Essay: Cost volume analysis in relation to container vehicle products

Module Title: Accounting and Financial Management

“All the work contained within is my own unaided effort and conforms to the University’s guidelines on plagiarism.”

This essay will look at Cost Volume Analysis in relation to Container Vehicle Products; the essay will attempt to explain the relevance of CVP to the Management Team.

Cost Volume Profit CVP analysis is a renowned Management technique that attempts to explain the connection between the cost and revenue functions within a company. Often company senior’s use the technique to get a more thorough understanding of the impact of altering product costs and or changing production volumes etc.

Using CVP analysis can help to decide the optimum sales volume to achieve a profitable outcome, or alternatively the variety of products that a company is capable of producing to achieve the most advantageous profits for the business.

CVP analysis employs linear cost and revenue functions within some specified time period and range of operations.

Break-even analysis is one of the CVP tools likely to be used in a production environment by the Management team. To use break-even analysis it is necessary to understand the 2 types of cost within the business, those, which are “variable”(fig2) and will vary with the production volumes and those which are “fixed”(fig1), that are incurred regardless of the volume of production.

Although the majority of costs can be classified as fixed or variable, a more detailed analysis of the variable costs will reveal two distinct types of costs, “direct-variable” and “semi variable” costs.

Fixed Costs

Are those costs, which are not dependant on changing levels of output i.e. rent, insurance, depreciation, wages etc?

Direct Variable Costs

Those costs identified as direct variable costs are determined by the production of a specific product or service and aligned to a distinct department. Pay and benefits of those individuals employed on the specific product line and the materials used to manufacture that product are examples of direct attributable costs.

Those costs that cannot be directly linked to specific products or product lines, but do vary with the volume of output are classified as “semi” variable costs. “When student numbers on a particular course are between 1 and 299, they can be accommodated in a single examination hall under the supervision of one team of invigilators at a cost of �2,000. However, if three hundred students are enrolled, they can no longer be accommodated at a single examination venue. Semi-variable costs will thus increase to �4,000, because it will be necessary to hire a second hall and employ additional invigilators. These step-like increases occur every time student numbers increase by three hundred”

www.articlesbase.com/managementarticles/howtop-managersuse-indirect-cost-control-strategies-396376.html.

Semi- Variable costs

Break-Even Analysis

BEA can be used to help the decision making process on such things as the volume of parts to be manufactured, revenue to be spent on advertising, continuation of an old product line, headcount increase or reduction and product pricing etc. “we can ask a whole set of “what-if” questions about how increases and decreases in the sales price, unit variable costs, sales mix and fixed costs would affect the outcome”-
Management Accounting: Concepts, Techniques & Controversial Issues James R. Martin.

The “break- even” point is the stage at which the business is either profitable or loss making.

Line B shows the cost of doing nothing i.e. the fixed costs, line C shows the total cost of doing a certain amount of activity, and this cost is variable plus fixed costs. The income line shows a breakeven point at point P (total sales income equals total costs). Where the volume of activity is below the BEP a loss will be incurred. “Accounting An Introduction”. Eddie McLaney, Peter Atrill.

At BEP Total sales revenue = Total costs

At all other points either the total sales revenue will exceed the total costs or the other way round. “Accounting An Introduction”. Eddie McLaney, Peter Atrill.

“Eddie McLaney, Peter Atrill” discuss how the BEP can help with the decision making process of capital investment. By adding in the additional fixed costs in to the BEP calculation it will become clear as to the required volume of product sales to break even before and after investment, and the profit per product. Although creating the chart is more labour intensive than merely doing the calculation, the visual impact can be very powerful when attempting to acquire investment.

