final paper man econ

6

Managerial Cost Analysis and Estimation

Learning Objectives

After reading this chapter, you should be able to:

• Distinguish between incremental costs and nonincremental costs for the purposes of managerial decision making.

• Identify incremental revenues as distinct from nonincremental revenues.

• Conduct contribution 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.

• Estimate unknown cost values for particular output levels using known cost data from other output levels using extrapolation, interpolation, and gradient analysis techniques.

• Use regression analysis to find which form of the cost function (linear, quadratic, or cubic) provides the line of best fit to the data and thus the most reliable cost estimates for managerial decision making.

©Peter Scholey/Getty Images

dou70192_06_c06_175-202.indd 175 11/1/12 4:26 PM

CHAPTER 6Section 6.1 Contribution Analysis

Introduction

In this chapter, we continue with the general topic of cost analysis for managers and will introduce several cost estimation and forecasting 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 relating to any particular decision that is expected to generate revenues, this will allow the decision to be profit maximizing (or loss minimizing). In this chapter, we continue our focus on cost minimization (for any given output and quality level) and will turn our attention to the profit-maximizing issues in the following chapter.

In the first half of this chapter, we will focus on “contribution analysis” whereby deci- sions are evaluated based on the financial contribution they make to the firm’s overheads and profits after covering the variable costs associated with those decisions. Contribution analysis allows managers to make choices among competing alternatives—their decision choices—such that the alternative chosen is the one that contributes most to the firm’s overheads and profits. Thus, contribution analysis is concerned with accurately estimat- ing the incremental costs and incremental revenues of each decision alternative such that the best (profit-maximizing) option can be chosen.

In the second half of this chapter, we will explore the estimation of cost curves. We begin with some simple techniques and continue on to apply the regression analysis technique that was introduced in Chapter 4. Estimating cost values for the current period involves collecting 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 project- ing that/those values forward or backward to predict the TVC, AVC, and MC levels for any particular output level in the current production period.

6.1 Contribution Analysis

The contribution of a decision is defined as the excess of incremental revenues over incremental costs, and it is called the contribution because it contributes to the firm’s fixed and unavoidable costs, and also to profits if total revenues are more than total costs. Contribution 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 conse- quential to a decision made, while incremental revenues are the revenues that are con- sequential 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 contribution made towards fixed costs and profit is $4 per unit. This differ- ence between incremental costs (which equal AVC in this case), and incremental revenue (which equals price in this case), is known as the contribution 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

dou70192_06_c06_175-202.indd 176 11/1/12 4:26 PM

CHAPTER 6Section 6.1 Contribution Analysis

not equal to price, as we shall see in the next chapter. More- over, decisions made by man- agers usually cause both fixed and variable costs to change, since the implementation of a new decision (e.g., to expand production) may cause not only extra direct labor and direct material costs but also require the purchase of additional capi- tal assets and the employment of salaried managers and other workers who must be treated as fixed costs since their sala- ries must be paid regardless of output levels. Thus, incremental costs are defined as the change in total costs that result from a particular decision.

But isn’t that the definition 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 production method that will allow produc- tion of the same output level at a lower AVC level.

Incremental costs must be accurately identified for good decision making. All costs that change must be included, and costs that do not change must not be included. For exam- ple, capital assets that have been idle with no alternative 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 alternative product, and the decision at hand would require them to be used elsewhere, we have to include the foregone contribu- tion 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 utilizing 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 calculation of incremental costs. However, if the truck would have been used to carry groceries to earn $200 during that time, 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 often called relevant costs since they are the costs that are relevant to the decision that is to be made, as distinct 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.

©iStockphoto/Thinkstock

Fixed and variable costs are incremental to decisions made by managers. Incremental costs represent the change in total costs resulting from a particular decision.

dou70192_06_c06_175-202.indd 177 11/1/12 4:26 PM

CHAPTER 6Section 6.1 Contribution Analysis

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 deci- sion to be made.

The second category of incremental costs is opportunity costs. Raw materials or compo- nents 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. Alternatively, if an item in inventory has little 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 implications for future costs, such as repairs and maintenance to equipment, vehicles, or other capital assets that will be necessitated as a result of their utilization 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 sum- mary of the various costs that are relevant or irrelevant for managerial decision making. Note that by relevant and irrel- evant we mean with respect to the decision to be made. If a cost is a consequence of the deci- sion to be made, it is a relevant or incremental cost. Some cur- rent period and future costs are irrelevant because the firm is committed 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.

©Brand X Pictures/Thinkstock

Future costs of repairs and maintenance must be taken into consideration when purchasing equipment.

dou70192_06_c06_175-202.indd 178 11/1/12 4:26 PM

CHAPTER 6Section 6.1 Contribution Analysis

Table 6.1: Summary of cost concepts for decision making

Relevant costs 5 incremental costs Irrelevant costs 5 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 Contribution foregone on the best alternative 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 depreciation 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 deci- sions 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 pro- ceed to work through some realistic business examples using contribution analysis and will examine three main types of contribution analysis with these suggestive names: Proj- ect A versus Project B, make-or-buy decisions, and take-it-or-leave-it decisions.

