BUS 640 Week 3 Discussion Responses NEEDED

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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.

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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-

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

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

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

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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.

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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.

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?

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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" 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

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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

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

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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.

© iStockphoto/Thinkstock

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

  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.

© iStockphoto/Thinkstock

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 + β2Q2 + β3Q3 (6-1)

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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)

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

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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:

TVC = α + β1Q + β2Q2 (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

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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) ]

6. Another form of the TVC func�on that we might have considered is TVC = a + β2Q2, which would imply MC = 2β2Q. I did run that regression equa�on to find that while the Q2 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 + β2Q2 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/boo

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.

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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/boo

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/boo

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/boo

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/boo

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/boo

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/boo

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/boo

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/boo

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/boo

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

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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?

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?

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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

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/AU

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/AU

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/AU

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/AU

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/AU

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/AU

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/AU

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).

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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/AU

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/AU

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/AU

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/AU

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/AU

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/AU

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/AU

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/AU

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/AU

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/AU

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/AU

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.