course, it

would be prudent for the sales manager to set the purchasing agent’s expecta�ons at an appropriate level by sta�ng that this is to be understood as a once-off deal to kick start a business rela�onship and that future dealings would be expected to allow Idaho a be�er profit margin.

In Chapters 1 and 2, we discussed decisions that had cost and revenue outcomes over mul�ple periods into the future, and we saw that we needed to express those future outcomes in expected present value terms. In those chapters, we spoke of profits in the first and subsequent �me periods that had to be discounted back to present value. We now know that it is the contribu�on, rather than an accoun�ng measure of profits (which might reflect an inaccurate alloca�on of overhead costs), that is important for managerial decision making.

1. Just as an opportunity cost is a revenue (or in this case a contribu�on) that must be foregone if a decision is taken, an opportunity revenue is a cost (or a nega�ve contribu�on) that is avoided if a decision is taken. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec6.1#return1) ]

2. The difference between the distribu�on price and the wholesale price provides the contribu�on margin for the distributor, and the difference between the wholesale price and the retail price provides the contribu�on margin for the retailer, assuming no other incremental costs—all other costs of the distributor and the retailer are fixed salary or other unavoidable costs. For example, the distributor’s price might be $50 and the distributor might mark this up by 50% to sell it to the retailer at the wholesale price of $75. The retailer might then mark up the wholesale price by 100% to sell it to the end user at the retail price of $150. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec6.1#return2) ]

3. That is, 7,000 units x $29.17 means $204,167 addi�onal opportunity cost that nearly wipes out the $208,333 contribu�on to be made by the deal if there is no replacement of exis�ng sales. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec6.1#return3) ]

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6.2 Cost Estimation Methods In this sec�on, we will examine several methods for es�ma�ng the level of costs per unit based on data collected from the firm’s prior produc�on experience. In the short- run context of business decision making, we are primarily concerned with the behavior of variable costs, but we also know that changes in a fixed cost category might be necessitated by a par�cular decision. We start with simple extrapola�on and later proceed to more complex, but probably more accurate, measures.

Extrapolation of Prior Observations

Extrapola�on means to impute values to a variable outside the range of previous data observa�ons. Extrapola�on is achieved by projec�ng (or extending) the rela�onship that is iden�fied between the output level and the cost level inside the range of data observa�ons to output levels outside the range of previously observed output levels.

So, if average variable cost (AVC) was observed to be constant over a range of output levels, we could make the simple assump�on that it will remain constant at that level for rela�vely small changes in output levels that are both higher and lower than the range of observa�ons for which we have prior data. In the le�-hand side in Figure 6.1, we show a situa�on where prior data indicates that AVC was about $4 at produc�on rates of both 3,000 and 4,000 units of output. These observa�ons are indicated by the stars. The broken lines to the le� and to the right of the observed data points indicate our extrapola�on of AVC to both higher and lower levels of output.

Another example of extrapola�on is shown on the right-hand side in Figure 6.1. Suppose we have (in a different produc�on process) observed that marginal cost (MC) increased from $4 to $5 when the output rate was increased from 3,000 to 4,000 units per day. Accordingly, we can extrapolate this data to es�mate that MC is likely to increase by another dollar to $6 per unit if output is increased to 5,000 per day.

Figure 6.1: Examples of cost extrapola�on given prior cost observa�ons

No�ce that the interrela�onships of the cost concepts mean that, if we have data on AVC at par�cular output levels, we can deduce the value of TVC; or conversely, if we know TVC and the output level, we can deduce the value of AVC. If we have two or more values of AVC or TVC we can deduce the value of MC since MC is equal to ΔTVC/ ΔQ, or the rate of change of TVC. Similarly, if we have total cost (TC) data points we can deduce the value of short-run average costs (SAC) and MC, and also by subtrac�on of TVC from TC, we can find total fixed costs (TFC) and average fixed costs (AFC). Deducing the shape of the costs curves from a limited amount of data generated by past produc�on experience will allow the manager to make more accurate es�mates of the incremental cost associated with Project A versus Project B, make-or-buy, and take-it- or-leave-it decisions.

Interpolation Between Prior Observations

While extrapola�on means making es�mates outside the data range, interpola�on means making es�mates inside the data range. In Figure 6.1, we implicitly interpolated between the data points by drawing a straight line between the data points. In other situa�ons it will be clear the rela�onship cannot be a linear one but must be curvilinear. We saw in Chapter 5 that the law of variable propor�ons, also known as the law of diminishing returns, will cause cost curves to bend in predictable ways. Suppose we have a produc�on situa�on where data has been collected twice. The first data point, when the output rate was 1,600 units per period, measured TVC as $6,400 and deduced AVC to be $4, and a second data collec�on at output rate 3,800 measured TVC as $15,200 and deduced that AVC was s�ll $4. A�er the second data collec�on, however, MC was es�mated to be TVC/Q = $6,230/100 = $6.23 by calcula�ng the costs of direct materials, direct labor, and variable overheads for the last batch of 100 units of output produced. This observa�on, that MC is substan�ally above AVC, should immediately ring alarm bells in the manager’s mind. If the data is accurate, it must mean that AVC is rising, and if AVC is s�ll at the same level ($4) as it was before then AVC must have fallen and then risen between the known data points. In Figure 6.2, we use our knowledge of the law of variable propor�ons to interpolate between the known data points and show curved lines represen�ng our es�mates of the AVC and MC values between the known data points.