Project A Versus Project B Decisions Managers often have to choose between two or more projects that are both potentially profitable because they do not have the productive capacity or the funding to handle both projects at the same time. 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 potential projects that were available for implementation at that time. The appropriate method for choosing between competing projects is contribution analysis—profits will be maximized by choosing the project that contributes the most towards overheads and profits.

Suppose a firm is considering implementation 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, vari- able 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

dou70192_06_c06_175-202.indd 179 11/1/12 4:26 PM

CHAPTER 6Section 6.1 Contribution Analysis

costs, and, thus, apparently makes a profit of $4,000. It might seem that Project B is supe- rior 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 $20,000 Costs Materials $2,000 Direct labor 6,000 Variable overhead 4,000 Allocated overhead 6,000 Total costs $18,000 Profit $2,000

Revenues $18,000 Costs Materials $5,000 Direct labor 3,000 Variable overhead 3,000 Allocated overhead 3,000 Total costs $14,000 Profit $4,000

When contribution analysis is applied to the above choice situation the result may be surprising. In Table 6.3, we show only the incremental costs and revenues and see that the contribution 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 contribution to overheads and profit and should be chosen for implementation by the profit-maximizing firm.

Table 6.3: Contribution analysis for Project A and Project B decision

Project A Project B

Incremental revenues $20,000 Incremental costs Materials $2,000 Direct labor 6,000 Variable overhead 4,000 Incremental costs $12,000 Contribution $8,000

Incremental revenues $18,000 Incremental costs Materials $5,000 Direct labor 3,000 Variable overhead 3,000 Incremental costs $11,000 Contribution $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 allocation rule of “100% of direct labor costs.” Simple rules like that almost certainly do not correctly reflect the relevant costs of undertaking any project. Contribution analysis allows an inci- sive look at the actual changes in the costs and revenues that would follow the decision to choose one project over the other.

dou70192_06_c06_175-202.indd 180 11/1/12 4:26 PM

CHAPTER 6Section 6.1 Contribution Analysis

Also, note that in this simple example we implicitly assumed there were no opportunity costs or opportunity revenues1 and no future costs or revenues associated with either project. In this case, there is only $1,000 difference in the contributions of the two proj- ects, so the decision to implement Project A is very sensitive to the assumption of zero opportunity and future costs and revenues. This calls for sensitivity analysis, an analysis of the impact of inaccurate data on the desirability of the decision made. A decision is called sensitive to the assumptions on which it is based if a relatively small change in those assumptions would cause a different decision to be made. It is useful to express the degree of sensitivity in terms of the percentage variation 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 proportion (or percentage) by which the incremental costs of Project A could increase without reducing its contribution to that of the next-best alternative (Project B). In this case it is $1,000/$12,000 5 8.33%, assuming the costs of Project B are calculated accurately. In practice, decision makers must be careful to assess whether there are any opportunity or future implications of their decision. If they cannot easily estimate these additional costs they should consider, if such additional costs are thought to exist, whether they are likely to be more than 8.33% (in this case) higher than the initial estimate. Where the sensitivity percent- age is relatively high, for example 40%, we say the decision is relatively insensitive to the accuracy of the cost estimates of incremental costs, unless the costs involved are likely to be highly volatile (e.g., the pump price of gasoline).

Make-or-Buy Decisions The next category of decision that involves incre- mental 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 main- tenance work (using current employees and pur- chasing the necessary equipment and supplies) and compare the incremental cost of this with the alternative solution of having an outside firm supply the maintenance and cleaning work.

In Table 6.4, we consider the make-or-buy prob- lem facing Wilson Tools. Wilson Tools manu- factures high-quality power tools such as drills, jigsaws, and sanders. All these tools require the

1. Just as an opportunity cost is a revenue (or in this case a contribution) that must be foregone if a decision is taken, an opportunity revenue is a cost (or a negative contribution) that is avoided if a decision is taken.

©Thinkstock Images/Thinkstock

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.

dou70192_06_c06_175-202.indd 181 11/1/12 4:26 PM

CHAPTER 6Section 6.1 Contribution Analysis

same roller-bearing unit, which the company manufactures in its own bearing depart- ment. 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 additional 7,500 units a month by supplying its tools to a chain of hardware stores in another state. Wilson could produce the additional 7,500 bearings in its bearing depart- ment, but this additional output would congest operations somewhat, so management is considering having the additional roller-bearing units supplied by a specialist bearing manufacturer. It is estimated that it will require an additional 15% in direct labor costs and an additional 12% in total materials costs to make the bearings in-house. No additional 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 specifications and submitted 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 additional 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 estimated incremental cost of direct labor is 15% of $126,390, or $18,959 per month. The estimated 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 produc- ing 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 deci- sion 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 administration 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 assumptions, the manager should first apply sensitivity analysis to the assumption 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 variation 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

dou70192_06_c06_175-202.indd 182 11/1/12 4:26 PM

CHAPTER 6Section 6.1 Contribution Analysis

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 Considerations

Quality. A number of other considerations 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 comparative 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 reduction in the market reputation enjoyed by Wilson Tools. On the other hand, if the quality of the specialist manufacturer’s bearing was significantly better, 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 relations. Wilson Tools may be nearing its full capacity output level and, if demand continues to grow, may need to expand its plant in the near future. An alternative strategy would be to establish a relationship with a specialist supplier of bearings to allow it to meet demand for its power tools in the future. The “buy” alternative gives it the opportunity to both test out a potential longer term supplier and to start building a mutually beneficial long-term supply relationship or work towards a potential strategic alliance that might be desirable in the future.