The amount of profit generated per part above the defined target sales figure will also be identified from the Break Even formulae; this is defined as the contribution per unit (sales revenue per unit less variable costs per unit)-once fixed overheads have been covered, further contribution is straight profit. “Total revenue, or sales pounds, less total variable costs equals the total contribution margin. Contribution margin is the revenue over and above the variable costs that contributes towards covering the fixed costs and also towards providing a profit after the fixed costs have been covered. Practically any cost-volume-profit problem can be solved with the last equation stated above and an understanding of the concepts involved”

“The Margin of safety is the degree of output that lies above the BEP “Mathematically, the margin of safety is: MS = Sales� – Break-even sales�

When sales are above the break-even point, the margin of safety is positive. When sales are below the break-even point, the margin of safety is negative” -Management Accounting: Concepts, Techniques & Controversial Issues James R. Martin.

Eddie McLaney and Peter Atrill discuss the direct link between the selling price of the product, the variable price per product and the fixed price per product. Prior to any future investment and the inevitable increasing of fixed costs, decisions will need to be made around the impact of this compared to the additional profit per product generated.

“Operating Gearing is the relationship between contribution and fixed costs” “Accounting An Introduction”. Eddie McLaney, Peter Atrill.

A business that has invested heavily in capital equipment would therefore have a high operating gearing in comparison to one with less automation

“Operational gearing is the effect of fixed costs on the relationship between sales and operating profits. If a company has no operational gearing, then operating profit would rise at the same rate as sales growth (assuming nothing else changed). Operational gearing is simple and important – and often neglected. High fixed costs increase operational gearing. Consider two companies with different cost structures but the same profits.

	Company A	Company B
Sales	1,000,000	1,000,000
Variable Costs	700,000	800,000
Fixed Costs	200,000	100,000
Operating profit	100,000	100,000

At this point both companies have the same sales and the same costs, and therefore the same operating profit. Now suppose both companies increase sales by 50%

	Company A	Company B
Sales	1,500,000	1,500,000
Variable Costs	1,050,000	1,200,000
Fixed Costs	200,000	100,000
Operating profit	250,000	200,000

The company with the higher operational gearing, A, makes 2.5� as much profit as it did before the 50% increase in sales, whereas B has only doubled its profits. Operational gearing is this effect on operating profit” http://moneyterms.co.uk/operational_gearing/

Again this theory allows visibility of a companies’ vulnerability to a down turn in sales if high investment in capital was to be made, compared to the lower profit margins available if no new investment is made.

The following symbols are used below to illustrate the various techniques used in cost-volume-profit analysis.

P = Sales price.

V = Variable costs per unit. Note: This is not inventory cost because it includes both variable manufacturing costs as well as variable selling and administrative expenses.

X = The number of units produced and sold. A unit is a common way to describe an output, but an output may be expressed in pounds, gallons, board feet, cubic feet, etc.

TR = S = Total revenue, or sales dollars.

TVC = Total variable costs = VX

TFC = Total fixed costs.

TC = Total costs = TFC + TVC.

P-V = Contribution margin per unit. This is the amount of sales revenue that each unit provides towards covering the fixed costs and providing a profit, i.e., what’s left over after the variable costs associated with the unit have been covered.

TCM = Total contribution margin = (P-V)(X).

CMR = (P-V)�P = (TR-TVC)�TR = (PX-VX)�PX = 1-(V�P). These are just different ways to define the contribution margin ratio. They all work because the functions are linear.

There are many algebraic equations illustrated on the next several pages that may appear to require memorization. However, every equation is simply a variation of the following basic concepts:

Total Revenue = Total Cost + Profit

TR = TC + NIBT

TR = TFC + TVC + NIBT

TR – TVC = TFC + NIBT

TCM = TFC + NIBT

COST-volume-profit (CVP) analysis looks at optimising productivity and delivery using the available assets. A variety of costing tools are available under the umbrella of the CVP framework, including break-even analysis, trans-fer pricing systems and mathematical programming methods for choosing op-timal sales mix and the optimal extent of processing joint products. Initially all CVP models were deterministic, as-suming demand and cost structures to be known with certainty. However, fol-lowing the significant developments dur-ing the last two decades in the area of economic decision making under un-certainty, attempts have been made to relax the certainty assumption of CVP models. These attempts [Hilliard and Leitch, 1975; Jaedicke and Robichek, 1964; Morrison and Kaczka, 1969] took the form of treating various variables, such as volume of sales, product prices and cost, as random. Assuming these variables were subject to well-defined distributions, risk measures, such as the probability of failing to achieve a speci-fied profit level, were derived from prob-ability and tolerance intervals. From a decision theory perspective, the CVP models can be viewed as differ-ent simplification levels of the firm’s short-run output decision. Certainty models probably represent the extreme level of simplification, since they assume that future values of the decision-relevant variables are perfectly predictable. Avail-able CVP models, which incorporate un-certainty, simplify the decision process by ignoring the decision makers’ attitudes toward risk, or equivalently, assuming risk neutrality. Obviously, given cost considerations, any realistic decision The comments of David Solomon’s, members of the accounting workshop at the Wharton School, Univer-sity of Pennsylvania and those of referees are acknowledged-edged gratefully. ZviAcar, Amir Barnea and Baruch Lev are members of the Faculty of Management at Tel-Aviv University. 137

138 The Accounting Review, January 1977 model must be simplified to some extent. 1 The decision makers’ problem is to de-termine the optimal level of simplifica-tion. However, determination of the optimal simplification, or equivalently, evaluation of the different payoffs and consequences of alternative simplified models, requires the construction of complete models. This is the main pur-pose of the current study: to present a comprehensive CVP analysis under un-certainty by combining the probability characteristics of the environmental vari-ables with the risk preferences of decision makers. Given the comprehensive model, an improved evaluation of model simpli-fication is made possible. Combining uncertainty with risk pref-ence function also provides new interest-ing insights, such as the effect of fixed cost changes on optimal short-run out-put.2 As is well known, the basic premise of conventional CVP models (a premise which is, of course, derived from eco-nomic optimization concepts under cer-tainty) is the irrelevance of fixed costs in determining optimal output, since they do not affect marginal cost and revenues. 3 As will be shown below, under uncer-tainty and risk aversion, fixed cost changes affect the firm’s short-run out-put decision. This and other insights can be furthered by examining the following decision model. THE DECISION MODEL Assume a single product, price-taker firm whose profit function is: ft = ( – c)x -k.(1 The firm thus produces the quantity x at a fixed cost k, and variable cost C(x) which, for simplicity, is assumed to equal cx, i.e., constant marginal costs. The only random element in the system is assumed to be the selling price f, which is non-negative and satisfies p = E@) + U = p + a. (2) The density function of a, f(i7), is known, with E(a)= 0 and Var(a) = c. Since price is random, so are the firm’s profits ;4 therefore, the generally accepted firm objective of profit maximization is no longer valid under uncertainty, as the maximization of a random variable is meaningless. Following the mainstream of economic research under uncertainty [e.g., Dhrymes, 1964; Leland, 1972; Lint-ner, 1970; Sandmo,1971], we assume that the firm’s decision makers possess a Von Neumann-Morgenstern utility of profit function, U = U(ft), with U'(fi) > O (non-satiation), U”(fi)<O (risk aversion) and determine optimal output by maximiz-ing the mathematical expectation of the utility of profits, E{ U(if)} .5 1 See [Demski, 1972, chs. 2-3, and Feltham, 1972, ch. 5] for a comprehensive discussion of complete and simplified decision models. 2 Fixed costs are, by definition, invariate with respect to output changes within a relevant range, however, they may vary across states of nature. Thus, for example, in a depression property taxes might be lowered (or in-vestment incentives increased) as a result of fiscal policy. This point often is overlooked in the literature on CVP analysis, which generally treats fixed resources and fixed costs as synonyms. In the short run, some resources (e.g.. plant) are fixed, yet the cost associated with these resources generally will vary across states of nature. Hence, we see the significance of the fixed-cost effect on the short-run output decision. 3 The well-known concept of contribution margin” (i.e., the difference between price and variable costs) and the decision rules derived from it are based on this premise. For example, a customer’s order should be ac-cepted if its contribution margin is positive. 4 Recall that the firm produces a single product, hence it cannot insure” itself completely against the risk of unexpected variations in product prices (represented by vu). A multiproduct firm can eliminate (diversify) part or all of the risk associated with random prices, depend-ing on the correlations among price changes of different products. We also assume that there are no insurance markets for these purposes. ‘ Obviously the performance evaluation of decision makers should be a function of both expected profit and risk. This is analogous to the two-parameter evaluation of portfolio performance e.g., Fama, [1972], except that in our case the “price” of risk is derived from decision makers’ utility function.