Figure 6.2: Interpola�on of cost data between known data points

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As with extrapola�on, interpola�on provides a best es�mate given the data available but may not be especially accurate. For example, in Figure 6.2, the AVC curve might be more or less U-shaped than we have shown, and the lowest point of the AVC curve (where the MC curve intersects) might be nearer to the 1,600 output rate or nearer to the 3,800 output rate. Thus, the manager must exercise judgment, u�lizing other sources of informa�on perhaps, to sketch in the AVC and MC curves. But the most important message to the manager from this cost es�ma�on exercise is that MC is rising rapidly, pulling AVC upwards. But, as we saw in Chapter 5, the average fixed cost (AFC) will be falling so that the sum of AVC and AFC (i.e., short-run average cost, SAC) might s�ll be falling at the higher output rate. The manager must pay close a�en�on to this issue to decide whether higher output rates should be avoided or whether considera�ons should be given to installing a larger size of plant if economies of scale are indeed available.

Gradient Analysis Using Known Data Points

A gradient is the slope at which the ver�cal eleva�on of a line or surface changes over a horizontal distance. That is a fancy way of saying that "slope equals rise over run." In the context of cost curves, the rise will be the change of a cost value (e.g., TC or TVC) and the run will be the change in the output rate. Marginal cost is a gradient, of course, in the case where the horizontal change is only one unit of output (or the average MC per unit over a rela�vely small DQ). More commonly in cost es�ma�on we find that output data will not be available as con�nuous data (i.e., not available for one-unit increments of output), but that data will be collected periodically at different �mes when output levels are discretely different and we need to interpolate the cost values in between the known data points, as we did above. Gradient analysis involves (possibly nonlinear) interpola�on between mul�ple data points.

Suppose weekly data has been collected for total fixed and variable costs for various output levels over five weeks as shown in Table 6.7. To calculate the gradient of the cost curves these data must first be rearranged in ascending order of output, that is, from the smallest to the largest. It is a simple ma�er to calculate the average cost levels for SAC, AVC, and AFC by simply dividing the TC, TVC, or TFC figures by the relevant output level. For decision-making purposes, the manager will be most interested in the behavior of marginal costs and will want to derive an es�mate of MC at various output rates. We know that MC is the change in TC (or TVC) for a one-unit change in output, but the changes in output are much larger in our data. Accordingly, we must es�mate MC as the average change in TC over the output interval by taking the gradient of TC or TVC with respect to output. In Table 6.7, we es�mate MC at four output rates by evalua�ng the ra�o ΔTVC/ΔQ for each output interval, where Δ (as usual) symbolizes a discrete change in the variable concerned.

Table 6.7: Gradient analysis to es�mate marginal cost levels

Produc�on period Output rate (Q) TVC ($)

AVC ($/Q)

ΔTVC ($)

ΔQ (Q)

MC=ΔTVC/ΔQ ($/Q)

Week 4 4,500 27,000 6.00 6,600

3,775

4,625

6,750

1,500

500

4.40

7.55

9.25

13.50

Week 3 6,000 33,600 5.60

Week 5 6,500 37,375 5.75

Week 1 7,000 42,000 6.00

Week 2 7,500 48,750 6.50

No�ce that the four es�mates of MC in Table 6.7 are shown in the middle of the intervals between the five observa�ons. This is more evident in Figure 6.3 where we show the AVC data points and the es�mated MC curve as the broken line joining the four gradient values that were calculated in Table 6.7. Note that the loca�on of the MC curve is more reliable in this case, compared with our earlier interpola�on exercise where the curvature of the AVC and MC curves was chosen arbitrarily. By placing the es�mated MC value in the middle of the output interval, we can gain a more accurate es�mate of the MC values for all output rates between the known data points.

Figure 6.3: Gradient es�ma�on of the marginal cost curve from known TVC data

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Most businesses output levels vary up and down from week to week as orders come in from retailers to replenish their stocks.

Cost Estimation Using Regression Analysis

Note that for both interpola�on and gradient analysis, we have essen�ally sketched in a line of best fit to join the calculated or es�mated data points. As we saw in Chapter 4, we can u�lize regression analysis to find the line of best fit to data points, and this is especially useful when we have a larger number of output and cost observa�ons and it is not so easy to see the shape of the rela�onship between output and costs. Suppose we have 12 weeks of data on the weekly output and total variable costs of an ice cream factory, as presented in Table 6.8.