Labor relations. 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 satisfaction. As a consequence, worker productivity (marginal and average product) may fall resulting in a rise in average variable and marginal costs per unit of output, thus, potentially nullifying the cost advantage of the “make” option.

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

Of course, these additional considerations may be difficult (or costly) to quantify. Thus, the manager will need to exercise judgment and, perhaps, make a calculated gamble to decide whether to make or buy, particularly when the incremental costs of both options are relatively close together.

Take-It-or-Leave-It Decisions In other situations, the manager might be faced with an offer that is non-negotiable and must decide whether to accept or decline that offer. For example, a prospective buyer might offer a fixed sum of money for particular capital asset, such as land, buildings, or piece of equipment owned by the firm, or for a particular quantity 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 potential customer might say, “I can get this (e.g., car) for $X from another supplier. If you can beat

dou70192_06_c06_175-202.indd 183 11/1/12 4:26 PM

CHAPTER 6Section 6.1 Contribution Analysis

that price you have a deal.” The manager’s task is to evaluate the contribution of the offer and compare it with the status quo—if the deal offers a positive contribution to overheads and profits it will be profit-maximizing to take the offer since it will contribute additional 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 positioning satellite) devices that allow pedestrians and drivers to navigate unfamiliar streets or highways and to find their way to specific destinations. Normally, the com- pany 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 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- tion level of the X1 model is nearly at full capacity at 160,000 units, but he could supply the additional 20,000 units by foregoing production and sale of 5,000 units of the more sophisticated and more expensive X2 model. Pertinent data relating to these two models is shown in Table 6.5. Because of the automated production 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, particularly 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 cus- tomer (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 negotiate for a better 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

2. The difference between the distribution price and the wholesale price provides the contribution

margin for the distributor, and the difference between the wholesale price and the retail price provides the contribution 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.

dou70192_06_c06_175-202.indd 184 11/1/12 4:26 PM

CHAPTER 6Section 6.1 Contribution Analysis

Because AVC is expected to be constant over a wide range of output levels for both prod- ucts, 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 over- heads. Note that Idaho Instruments appears to have a very simple rule for the allocation 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 nor- mal 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 contribution per unit to overheads and profit (i.e., the contribution 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 1 $10 5 $35.

So, to produce 20,000 more units of model X1 will cause an incremental cost of 20,000 times 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 production 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 times the $25 AVC of the X2 model, that is $125,000, but nei- ther will the firm receive the foregone sales revenue of 5,000 times the $60 price, that is $300,000. Thus, taking the deal will also cause a net reduction of $175,000 (i.e., $300,000 minus $125,000) in the contribution of the X2 model towards the overheads and profits of the company. Alternatively, and more quickly, we could multiply X2’s contribution mar- gin ($35) by 5,000 units to find the $175,000 figure for the contribution foregone if the sales manager decides to take the deal. Similarly, the foregone contribution from the 4,000 units of the X1 model that is expected to be lost is 4,000 times its contribution 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: Contribution analysis of the take-it-or-leave-it offer

Incremental revenue Sale of 20,000 units of X1 at $40 each $800,000

Incremental costs Variable costs of 20,000 units of X1 at $20.83 each $416,667 Foregone contribution of 5,000 units of X2 at $35.00 each $175,000 Foregone contribution of 4,000 units of X1 at $29.17 each $116,667 Total incremental costs $708,333 Contribution to overheads and profit $ 91,667

You can see that there will be a net contribution of $91,667 to the overheads and profit of Idaho Instruments if the sales manager takes the deal—despite the fact that initially 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 contribution from 4,000 units of the X1 model. But this demon- strates the beauty of contribution analysis: It cuts through arbitrary overhead cost allo- cations and pricing rules to focus only on what costs and revenues actually change as a result of making a particular decision.

dou70192_06_c06_175-202.indd 185 11/1/12 4:26 PM

CHAPTER 6Section 6.1 Contribution Analysis

Other Considerations

The preceding analysis is subject to some simplifying assumptions of course, and the sales manager must consider these before getting back to the purchasing manager. The first issue is the sales manager’s assumption that 4,000 units of the X1 model that will be placed directly into this retail store will simply replace or cannibalize existing sales, that is, will be sold via this new retailer instead of through Idaho’s regular distribution chan- nel and by the existing 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 distribution channels. Applying sensitivity analysis to his assumption he could make a few simple calculations and find that the critical ratio 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 continuing to supply the market through the regular distribution chan- nels, and so, the decision should be reversed.3 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 little 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 potential new customers who would potentially 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 proportion of this 20,000-unit deal that will simply cannibalize existing sales and make the decision accordingly.