Adar, Barnea and Lev 139 The comparative statics of this maxi-mization problem recently was derived by Sandmo [ 1971 ] and Leland [ 1972 ]. How-ever, this general derivation does not lend itself to the compact and simple frame-work which is a major advantage of CVP models (e.g., the general model cannot be presented graphically, and it does not provide a compact risk measure). Accord-ingly, we adopt the Tobin-Markowitz- Lintner “mean standard deviation” framework6 and assume that decision makers rank alternative pairs of ex-pected profit, a, and standard deviation of profit, As, according to the utility func-tion G(a,, 7-); Ga < 0, G'- > 0. (3) The values of ft and ad are obtained readily from (1) and (2): 7i = (pc)x -k. (4) as = x a. (5) Thus, both 7i and ad depend on output. Obviously, given market and technologi-cal conditions, not all (7, a.) combina-tions are feasible to the firm. The set of feasible (7, a.) combinations open to the firm can be derived by substituting (5) into (4), which yields: r [ c)/],-k = ban-k, (6) where b = (p- Consequently, optimal firm behaviour implies maximizing (3) subject to the constraint (6).7 This maximization prob-lem is equivalent, under certain circum-stances, to the maximization of E[U(fi)] with respect to output.8 By setting the Lagrangian L=G m (axiizn, )-e div th-ba + k], (7) and maximizing, we derive the following first- and second-order conditions: G +? biO>b G, (8) 7T- ba + k = 0 G1 1 G12 b D= G21 G22 -1 >0 (9) where b -1 G, = G’a, = 8G( )/8a,; G2 = G’ = 8G( )/08; G = G “a, -= 02G( )/Oa0 7-; = -= 2G( )/f, etc. The ratio – G1/G2 in (8) is the slope of the indifference curve of G( ), and b is the slope of the linear constraint (6). Condition (9) implies that the indiffer-ence curves of G( ) are convex. Figure 1 presents graphically this opti-mal short-run output decision for the firm. In the first quadrant, the optimal * as* combination is determined at the tangency point between the constraint line originating from k and the indiffer-ence curve II (i.e., the first-order con-dition (8) of b = – G1/G2). Optimal out-put, x*, is determined in the second and fourth quadrants, using expressions (4) and (5). Specifically, the slope of the downward sloping straight line in the second quadrant originating from the intersection of the axes is au. Hence, by expression (5), the downward perpen-dicular (broken line) from a* intersects 6 The motives of this shift from the general to the mean-standard deviation framework are the same as those underlying the widespread use of the mean-standard deviation portfolio model. 7 Note the role of the fixed cost, k, in the determination of the feasible set (6). 8 As is well-known from the portfolio literature, this equivalence holds when U(Tr) is quadratic, or when all the moments of flu) depend solely on the mean and standard deviation (e.g., when ia, the random element of price is distributed normally). In other cases, this proce-dure is only an approximation.

140 The Accounting Review, January 1977 FIGURE I OPTIMAL SHORT-RUN OUTPUT I II III I~~~~~~~~ I x I (p_- C ~~ ~ ~~ ~~~h =(p- c)lia.I(, x a7T~~~~~ 1 u I~~~~~~ I I~~~~~~~~ this line at x*. The slope of the upward sloping straight line in the fourth quad-rant originating from k, is p- c. There-fore, by expression (4), the horizontal (broken) line from -* intersects this up-ward sloping line at x*.9 Optimal production decision thus de-pends on decision makers’ risk attitudes (the indifference map), as well as on the fixed cost, k, and the slope of the con-straint line, b, which can be interpreted as the price of risk, in term of the expected contribution margin (p– c), per unit of standard deviation of price, v. COST-VOLUME-UTILITY ANALYSIS The model developed in the preceding section provides for an extension of con-ventional CVP analysis to a cost-volume-utility (CVU) analysis under price uncer- 9 Note that the lower part of the vertical axis measures both output, x, and fixed cost, k. This procedure is used in graphical models for compact presentation.