Table 6.8: Data on output and TVC levels for an ice cream factory Week Ending Output (gallons) Total variable costs ($)

Sept. 7 7,300 5,780

Sept. 14 8,450 7,010

Sept. 21 8,300 6,550

Sept. 28 9,500 7,620

Oct. 5 6,700 5,650

Oct. 12 9,050 7,100

Oct. 19 5,450 5,060

Oct. 26 5,950 5,250

Nov. 2 5,150 4,490

Nov. 9 10,050 7,520

Nov. 16 10,300 8,030

Nov. 23 7,750 6,350

No�ce that the output levels vary up and down from week to week as orders come in from retailers to replenish their stocks of ice cream. We can see that output and TVC are posi�vely related, but is this posi�ve rela�onship a linear rela�onship (implying constant MC) or a curvilinear rela�onship (implying falling and/or rising MC)? Knowing what we know about the law of variable propor�ons, namely that for equal increments of the variable inputs the output level will increase first at an increasing rate and later at a decreasing rate, our default assump�on ought to be that the line of best fit to the TVC data is most likely to be a cubic func�on, taking the form:

TVC = α + β1Q + β2Q 2 + β3Q

3 (6-1)

Where the parameter α represents the unknown factors not explained by the independent variables (Q, Q2 and Q3), and the β’s are the es�mated regression coefficients to the independent variables. If the line of best fit does prove to be a cubic func�on, then we can es�mate the MC func�on as the rate of change of the TVC curve, which is mathema�cally equivalent to the first deriva�ve of the TVC func�on, or:

MC = δTVC/δQ = β1 + 2β2Q + 3β3Q (6-2)

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As you can see, and consistent with our analysis in Chapter 5, a cubic TVC func�on will give rise to a quadra�c MC func�on, which will be U-shaped, falling at first due to

increasing returns to the variable inputs and later rising due to diminishing returns to the variable inputs.4 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec6.2#footernote4)

The U-shape means we should expect a nega�ve value for the β2 coefficient and posi�ve values for the β1 and β3 coefficients—these would cause the MC curve to start

with a ver�cal axis intercept of β1, fall as output levels increase (at first) due to the nega�ve β2 coefficient, which is rela�vely large compared to the β3 coefficient, and then

rise later as the rela�vely large Q-squared values dominate the equa�on. Regression analysis will allow us to find es�mates of these β coefficients in the TVC func�on and, thus, we will be able to calculate the es�mated MC at any value of Q by plugging that value of Q into the MC expression given by equa�on 6-2.

In Chapter 4, when we introduced mul�ple regression analysis in the context of es�ma�ng the demand func�on, the independent variables were different drivers of demand. Here, in the context of cost es�ma�on, the independent variables are different variants of the same driver of costs—namely the output level. So, the three

independent variables on the right-hand side of the regression equa�on for TVC will be (i) the output level Q; (ii) the output level squared (Q2); and (iii) the output level

cubed (Q3). To conduct the regression analysis we first need to enter these data into columns in an Excel spreadsheet. To avoid huge numbers we will express the data in thousands of units, as shown in Table 6.9.

Table 6.9: Data set up for the regression of TVC against output TVC ($000s) Q (000s) Q2 (000s) Q3 (000s)

5.78 7.30 53.29 389.02

7.01 8.45 71.40 603.35

6.55 8.30 68.89 571.79

7.62 9.50 90.25 857.38

5.65 6.70 44.89 300.76

7.10 9.05 81.90 741.22

5.06 5.45 29.70 161.88

5.25 5.95 35.40 210.65

4.49 5.15 26.52 136.59

7.52 10.05 101.00 1,015.08

8.03 10.30 106.09 1,092.73

6.35 7.75 60.06 465.48

To do the regression analysis of this data, check that Statpro (or other sta�s�cs add-in program) has been added to your Excel so�ware by pulling down the Add-Ins tab to find it. (If it is not there, you will have to do an Internet search and download a copy.) When it is downloaded click on the Statpro name and select Regression. You will then

need to iden�fy which is the dependent variable (TVC) and the independent variables you want to enter into the regression equa�on, that is, Q, Q2 and Q3. Indicate where you want the results to be posted—below or adjacent to the data columns or in a separate worksheet. A�er you have indicated which of the independent variables are to be entered, allow the program to make the calcula�ons. A table showing the results will appear in the chosen area of the spreadsheet. This will include the value of the a

and the various β coefficients, as well as the coefficient of determina�on (R2), the standard error of es�mate (Se) and the standard errors of the coefficients (Sβ) sta�s�cs.

Your results table will look something like Table 6.10.

Table 6.10: Results from the regression analysis to es�mate a cubic TVC func�on Variable Coefficient Std err of coeffic. t-sta�s�c P-value

Intercept α = 2.8318 8.3176 0.3405 0.7423

Output (Q) β1 = 0.0377 3.839 0.0111 0.9914

Output squared (Q2) β2 = 0.0802 0.4462 0.1798 0.8612

Output cubed (Q3) β2 = −0.0035 0.0191 −0.1825 0.8598

Adjusted R2 0.9676

Standard error of es�mate 0.2031

These results indicate a very high adjusted R2 (i.e., adjusted for degrees of freedom), which might seem like a good result, but the rela�vely high standard errors of the coefficients, the rela�vely low t-sta�s�cs, and the very high P-values indicate that the cubic form of the line of best fit does not fit the data very well at all, and that using

this func�on to predict TVC, AVC, and MC values would be poten�ally unreliable.5 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec6.2#footernote5) So we can conclude that the line of best fit is perhaps not a cubic func�on, and that the law of variable propor�ons does not seem to be opera�ng over this range of output levels.