The second main issue is the relationship with the distribu- tor and the existing retailers. Undoubtedly, the distributor and the existing retailers will become aware that a new store has entered the market to sell Idaho’s X1 GPS devices. Hav- ing 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 attract cus- tomers away from the existing marketing channel. Losing sales (and share of the market) will possibly cause the distributor and existing retailers to become

3. That is, 7,000 units x $29.17 means $204,167 additional opportunity cost that nearly wipes out the $208,333 contribution to be made by the deal if there is no replacement of existing sales.

©iStockphoto/Thinkstock

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

dou70192_06_c06_175-202.indd 186 11/1/12 4:26 PM

CHAPTER 6Section 6.2 Cost Estimation Methods

unhappy with what they might feel is unfair competition, and they may take their busi- ness elsewhere in the future. This would cause a future reduction in demand through the regular marketing channels and potential 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 conservative assumption 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 contributions must be considered by the sales manager before making the deci- sion. Of course, it would be prudent for the sales manager to set the purchasing agent’s expectations at an appropriate level by stating that this is to be understood as a once-off deal to kick start a business relationship and that future dealings would be expected to allow Idaho a better profit margin.

In Chapters 1 and 2, we discussed decisions that had cost and revenue outcomes over multiple periods into the future, and we saw that we needed to express those future out- comes in expected present value terms. In those chapters, we spoke of profits in the first and subsequent time periods that had to be discounted back to present value. We now know that it is the contribution, rather than an accounting measure of profits (which might reflect an inaccurate allocation of overhead costs), that is important for managerial decision making.

6.2 Cost Estimation Methods

In this section, we will examine several methods for estimating the level of costs per unit based on data collected from the firm’s prior production experience. In the short-run context of business decision making, we are primarily concerned with the behav- ior of variable costs, but we also know that changes in a fixed cost category might be necessitated by a particular decision. We start with simple extrapolation and later proceed to more complex, but probably more accurate, measures.

Extrapolation of Prior Observations Extrapolation means to impute values to a variable outside the range of previous data observations. Extrapolation is achieved by projecting (or extending) the relationship that is identified between the output level and the cost level inside the range of data observa- tions 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 assumption that it will remain constant at that level for relatively small changes in output levels that are both higher and lower than the range of observations for which we have prior data. In the left-hand side in Figure 6.1, we show a situation where prior data indicates that AVC was about $4 at production rates of both 3,000 and 4,000 units of output. These observations are indicated by the stars. The broken

dou70192_06_c06_175-202.indd 187 11/1/12 4:26 PM

CHAPTER 6Section 6.2 Cost Estimation Methods

lines to the left and to the right of the observed data points indicate our extrapolation of AVC to both higher and lower levels of output.

Another example of extrapolation is shown on the right-hand side in Figure 6.1. Suppose we have (in a different production 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. Accord- ingly, we can extrapolate this data to estimate that MC is likely to increase by another dol- lar to $6 per unit if output is increased to 5,000 per day.

Figure 6.1: Examples of cost extrapolation given prior cost observations

$/Q

4

0 3 4

AVC

MC

Q/t (‘000s) Q/t (‘000s) 0 3 4 5

4

5

6

$/Q

Notice that the interrelationships of the cost concepts mean that, if we have data on AVC at particular 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 DTVC/DQ, 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 subtraction 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 production experience will allow the manager to make more accurate estimates 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 extrapolation means making estimates outside the data range, interpolation means making estimates 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 situations it will be clear the relationship cannot be a linear one but must be curvilinear. We saw in Chapter 5 that the law of variable proportions, also known as the law of diminishing returns, will cause cost curves to bend in predictable ways. Suppose we have a produc- tion situation 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,

dou70192_06_c06_175-202.indd 188 11/1/12 4:26 PM

CHAPTER 6Section 6.2 Cost Estimation Methods

and a second data collection at output rate 3,800 measured TVC as $15,200 and deduced that AVC was still $4. After the second data collection, however, MC was estimated to be DTVC/DQ 5 $6,230/100 5 $6.23 by calculating the costs of direct materials, direct labor, and variable overheads for the last batch of 100 units of output produced. This obser- vation, that MC is substantially 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 still 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- tions to interpolate between the known data points and show curved lines representing our estimates of the AVC and MC values between the known data points.

Figure 6.2: Interpolation of cost data between known data points

$/Q

AVC

MC

Q/t 0 32

$6.23

$4.00

1 64 5

As with extrapolation, interpolation provides a best estimate 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, utilizing other sources of information perhaps, to sketch in the AVC and MC curves. But the most important message to the man- ager from this cost estimation 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 still be falling at the higher output rate. The manager must pay close attention to this issue to decide whether higher output rates should be avoided or whether considerations should be given to installing a larger size of plant if economies of scale are indeed available.

dou70192_06_c06_175-202.indd 189 11/14/12 4:08 PM

CHAPTER 6Section 6.2 Cost Estimation Methods

Gradient Analysis Using Known Data Points A gradient is the slope at which the vertical elevation 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 relatively small DQ). More commonly in cost estimation we find that output data will not be available as continuous data (i.e., not available for one-unit increments of output), but that data will be collected periodically at different times 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) interpolation between multiple data points.