Adar, Barnea and Lev 141 FIGURE 2 Two ALTERNATIVE PLANS 7t 0k ”if ( 0~~~~~~ 2~~~~~~~ k~~~~~ k~~~~ N a~~~~~~~~~~~~~~~~X tainty. Such a CVU model can be used for an ex-ante examination of conse-quences of various alternative plans under management control. While CVP analysis considers the consequences of various price-quantity relationships on profit, the scope of CVU analysis is ob-viously broader. For example, suppose that management considers an invest-ment in market research intended to de-crease price uncertainty, v. In Figure 2, this plan is represented by a new con-straint line k’al, with a lower intercept and a higher slope than the current line k’a0. This reflects an increase in fixed cost (due to the increase in market re-search) and decrease in v”. Given man-agement’s particular preferences (re-flected by the shape of the indifference curves), it is obvious that the proposed project should be rejected, since it re-sults in a lower utility level at the new optimum. In addition, the model shows that the maximal increase in fixed costs

142 The Accounting Review, January 1977 warranted by the proposed decrease in price uncertainty, v”, is (k2-0k), where k2 is the intercept of a constraint line parallel to k1 a’ and tangent to the origi-nal indifference curve. The consequences of alternative plans that affect expected price, marginal cost, fixed costs and price uncertainty, or any combination of those, can be evaluated in a similar manner. Notice that the effect of such plans on optimal output can be determined readily in the second or the fourth quadrants of Figure 2, de-pending on the changes considered.’0 The construct of Figure 2 can be trans-formed into a different graph providing for a more efficient ex-ante examination of alternative investment-production plans. Specifically, in Figure 2 the firm is indifferent between the two (b, k) com-binations represented by constraints k0aO and k2a2, since in both cases

COST-VOLUME-PROFIT ANALYSIS ADJUSTED FOR LEARNING* E. V. McINTYREt A model is developed for cost-volume-profit analysis which incorporates a nonlinear cost function to express the effects of employee learning. Sensitivity analysis is applied to the model to assess the impact of estimation errors in the learning rate and steady-state production time on estimated profit and break-even quantities. The paper also examines the effects on the model of (1) alternative accounting treatments of production-related costs, and (2) continuous learning due to employee turnover. Cost-volume-profit (CVP) analysis is a well-known managerial tool that attempts to specify a firm’s cost and revenue functions and the relationships between the two. It is used by managers, accountants, investment analysts, and other interested persons to examine the effects on profit of changes in costs, volume, selling price, product mix and related factors. Traditional CVP analysis employs linear cost and revenue functions within some specified time period and range of operations. The use of linear cost functions implicitly assumes, among other things, that a firm’s labor force is either a homo- geneous group or a collection of homogeneous subgroups in a constant mix, and that total production changes in a linear fashion through appropriate increases or de- creases of seemingly interchangeable labor units. When production involves new products or designs which are labor intensive and require complex work skills, learning becomes a factor and the above assumptions may not be warranted. Even for established products these assumptions may not be justifiable if turnover is high and complex labor tasks must be learned by new employees. The purpose of this paper is to show how alternative assumptions regarding a firm’s labor force may be utilized by integrating conventional CVP analysis with learning curve theory.’ Explicit considera- tion of the effects of learning, where such effects are considered material, should substantially enrich CVP analysis and improve its use as a tool for planning and control of operations.2 Specifically, this paper examines (1) the effects of learning on the break-even equation and solution, (2) the use of sensitivity analysis to see how errors in estimating learning parameters change estimated profit and break-even quantities, (3) some effects of alternative accounting treatments of learning-related production costs, and (4) the impact on periodic profit of continuous learning due to employee turnover. The Break-Even Equation and Solution A number of mathematical models have been used to give formal recognition to the fact that people generally require less time to perform a complex task after they have * Accepted by Samuel Eilon, former Departmental Editor for Production Management; received January 11, 1977. This paper has been with the author 1 month, for 1 revision. t Florida State University. I The incorporation of learning curves and break-even analysis is discussed briefly by Brenneck [8], Harvey [10], and Pegels [16]. The present paper develops further this combination of managerial tools. Learning curve models have also been used in conjunction with other types of managerial analyses, such as capital budgeting, variance analysis, inventory management, and pricing policy. A partial listing of articles relating to these and other applications is contained in Morse [13]. Some other articles of interest are Abernathy and Baloff [1], Baloff and McKersie [5], Bhada [7], Bump [9], Keachie and Fontana [11], Morse [14], [15], and Summers and Welsch [19]. 2 The significance of learning or “start-up” costs to various industries is documented in Baloff and Kennelly [4]. 149 Copyright ?) 1977, The Institute of Management Sciences