Thus, we need to find a line of best fit that is a be�er fit to the data. To find whether only diminishing returns are evident, we would repeat the regression analysis to es�mate TVC as a quadra�c func�on of output, namely:

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TVC = α + β1Q + β2Q 2 (6-3)

If this quadra�c func�on is an acceptably-reliable line of best fit, this would imply a linear MC func�on of the form:

MC = β1 + 2β2Q (6-4)

Repea�ng the regression analysis, but this �me entering only Q and Q2 as independent variables, we find the results shown in Table 6.11. Again we get a strong R2 result but it is unreliable since the t-sta�s�cs are all too low and the P-values are too high. We note that the 0.0956 P-value for the Output (Q) variable indicates that output is a "marginally significant" determinant of TVC in this form of the TVC equa�on, indica�ng that we could be confident at the 90.44% level that TVC is a func�on of output, but

the unreliability of the other independent variable (Q2) renders the quadra�c es�mate unreliable as well.

Table 6.11: Results from the regression analysis to es�mate a quadra�c TVC func�on Variable Coefficient Std err of coeffic. t-sta�s�c P-value

Intercept α = 1.3354 1.3060 1.0225 0.3332

Output (Q) β1 = 0.6514 0.3499 1.8614 0.0956

Output squared (Q2) β2 = 0.0011 0.0226 −0.0468 0.9637

Adjusted R2 0.9711

Standard error of es�mate 0.1918

So once more we ask Excel to calculate a regression equa�on, this �me using a simple bivariate equa�on of the form:

TVC = α + βQ (6-5)

which, if reliable, would mean that

MC = β (6-6)

The regression results for this simple linear TVC func�on are shown in Table 6.12. At last, this form of the TVC func�on provides a reliable es�mate of the coefficients, and

we can be highly confident (above the 99% level, according to the P-values) that TVC is a simple linear func�on of output. Note that the explanatory power (adjusted R2) is slightly be�er than it was for the other two forms of the regression equa�on, and also that the standard error of es�mate is smaller than it was for the other two forms of

the TVC func�on that we es�mated.6 (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec6.2#footernote6)

Table 6.12: Results from the regression analysis to es�mate a linear TVC func�on Variable Coefficient Std err of coeffic. t-sta�s�c P-value

Intercept α = 1.3953 0.2505 5.5703 0.0002

Output β = 0.6351 0.0313 20.3020 0.0000

Adjusted R2 0.9763

Standard error of es�mate 0.1820

So what does all this mean for the ice cream factory? The regression results in Table 6.12 indicate that the TVC = 1.3953 + 0.6351 Q. That is, the line of best fit to the TVC func�on intersects the ver�cal axis at $1,395.30 and slopes upward at $0.6351 per gallon of ice cream manufactured. Do not be concerned about the posi�ve intercept value for the TVC curve—our data had no output values anywhere near zero—the intercept value simply serves to li� up the TVC curve so it passes through the data points at the correct height. We are more concerned with the slope of the TVC curve in the relevant range of our data observa�ons, which provides our es�mated value for marginal costs, and which we have es�mated to be constant across the range of output values contained in the data at about 63 cents per gallon. Perhaps diminishing returns will later set in (at higher output levels) but they are not evident in the output range represented by the data we have.

So, the manager of the ice cream factory now knows that she can reliably es�mate the marginal cost of ice cream at $0.63 per gallon for any volume of output within the observed data range (i.e., interpola�on) or for rela�vely small extrapola�ons outside the observed data range (i.e., less than 5,150 gallons or more than 10,300 gallons; see Table 6.8). Pricing, make-or-buy, and take-it-or-leave-it decisions can be made based on this es�mate of the marginal cost (which is also the incremental cost of an extra gallon of ice cream in this case because there were no varia�ons in fixed costs associated with the varia�ons in the output levels).