Suppose weekly data has been collected for total fixed and variable costs for various out- put 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 matter 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 estimate 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 estimate 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 estimate MC at four output rates by evaluating the ratio DTVC/DQ for each output interval, where D (as usual) symbolizes a discrete change in the variable concerned.

Table 6.7: Gradient analysis to estimate marginal cost levels

Production period

Output rate (Q)

TVC ($)

AVC ($/Q)

DTVC ($)

DQ (Q)

MC5DTVC/ DQ ($/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

Notice that the four estimates of MC in Table 6.7 are shown in the middle of the intervals between the five observations. This is more evident in Figure 6.3 where we show the AVC data points and the estimated MC curve as the broken line joining the four gradient values that were calculated in Table 6.7. Note that the location of the MC curve is more reliable in this case, compared with our earlier interpolation exercise where the curvature of the AVC and MC curves was chosen arbitrarily. By placing the estimated MC value in the middle of the output interval, we can gain a more accurate estimate of the MC values for all output rates between the known data points.

dou70192_06_c06_175-202.indd 190 11/1/12 4:26 PM

CHAPTER 6Section 6.2 Cost Estimation Methods

Figure 6.3: Gradient estimation of the marginal cost curve from known TVC data

$/Q

$6

AVC

MC

Q/t 0 321 6 874 5

Cost Estimation Using Regression Analysis Note that for both interpolation and gradient analysis, we have essentially sketched in a line of best fit to join the calculated or estimated data points. As we saw in Chapter 4, we can utilize regression analysis to find the line of best fit to data points, and this is especially useful when we have a larger number of out- put and cost observations and it is not so easy to see the shape of the relationship between out- put and costs. Suppose we have 12 weeks of data on the weekly output and total variable costs of an ice cream factory, as pre- sented in Table 6.8.

©iStockphoto/Thinkstock

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

dou70192_06_c06_175-202.indd 191 11/1/12 4:26 PM

CHAPTER 6Section 6.2 Cost Estimation Methods

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

Notice 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 positively related, but is this positive relationship a linear relationship (implying constant MC) or a curvilinear relationship (implying falling and/or rising MC)? Knowing what we know about the law of variable proportions, namely that for equal increments of the vari- able inputs the output level will increase first at an increasing rate and later at a decreas- ing rate, our default assumption ought to be that the line of best fit to the TVC data is most likely to be a cubic function, taking the form:

TVC 5 a 1 1Q 1 2Q 2 1 3Q

3 (6-1)

Where the parameter a represents the unknown factors not explained by the indepen- dent variables (Q, Q2 and Q3), and the ’s are the estimated regression coefficients to the independent variables. If the line of best fit does prove to be a cubic function, then we can estimate the MC function as the rate of change of the TVC curve, which is mathematically equivalent to the first derivative of the TVC function, or:

MC 5 TVC/Q 5 1 1 22Q 1 33Q (6-2)

As you can see, and consistent with our analysis in Chapter 5, a cubic TVC function will give rise to a quadratic MC function, which will be U-shaped, falling at first due to increas- ing returns to the variable inputs and later rising due to diminishing returns to the vari- able inputs.4 The U-shape means we should expect a negative value for the 2 coefficient

4. In case your math is rusty, we have used the power rule to find the derivative 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 derivative of aXb 5 baXb-1.

dou70192_06_c06_175-202.indd 192 11/1/12 4:26 PM

CHAPTER 6Section 6.2 Cost Estimation Methods

and positive values for the 1 and 3 coefficients—these would cause the MC curve to start with a vertical axis intercept of 1, fall as output levels increase (at first) due to the negative 2 coefficient, which is relatively large compared to the 3 coefficient, and then rise later as the relatively large Q-squared values dominate the equation. Regression anal- ysis will allow us to find estimates of these  coefficients in the TVC function and, thus, we will be able to calculate the estimated MC at any value of Q by plugging that value of Q into the MC expression given by equation 6-2.

In Chapter 4, when we introduced multiple regression analysis in the context of estimat- ing the demand function, the independent variables were different drivers of demand. Here, in the context of cost estimation, 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 equation 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 regres- sion 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 statistics add-in program) has been added to your Excel software 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 identify which is the dependent variable (TVC) and the independent variables you want to enter into the regression equation, 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. After you have indicated which of the independent variables are to be entered, allow the

dou70192_06_c06_175-202.indd 193 11/1/12 4:26 PM

CHAPTER 6Section 6.2 Cost Estimation Methods

program to make the calculations. 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  coefficients, as well as the coefficient of determination (R2), the standard error of estimate (Se) and the standard errors of the coefficients (S) statistics. Your results table will look something like Table 6.10.