150 E. V. McINTYRE acquired some familiarity and experience with the related work skills. These models are typically called learning curve models, although many factors other than pure learning may be responsible for the observed decrease in time and costs which learning-curve models attempt to capture.3 The reduction in production time associated with increased production can be expressed either as a decrease in the time of the marginal or xth unit, or as a decrease in the cumulative average time of x units. Models exist for either mode of expression. For illustrative purposes, we utilize the average-time model, which states that cumulative average time per unit will be reduced by 1 – R every time production doubles. In different words, the model states that cumulative average time is R percent of what it was prior to the last doubling of production. The model may be expressed more formally as follows. y = axb, (1) where y = the cumulative average production time per unit after x units have been produced, a = the time required to produce the first unit, x = cumulative production, b =index of learning = log R/log 2 = ln R/ln 2, R = learning rate. A derivation of this model, some of its characteristics, and a comparison with the marginal-time model are given in the Appendix. Although the model is expressed in terms of average time, an expression can be derived easily for marginal time. We will find it useful to employ such an expression subsequently. For ease of reference, additional notation which is used repeatedly throughout the paper is listed and defined below. Notation used only once will be defined when it is introduced in the development of the model. ‘g= profit. p = selling price per unit less all variable costs other than labor costs. c = labor costs per unit of time. f = fixed costs per period. n = number of simultaneously operating production teams. t,.= steady-state marginal time. x5= number of units produced until steady-state conditions are reached. y, = average time per unit for the first x5 units. TI = total production time for the first xs units. XB = break-even number of units. XT = total estimated production or a specified number of units over which learning costs are amortized. Learning curve models such as (1) could be useful in CVP analysis within the framework of either a project or a time period orientation. For projects requiring production of a specified number of units, learning curve models may be used to improve estimates of project profit and project break-even quantities. Evaluations of new products with continuing sales over an indefinite number of time periods may incorporate learning curves to refine estimates of periodic profit and break-even points both during and after the learning process. Both project and period applica- tions are potentially useful and interesting; however, the primary concern of this 3 Further discussion of this point is provided by Bhada [6] and Andress [2]. In spite of the danger of oversimplification, in the interest of conciseness “learning” will be used in this paper to describe the reduction in unit time related to increased production.