4. In case your math is rusty, we have used the power rule to find the deriva�ve of the TVC curve, because the independent variables included variables that were raised to the power 2 (squared) and 3

(cubed). The power rule says that the deriva�ve of aXb = baXb-1. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec6.2#return4) ] 5. The t-sta�s�cs need to be somewhere close to 2.0 (or larger) to allow us to be confident (at the 95% level of confidence or be�er) that the variable is a sta�s�cally significant determinant of TVC. The P-

values indicate the level of significance for each independent variable; for example, a P-value of 0.05 would indicate that we could be confident at the 95% level—the confidence level is given by 1 minus the P-value. As you can see in Table 6.10, the P-values are way too high to allow us to hold any reasonable level of confidence in this par�cular es�mate of the TVC curve. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec6.2#return5) ]

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6. Another form of the TVC func�on that we might have considered is TVC = a + β2Q 2, which would imply MC = 2β2Q. I did run that regression equa�on to find that while the Q

2 variable is sta�s�cally

significant above the 99% level of confidence, the R2 was marginally lower and the standard error of es�mate was higher, so that the best line of best fit is the linear TVC equa�on given by equa�on 6-5. [return (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/sec6.2#return6) ]

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Summary

In this chapter, we have been concerned with cost es�ma�ons and calcula�ons for decision-making purposes. We began with contribu�on analysis, which required the accurate iden�fica�on of incremental costs and incremental revenues. We applied contribu�on analysis to three types of business decision problems, namely Project A versus Project B decisions, make-or-buy decisions; and take-it-or-leave-it decisions. To maximize the firm’s contribu�on to overheads and profits, the decision maker must consider all costs and revenues that vary as a result of the decision but only those that vary as a result of the decision. When the decision involves a varia�on in the output level, we start with changes in the costs of the variable inputs—how is TVC expected to change as a result of the proposed change in the output level? Then we must consider whether there will be any changes in the fixed costs (also known as overheads) that are a consequence of the decision to be made. If so, these are also incremental costs to be included in the contribu�on analysis. These are the present-period explicit costs of the decision to be made. We must also consider implicit costs such as the opportunity cost of resources u�lized, which may include contribu�on foregone on other sales that cannot be made as a result of the decision; the cost of replacing owned items (to be used because of the decision) in inventory; or the market value of owned items or resources that might have alterna�vely been sold. Future implica�ons of the decision must also be considered. The decision at hand might cause future costs or future revenues to be incurred or received, such as loss or gain of future business due to changing the firm’s rela�onship with exis�ng customers (known as ill will or goodwill, respec�vely); impacts on the rela�onship between management and workers (known as employment rela�ons), which may affect future labor produc�vity; impacts on the business rela�onship with suppliers (i.e., supplier rela�ons); and impacts on future demand due to changes in the actual or perceived quality of the firm’s product.

In the second half of this chapter, we examined techniques to es�mate the shape and placement of cost curves, or, more importantly, to es�mate the values of cost categories at par�cular output rates. We began with simple extrapola�on of known data points, a method that is appropriate for es�ma�ng cost values that are outside the range of our data observa�ons. Extrapola�on means to simply extend the values of the observed data points in the direc�on they seem to be heading. We noted that extrapola�on becomes increasingly more unreliable the further one extrapolates outside the observed data points, due to the law of variable propor�ons (or diminishing returns) causing the extrapola�on to be inaccurate. We next considered interpola�on, or the es�ma�on of data points between known data points. A linear interpola�on is the simplest assump�on unless we have data to suggest a curvilinear interpola�on is more appropriate, such as diminishing returns to the variable inputs being evidenced by rising MC observa�ons. We then considered gradient analysis, which is simply interpola�on between data points when we have several data points. This allows us to more accurately sketch in a line of best fit to the data observa�ons.

With more data points, regression analysis can be used to find the sta�s�cal rela�onship between costs and output levels, but as we saw the choice of func�onal form and the reliability of the results obtained are of paramount importance. Because the law of variable propor�ons is likely to be present in any produc�on process, it makes sense to start the regression analysis of observed total cost (TC) or total variable cost (TVC) data by including squared and cubic output quan�ty terms in the regression equa�on.

Observa�on of the regression sta�s�cs (the adjusted R2, the standard error of es�mate, the standard error of the coefficient, the t-sta�s�cs and the P-values) will allow us to

judge whether the form of the regression equa�on is sufficiently reliable. Regardless of the adjusted R2 value, if any of the independent variables (Q, Q2 and/or Q3) are not significant at the 95% level of confidence (i.e., do not have t-sta�s�cs close to or above 2, or P-values less than 0.05) we cannot be confident that they explain the varia�on in the dependent variable (TVC) and thus, they should not be used in the predic�ve equa�on to es�mate levels of TVC for future levels of output.

If the cubic regression equa�on does not provide a reliable explana�on of the varia�on in TVC then we would revert to a quadra�c regression equa�on, effec�vely assuming that the range of data observed does not include the ini�al increasing returns to the variable inputs but only observes diminishing returns to the variable inputs. Re-running

the regression analysis in the form TVC = α + β1Q + β2Q 2 will provide new es�mates of the regression parameters (β1 and β2) and new regression sta�s�cs to indicate

whether the quadra�c form offers a more reliable es�mate of the rela�onship between TVC and the output level. Again we scru�nize the t-sta�s�cs and/or the P-values to

see which of the independent variables are reliable determinants at the 95% confidence level. If all the independent variables included (Q and Q2) are found to be reliable,

we can stop there, assuming the adjusted R2 is sufficiently high (say, above 0.7). If either of those independent variables is not significant at the 95% level, it behooves us to check for a simple linear rela�onship between TVC and Q (as we did in this chapter), and if we find this to be the most reliable explanatory equa�on then this is the form we should use for predic�ng future values of TVC for proposed output levels.