Table 6.10: Results from the regression analysis to estimate a cubic TVC function

Variable Coefficient Std err of coeffic. t-statistic P-value

Intercept a 5 2.8318 8.3176 0.3405 0.7423

Output (Q) 1 5 0.0377 3.839 0.0111 0.9914

Output squared (Q2) 2 5 0.0802 0.4462 0.1798 0.8612

Output cubed (Q3) 2 5 20.0035 0.0191 20.1825 0.8598

Adjusted R2 0.9676

Standard error of estimate 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 relatively high standard errors of the coefficients, the relatively low t-statistics, 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 function to pre- dict TVC, AVC, and MC values would be potentially unreliable.5 So we can conclude that the line of best fit is perhaps not a cubic function, and that the law of variable proportions does not seem to be operating over this range of output levels.

Thus, we need to find a line of best fit that is a better fit to the data. To find whether only diminishing returns are evident, we would repeat the regression analysis to estimate TVC as a quadratic function of output, namely:

TVC 5 a 1 1Q 1 2Q 2 (6-3)

If this quadratic function is an acceptably-reliable line of best fit, this would imply a linear MC function of the form:

MC 5 1 1 22Q (6-4)

5. The t-statistics need to be somewhere close to 2.0 (or larger) to allow us to be confident (at the 95% level of confidence or better) that the variable is a statistically 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 particular estimate of the TVC curve.

dou70192_06_c06_175-202.indd 194 11/1/12 4:26 PM

CHAPTER 6Section 6.2 Cost Estimation Methods

Repeating the regression analysis, but this time 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-statistics 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 sig- nificant” determinant of TVC in this form of the TVC equation, indicating that we could be confident at the 90.44% level that TVC is a function of output, but the unreliability of the other independent variable (Q2) renders the quadratic estimate unreliable as well.

Table 6.11: Results from the regression analysis to estimate a quadratic TVC function

Variable Coefficient Std err of coeffic. t-statistic P-value

Intercept a 5 1.3354 1.3060 1.0225 0.3332

Output (Q) 1 5 0.6514 0.3499 1.8614 0.0956

Output squared (Q2) 2 5 0.0011 0.0226 20.0468 0.9637

Adjusted R2 0.9711

Standard error of estimate 0.1918

So once more we ask Excel to calculate a regression equation, this time using a simple bivariate equation of the form:

TVC 5 a 1 Q (6-5)

which, if reliable, would mean that

MC 5  (6-6)

The regression results for this simple linear TVC function are shown in Table 6.12. At last, this form of the TVC function provides a reliable estimate of the coefficients, and we can be highly confident (above the 99% level, according to the P-values) that TVC is a simple linear function of output. Note that the explanatory power (adjusted R2) is slightly better than it was for the other two forms of the regression equation, and also that the standard error of estimate is smaller than it was for the other two forms of the TVC function that we estimated.6

6. Another form of the TVC function that we might have considered is TVC 5 a 1 2Q 2, which

would imply MC 5 22Q. I did run that regression equation to find that while the Q 2 variable is

statistically significant above the 99% level of confidence, the R2 was marginally lower and the standard error of estimate was higher, so that the best line of best fit is the linear TVC equation given by equation 6-5.

dou70192_06_c06_175-202.indd 195 11/1/12 4:26 PM

CHAPTER 6Summary

Table 6.12: Results from the regression analysis to estimate a linear TVC function

Variable Coefficient Std err of coeffic. t-statistic P-value

Intercept a 5 1.3953 0.2505 5.5703 0.0002

Output  5 0.6351 0.0313 20.3020 0.0000

Adjusted R2 0.9763

Standard error of estimate 0.1820

So what does all this mean for the ice cream factory? The regression results in Table 6.12 indicate that the TVC 5 1.3953 1 0.6351 Q. That is, the line of best fit to the TVC func- tion intersects the vertical axis at $1,395.30 and slopes upward at $0.6351 per gallon of ice cream manufactured. Do not be concerned about the positive intercept value for the TVC curve—our data had no output values anywhere near zero—the intercept value simply serves to lift 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 observations, which provides our estimated value for marginal costs, and which we have estimated 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 estimate the mar- ginal cost of ice cream at $0.63 per gallon for any volume of output within the observed data range (i.e., interpolation) or for relatively small extrapolations 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 estimate of the marginal cost (which is also the incremental cost of an extra gallon of ice cream in this case because there were no variations in fixed costs associated with the variations in the output levels).