Chapters 5 and 6 have been concerned with the cost side of the firm’s opera�ons, just as Chapters 3 and 4 were concerned with the demand (or revenue) side of the firm’s business. As you know, profit is the excess of revenues over costs. In the following two chapters, we will u�lize the concepts learned in the preceding chapters to consider the firm’s pricing decision on the presump�on that the firm’s objec�ve is to maximize profit.

Ques�ons for Review and Discussion

Click on each ques�on to reveal the answer.

1. List out all the categories of incremental cost that you can recollect from your reading of this chapter and provide examples of each one. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

Incremental costs may be present-period explicit costs, future-period explicit costs, or implicit costs (opportunity costs) in the present or future periods. For example, if I were to buy a new car, I might incur explicit costs of $30,000 today, $1,000 a year for scheduled maintenance services, and $2,000 a year for fuel. Implicit costs would include loss of salvage value (due to deprecia�on), of say $5000 the first year, $2500 the second year, $1250 the third year, and so on. Another implicit cost is the opportunity cost of the foregone interest on my $30,000 which could have earned (say) 5% interest, compounding annually.

2. Accountants are concerned with historical costs of resources and the alloca�on of some of these against current period revenues and consequently derive an accoun�ng measure of the firm’s profit. The economic profit of the firm is likely to differ from this. Please explain. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

Generally Agreed Accoun�ng Principles require that the historical cost of assets should be charged against revenues in subsequent years according to a straight-line or reducing balance deprecia�on method. This is necessary to allow investors to compare the profitability of different firms before making their investment decisions. The economic profit might be more or less than the accoun�ng profit depending on the difference between the deprecia�on charge and the actual salvage value of the asset (in the market for used equipment, for example). Economic profit would also consider other implicit and future costs that the accoun�ng conven�ons would ignore (but investors should consider).

3. Why does contribu�on analysis ignore the fixed overhead costs that financial accountants would want to include in the full cost of the firm’s product? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

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Contribu�on analysis is concerned with the es�ma�on of incremental economic costs and revenues to determine the change in profits due to a specific decision to be made. It essen�ally asks whether each decision will increase profit (or not) given that the firm has previously invested in par�cular assets and other resources. Accoun�ng profits are found a�er considering all explicit costs and revenues across all decisions over a specific produc�on period and seek to measure the return on the firm's investment for comparison with other investment opportuni�es.

4. How should future costs and revenues be included in the calcula�on of the contribu�on of a decision? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

Future costs and revenues that are incremental to the decision should be included in expected net present value (ENPV) terms in order to weight these future period and uncertain cash flows appropriately so they can be added to and compared with present-period certain costs.

5. Under what circumstances would a manager make a decision that ignores the future cost and revenue implica�ons of that decision? (There are many reasons so your thinking may range widely on this one.) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

If the chances of a future event were judged to be extremely small, and thus the ENPV of that event is expected to be trivial, the manager might ra�onally ignore that possible event rather than spend funds on search costs. Also, if the firm is expected to be bankrupt, or the manager expects to be re�red, promoted, or working elsewhere in the future, the manager might ignore the future cost and revenue implica�ons of a decision, focusing instead on the near-term consequences.

6. When is extrapola�on a sa�sfactory method of cost es�ma�on and when is it not? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

Extrapola�on is safe enough—i.e., the margin for error is rela�vely small—if the distance between the known data point and the es�mated data point is rela�vely small. The greater this distance, the greater the probability that the assumed rela�onship between the dependent variable and the independent variable(s) will not hold true.

7. Gradient analysis interpolates between known data points. This interpola�on may be linear or curvilinear. How do we know when we should fit a curvilinear line of best fit to the gradient data points? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

Linear interpola�on between a series of pairs of data points would result in a series of straight lines, joining with kinks at the known data points (if the overall rela�onship is not exactly linear). Such abrupt changes in the rela�onship between the dependent variable and the independent variable(s) are not at all likely, due to the con�nuous rela�onship (and the smooth rate of change) we expect between input and output rela�onships. Thus, rather than use straight lines that result in kinks, we bend the lines to depict a smoothly changing rela�onship between the variables.

8. Regression analysis of cost data does not interpolate between known data points—instead it es�mates a line of best fit to the observed data points, allowing for poten�al devia�ons from the line of best fit. Please explain. (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

Whereas interpola�on joins the data points, assuming them to be error free, regression analysis assumes that the data points may each contain random error and looks for a single line of best fit across all data points that minimizes the sum of the squared devia�ons (of the actual data point from the line of best fit) for the en�re data set.

9. How do we know that the func�onal form of the regression equa�on (i.e., a linear, quadra�c, or cubic func�on) is the best form of the regression equa�on for predic�ng cost levels at future output levels? (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

The func�onal form (linear, quadra�c or cubic) that best fits the data set is the one that has the highest R2 and each independent variable significant at the 95% (or higher) confidence level (as evidenced by the t-sta�s�cs having a value around 2 or greater) or the p-value being 0.05 or lower, which should be accompanied by rela�vely low standard errors of es�mate and standard errors of the coefficients.

10. When the regression equa�on predicts an es�mated value of TVC at a par�cular level of Q, how do we calculate the 95% confidence interval around that predicted value of TVC? (You may need to refer to Chapter 4 to refresh your memory about the standard error of es�mate.) (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/

The predicted value of TVC for any value of Q is obtained by inser�ng that value of Q into the regression equa�on (including Q2 and Q3 if the regression equa�on is quadra�c or cubic) and solving for TVC. Given that the sample data probably contains error terms, we find the 95% confidence interval by adding and subtrac�ng twice the standard error of es�mate (to and from the predicted value of TVC) to indicate a range of values into which the actual value of TVC is likely to fall 95% of the �me.

Decision Problems

1. The Muscle-Man Company (MMC) manufactures and assembles forkli� tractors and supplies parts to other forkli� manufacturers. It fabricates most of the component parts but buys the engines, hydraulic systems, wheels, and �res from suppliers. Demand es�mates indicate that MMC should increase produc�on level from 60 units to 70 units monthly. Sufficient capacity exists in most departments to allow this increase, except that produc�on of 10 extra chassis assemblies could be a�ained only by realloca�ng labor and equipment from fork assembly to chassis assembly. The fork assembly department currently produces 90 units monthly and supplies the extra 30 units to other forkli� manufacturers at $1,880 each. This department could only produce 10 more fork assemblies if the remainder of its labor and equipment is to be reallocated to build the extra 10 chassis assemblies, so the sale of fork assemblies to other manufacturers must be forgone. Alterna�vely, the extra 10 chassis could be purchased from a supplier, and the lowest quote is from Fenton Fabricators, for $3,050 per unit. The costs for the Chassis and Fork departments for a representa�ve month are as follows:

Costs Chassis department Fork department

Produc�on level Direct materials Direct labor Deprecia�on Allocated overheads (200% of direct labor) Total

60 46,500 63,000 7,500

126,000 243,500

90 20,700 40,500 5,000

81,000 147,200

a. Should MMC make or buy the 10 addi�onal chassis assemblies? b. What qualifica�ons would you add to your decision?

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2. The Rakita Racquets Company restrings tennis racquets, a business with highly seasonal demand. Given this seasonality, Rakita tries to keep its overheads low and uses largely casual labor. The owner-manager has kept a record over the past 12 months, as shown in the following table. During that �me the costs of casual labor and of other variable inputs (stringing materials, energy, and packaging) have remained constant, and because of the con�nual turnover of casual labor the produc�vity of labor has also remained more or less constant.

Month TVC ($) Racquets restrung

June July

August September

October November December January February March April May

35,490 42,470 48,980 52,530 37,480 33,510 31,850 27,860 22,160 19,520 25,960 32,980

4,500 5,575 6,300 6,525 5,325 4,050 2,850 2,450 1,525 925 1,925 3,500

a. Derive average variable cost (AVC) data from the data in this table. b. Use gradient analysis to provide an es�mate of 11 data points that seem to represent the MC curve over this range of outputs. Plot these data points and sketch in

es�mated MC and AVC curves that seem to best fit these data points. c. Suppose that demand is es�mated to move from its present (May) level of 3,500 units to 5,000 units next month (June). What is the incremental cost of mee�ng this

demand? d. Assuming that Rakita’s price to restring a racquet has been constant at $15 over the past year, and will remain at that level, what contribu�on to overheads and profit

can it expect in June?

3. The Tico Taco Company has es�mated its weekly TVC func�on from data collected over the past several months, as TVC = 435.85 – 1.835Q2 + 3.658Q3 where TVC represents thousands of dollars and Q represents thousands of boxes of tacos produced per week. The company is currently producing 2,000 boxes weekly and is considering expanding its output to 2,200 boxes weekly. To do this, it will have to hire another taco machine operator ($400 per week) and lease another taco machine ($200 per week).

a. Derive an expression for the marginal cost (MC) curve. b. Es�mate the incremental costs of the extra 200 boxes per week. c. Should Tico Taco expand its output? Why or why not? State all assump�ons and qualifica�ons which underlie your recommenda�on.

4. Scruples Footwear Design is a bou�que manufacturer of designer loafer shoes. The TVC func�on has been es�mated as TVC = 20Q + 0.00782Q2 and the demand func�on has been es�mated as Q = 1,346.55 – 27.495P where Q represents pairs of shoes and P is the price Scruples receives per pair of shoes. The coefficients of determina�on for these two regression equa�ons were 0.9638 and 0.9422, respec�vely. The standard error of es�mate was 286.22 for the cost func�on and 30.967 for the demand func�on. Its current price is $32.50 per pair (wholesale price) and it has been producing well below full capacity output levels, and its inventory levels are at the desired level of 100 pairs.