Summary

In this chapter, we have been concerned with cost estimations and calculations for decision- making purposes. We began with contribution analysis, which required the accurate iden- tification of incremental costs and incremental revenues. We applied contribution 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- tion 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 variation 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 contribution 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 utilized, which may include contribution

dou70192_06_c06_175-202.indd 196 11/1/12 4:26 PM

CHAPTER 6Summary

foregone on other sales that cannot be made as a result of the decision; the cost of replac- ing owned items (to be used because of the decision) in inventory; or the market value of owned items or resources that might have alternatively been sold. Future implications 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 relationship with existing customers (known as ill will or goodwill, respectively); impacts on the relationship between management and workers (known as employment relations), which may affect future labor productivity; impacts on the business relationship with suppliers (i.e., supplier relations); 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 estimate the shape and placement of cost curves, or, more importantly, to estimate the values of cost categories at particular output rates. We began with simple extrapolation of known data points, a method that is appropriate for estimating cost values that are outside the range of our data observations. Extrapolation means to simply extend the values of the observed data points in the direction they seem to be heading. We noted that extrapolation becomes increas- ingly more unreliable the further one extrapolates outside the observed data points, due to the law of variable proportions (or diminishing returns) causing the extrapolation to be inaccurate. We next considered interpolation, or the estimation of data points between known data points. A linear interpolation is the simplest assumption unless we have data to suggest a curvilinear interpolation is more appropriate, such as diminishing returns to the variable inputs being evidenced by rising MC observations. We then considered gradient analysis, which is simply interpolation between data points when we have sev- eral data points. This allows us to more accurately sketch in a line of best fit to the data observations.

With more data points, regression analysis can be used to find the statistical relationship between costs and output levels, but as we saw the choice of functional form and the reli- ability of the results obtained are of paramount importance. Because the law of variable proportions is likely to be present in any production process, it makes sense to start the regression analysis of observed total cost (TC) or total variable cost (TVC) data by includ- ing squared and cubic output quantity terms in the regression equation. Observation of the regression statistics (the adjusted R2, the standard error of estimate, the standard error of the coefficient, the t-statistics and the P-values) will allow us to judge whether the form of the regression equation 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-statistics close to or above 2, or P-values less than 0.05) we cannot be confident that they explain the variation in the dependent variable (TVC) and thus, they should not be used in the predictive equation to estimate levels of TVC for future levels of output.

If the cubic regression equation does not provide a reliable explanation of the variation in TVC then we would revert to a quadratic regression equation, effectively assuming that the range of data observed does not include the initial increasing returns to the vari- able inputs but only observes diminishing returns to the variable inputs. Re-running the regression analysis in the form TVC 5 a 1 1Q 1 2Q

2 will provide new estimates of the regression parameters (1 and 2) and new regression statistics to indicate whether the quadratic form offers a more reliable estimate of the relationship between TVC and

dou70192_06_c06_175-202.indd 197 11/1/12 4:26 PM

CHAPTER 6Questions for Review and Discussion

the output level. Again we scrutinize the t-statistics 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 indepen- dent variables is not significant at the 95% level, it behooves us to check for a simple lin- ear relationship between TVC and Q (as we did in this chapter), and if we find this to be the most reliable explanatory equation then this is the form we should use for predicting future values of TVC for proposed output levels.

Chapters 5 and 6 have been concerned with the cost side of the firm’s operations, just as Chapters 3 and 4 were concerned with the demand (or revenue) side of the firm’s busi- ness. As you know, profit is the excess of revenues over costs. In the following two chap- ters, we will utilize the concepts learned in the preceding chapters to consider the firm’s pricing decision on the presumption that the firm’s objective is to maximize profit.

Questions for Review and Discussion

1. List out all the categories of incremental cost that you can recollect from your read- ing of this chapter and provide examples of each one.

2. Accountants are concerned with historical costs of resources and the allocation of some of these against current period revenues and consequently derive an account- ing measure of the firm’s profit. The economic profit of the firm is likely to differ from this. Please explain.

3. Why does contribution analysis ignore the fixed overhead costs that financial accountants would want to include in the full cost of the firm’s product?

4. How should future costs and revenues be included in the calculation of the contribu- tion of a decision?

5. Under what circumstances would a manager make a decision that ignores the future cost and revenue implications of that decision? (There are many reasons so your thinking may range widely on this one.)

6. When is extrapolation a satisfactory method of cost estimation and when is it not? 7. Gradient analysis interpolates between known data points. This interpolation may

be linear or curvilinear. How do we know when we should fit a curvilinear line of best fit to the gradient data points?

8. Regression analysis of cost data does not interpolate between known data points— instead it estimates a line of best fit to the observed data points, allowing for poten- tial deviations from the line of best fit. Please explain.

9. How do we know that the functional form of the regression equation (i.e., a linear, quadratic, or cubic function) is the best form of the regression equation for predicting cost levels at future output levels?

10. When the regression equation predicts an estimated value of TVC at a particular 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 estimate.)

dou70192_06_c06_175-202.indd 198 11/1/12 4:26 PM

CHAPTER 6Decision Problems

Decision Problems

1. The Muscle-Man Company (MMC) manufactures and assembles forklift tractors and supplies parts to other forklift manufacturers. It fabricates most of the compo- nent parts but buys the engines, hydraulic systems, wheels, and tires from suppli- ers. Demand estimates indicate that MMC should increase production level from 60 units to 70 units monthly. Sufficient capacity exists in most departments to allow this increase, except that production of 10 extra chassis assemblies could be attained only by reallocating 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 forklift manufacturers at $1,880 each. This department could only produce 10 more fork assemblies if the remainder of its labor and equip- ment is to be reallocated to build the extra 10 chassis assemblies, so the sale of fork assemblies to other manufacturers must be forgone. Alternatively, 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- tive month are as follows:

Costs Chassis department Fork department

Production level Direct materials Direct labor Depreciation 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 additional chassis assemblies? b. What qualifications would you add to your decision?