Today the purchasing agent of a high-class chain store has asked for a special deal for what would be Scruples’ largest single order ever, namely 400 pairs of shoes. This represents a large opportunity for Scruples, since this order would allow its shoes to reach a na�onal market and would most likely cause substan�al growth of sales. The purchasing agent has offered only $28 per pair, however, and says "Take it or leave it!"

a. From the es�mated cost func�on, and given that fixed costs are $2,000 per week, calculate and plot the per unit cost curves that Scruples faces. b. What are the profit-maximizing price and output levels for Scruples shoes, in the absence of the deal offered by the chain store? c. What is the contribu�on from the chain-store deal, presuming that this deal is over and above the profit-maximizing price and output level? d. What do you recommend Scruples do, with respect to the proposed price change and the chain-store deal? e. What assump�ons and qualifica�ons underlie your recommenda�ons?

5. Over the past 12 months the Four Winds Novelty Company firm has recorded its Internet sales (equals its monthly output levels) and its monthly total variable costs (TVC) for a par�cular novelty item as shown in the following table. Sales have grown over this period with rela�vely few shocks due to uncontrollable weather, poli�cal and spor�ng events. This online retailer carries no inventories; when it receives a pre-paid online order from a customer, it simply buys the product from a supplier and ships it out to the customer.

Sales = Output TVC ($)

102,813 176,163 196,121 222,885 226,356 296,416 378,446 450,666 579,696 607,082 624,680 636,133

201,953 340,608 377,940 432,863 441,714 629,267 867,596

1,103,807 1,701,125 1,917,861 2,195,352 2,479,195

a. Using regression analysis, find an equa�on that best fits the data to represent the TVC func�on. b. At what sales/output level will average variable costs (AVC) reach a minimum? c. At what sales/output level will marginal costs (MC) reach a minimum? d. Es�mate the value of TVC for sales/output level 250,000 units and calculate the 95% confidence interval for your es�mate.

Key Terms

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Click on each key term to see the defini�on.

cannibalize exis�ng sales (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A situa�on where sales of a product, via a new distribu�on channel or new retail outlet, will replace or eat into the sales of the product through the pre-exis�ng channels and retailers.

contribu�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The excess of incremental revenues over incremental costs, rela�ng to a par�cular decision, is called the contribu�on because it contributes to pay for the firm’s fixed and unavoidable costs and also to profits if total revenues are more than total costs.

contribu�on analysis (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A process of assessing the incremental costs and incremental revenues associated with a decision to determine whether the la�er will exceed the former and thus whether the decision should in fact be made by a profit-maximizing firm.

es�ma�on of cost curves (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A process of es�ma�ng the values of costs, in par�cular categories of costs, at various output rates.

extrapola�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

To es�mate a data value (e.g., a cost level) that lies outside the range of previous data observa�ons by projec�ng, or extending, the rela�onship observed within the range of data points to higher or lower level of the independent variable (e.g., output).

future costs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The costs a firm might expect to incur in one or more future produc�on periods as a result of a decision made in the present or prior produc�on periods.

gradient (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A measure of the steepness (slope) between two points on a cost curve, calculated by the ra�o of the rise (increase in cost) over the run (increase in output level).

gradient analysis (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A process that involves calcula�on of the gradients by interpola�ng between sequen�al pairs of data points.

incremental revenues (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The change in total revenue that results from a par�cular decision.

interpola�on (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A process of assigning es�mated values to unknown points between two separate data points for which data is known.

irrelevant costs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A cost that is not relevant to a decision that is about to be made because it is not incremental to that decision, such as a sunk cost or an unavoidable cost.

labor rela�ons (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The state of the rela�onship that exists between the management of a firm and the other employees of the firm. Deteriora�on of this rela�onship may reduce the willingness of employees to raise or maintain their produc�vity in the produc�on process.

longer-term supplier rela�ons (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

The state of the rela�onship between the firm and its suppliers over the longer term. Deteriora�on of this rela�onship may cause suppliers to be unwilling to offer be�er deals, rapid delivery, or other discre�onary services.

present-period explicit costs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

Actual outlays of cash in the present produc�on period to pay for the variable and fixed inputs that are required to implement the decision that is made.

P-values (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

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Indicate the probability that the rela�onship between the dependent variable and one of the independent variables, as indicated by the regression equa�on, is not true. Thus a P-value of 0.5 indicates we can be confident at the 95% confidence level that the coefficient to an independent variable in the sample is a reliable es�mate of the true rela�onship in the popula�on as a whole.

relevant costs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A cost that is relevant to a decision to be made, and only incremental costs are relevant to the decision to be made.

sensi�vity analysis (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

An analysis of the degree to which the assump�ons underlying a decision to be made might be incorrect without this causing the decision to be the wrong one.

t-sta�s�cs (h�p://content.thuzelearning.com/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�ons/fm/books/AUBUS640.12.1/sec�o

A measure of the reliability of the coefficient to an independent variable in a regression equa�on. The t-sta�s�cs need to be about 2.0 or be�er to allow confidence at the 95% level of confidence that the variable is a sta�s�cally significant determinant of the dependent variable.