2. The Rakita Racquets Company restrings tennis racquets, a business with highly sea- sonal 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 time the costs of casual labor and of other variable inputs (stringing materials, energy, and packaging) have remained constant, and because of the continual turnover of casual labor the productivity 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

dou70192_06_c06_175-202.indd 199 11/1/12 4:26 PM

CHAPTER 6Decision Problems

a. Derive average variable cost (AVC) data from the data in this table. b. Use gradient analysis to provide an estimate of 11 data points that seem to repre-

sent the MC curve over this range of outputs. Plot these data points and sketch in estimated MC and AVC curves that seem to best fit these data points.

c. Suppose that demand is estimated to move from its present (May) level of 7,000 units to 10,000 units next month (June). What is the incremental cost of meeting 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 contribution to overheads and profit can it expect in June?

3. The Tico Taco Company has estimated its weekly TVC function from data collected over the past several months, as TVC 5 435.85 – 1.835Q2 1 3.658Q3 where TVC rep- resents 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 consider- ing 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. Estimate the incremental costs of the extra 200 boxes per week. c. Should Tico Taco expand its output? Why or why not? State all assumptions and

qualifications which underlie your recommendation.

4. Scruples Footwear Design is a boutique manufacturer of designer loafer shoes. The TVC function has been estimated as TVC 5 20Q 1 0.00782Q2 and the demand func- tion has been estimated as Q 5 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- tion for these two regression equations were 0.9638 and 0.9422, respectively. The standard error of estimate was 286.22 for the cost function and 30.967 for the demand function. Its current price is $32.50 per pair (wholesale price) and it has been produc- ing 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 national market and would most likely cause substantial growth of sales. The purchasing agent has offered only $28 per pair, however, and says “Take it or leave it!”

a. From the estimated cost function, 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 contribution 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 assumptions and qualifications underlie your recommendations?

dou70192_06_c06_175-202.indd 200 11/1/12 4:26 PM

CHAPTER 6Key Terms

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 particular novelty item as shown in the following table. Sales have grown over this period with relatively few shocks due to uncontrollable weather, political and sporting 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 5 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 equation that best fits the data to represent the TVC function.

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. Estimate the value of TVC for sales/output level 250,000 units and calculate the

95% confidence interval for your estimate.

Key Terms

cannibalize existing sales A situation where sales of a product, via a new distri- bution channel or new retail outlet, will replace or eat into the sales of the product through the pre-existing channels and retailers.

contribution The excess of incremental revenues over incremental costs, relating to a particular decision, is called the con- tribution 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.

contribution analysis A process of assess- ing the incremental costs and incremental revenues associated with a decision to determine whether the latter will exceed the former and thus whether the decision should in fact be made by a profit- maximizing firm.

estimation of cost curves A process of estimating the values of costs, in particular categories of costs, at various output rates.

extrapolation To estimate a data value (e.g., a cost level) that lies outside the range of previous data observations by projecting, or extending, the relationship observed within the range of data points to higher or lower level of the indepen- dent variable (e.g., output).

dou70192_06_c06_175-202.indd 201 11/1/12 4:26 PM

CHAPTER 6Key Terms

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

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

gradient analysis A process that involves calculation of the gradients by interpo- lating between sequential pairs of data points.

incremental revenues The change in total revenue that results from a particular decision.

interpolation A process of assigning esti- mated values to unknown points between two separate data points for which data is known.

irrelevant costs A cost that is not relevant to a decision that is about to be made because it is not incremental to that deci- sion, such as a sunk cost or an unavoid- able cost.

labor relations The state of the relation- ship that exists between the management of a firm and the other employees of the firm. Deterioration of this relationship may reduce the willingness of employees to raise or maintain their productivity in the production process.

longer-term supplier relations The state of the relationship between the firm and its suppliers over the longer term. Dete- rioration of this relationship may cause suppliers to be unwilling to offer better deals, rapid delivery, or other discretion- ary services.

present-period explicit costs Actual outlays of cash in the present production period to pay for the variable and fixed inputs that are required to implement the decision that is made.

P-values Indicate the probability that the relationship between the dependent vari- able and one of the independent variables, as indicated by the regression equation, 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 esti- mate of the true relationship in the popu- lation as a whole.

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

sensitivity analysis An analysis of the degree to which the assumptions underly- ing a decision to be made might be incor- rect without this causing the decision to be the wrong one.

t-statistics A measure of the reliability of the coefficient to an independent variable in a regression equation. The t-statistics need to be about 2.0 or better to allow con- fidence at the 95% level of confidence that the variable is a statistically significant determinant of the dependent variable.

dou70192_06_c06_175-202.indd 202 11/1/12 4:26 PM