COST-VOLUME-PROFIT ANALYSIS ADJUSTED FOR LEARNING 151 paper is with the second situation, i.e., the use of learning curve theory to refine periodic CVP analysis. The analyses that follow retain the usual assumptions of CVP analysis with the addition of nonlinear cost functions to express the effects of learning. With these assumptions, the profit equation for the initial period of production for a product subject to the learning function of (1) can be written as4 5 px – cyx – f. Sub- stituting for y, we obtain Tr=px-caXb+1 (2) (2) incorporates a learning factor which reduces average production costs over time, but in so doing it implicitly assumes a fixed labor force. In (2) production Etime decreases at a steady rate, which implies that the applicable labor force is collectively progressing down the learning curve as a single unit. It is desirable to continue to treat labor as a variable cost, but if learning is an important factor, we can no longer allow for variability with the usual CVP assumption of homogeneous labor units that increase or decrease production at constant marginal costs. To retain labor variability in the model, explicit assumptions are needed with respect to the timing and composition of changes in the firm’s labor force. For planning purposes it may be sufficient to consider variability in the labor force as occurring at the beginning of the planning period. This type of variability can be introduced into the model through the notion of parallel or simultaneous production runs which can be varied at the beginning of the period. The resulting profit function for the initial period of production with n production processes operating simul- taneously is as follows: qT = px – nca(x/n)b + f; (3) that is, x units will be produced by n labor teams of one or more employees each, with each team producing x/n units. Observe that when additional production teams are added, more units are produced during a given time period; however, the average time for a given number of units increases because more employees are producing while they are learning. If different skill levels of employees produce different learning parameters between production runs, (3) can be modified to accommodate such differences by summing across average skill levels and resulting output rates, i.e., n n qT = pE Xi- C E aixl: ‘-f. (4) i-I i=l In the preceding equation. ai and bi denote the parameters applicable to the average skill level of the ith production run, and xi represents the output of the ith run in a given period of time. Because we are considering the initial period of production for a new product subject to a learning process, we will assume in subsequent formulations that, on average, each production line starts with approximately the same degree of specific skills. For this purpose we will utilize (3) rather than (2), since (3) reduces to (2) when n = 1. 4 If other variable manufacturing costs are incurred in proportion to labor hours, c may be modified to include these costs. If some variable costs are related to labor hours and decline per unit at different rates, it may be possible to apply separate learning rates or models to each type of cost. S Fixed costs may be fixed over certain ranges or production, but may vary with recruiting and training efforts. If so, there may be a dependency between fixed costs and the learning rate, R. A somewhat similar dependency between the parameters a and b was noted by Baloff [3].

152 E. V. MCINTYRE Estimated cumulative profit at a given level of production and sales can be determined directly from (3). To estimate the quantity of output where cumulative production will initially break even, (3) may be set equal to zero and solved for x. Due to the exponential form of (3) and the presence of fixed cost, it is not possible to present a general solution for this break-even quantity. The cumulative break-even quantity may be determined for specific values of the parameters, however, by using the method of successive approximation to solve (3) for 7r = 0. Steady-State Conditions It is unreasonable to expect that learning, or the reduction in production time per unit, will continue indefinitely for a given product. At some point production should reach steady-state conditions for seasoned employees. For example, Baloff and Kennelly [4] report that in a study of twenty-four process start-ups in the steel industry, steady-state productivity was achieved in periods varying from 2 to 43 months. Lengthy breaks in production and employee turnover (discussed later) may cause a departure from steady-state production; nevertheless, the basic model must be modified if it is to represent reasonably the changed conditions applicable to a steady-state situation. To make useful projections of profit and break-even levels under steady-state production, estimates are needed of marginal production time and cumulative produc- tion at the point where steady-state conditions are initially reached. Given an independent estimate of steady-state marginal time, the number of units required to reach steady-state production can be estimated directly from the learning curve equation. Alternatively, given an independent estimate of the units to be produced before learning stops, steady-state marginal time can be estimated from the learning- curve model. Either quantity may be difficult to estimate directly. For the following exposition we will assume the first alternative, since post-learning marginal time may be susceptible to independent estimation by analysis of component work tasks involved in the production process. Recalling that t, represents steady-state marginal time, xs represents the number of units produced until steady-state conditions are reached, ys represents the average time per unit for the first xs units, and Ts represents the total production time for the first xs units, we can estimate x5 as follows: Ts =ysx= axsb+1, and (5) ts = dTs/dxs = (b + I)axsb = (b + I)ys. (6) Solving (6) for xs, we obtain xS tsl(b + lI)a ]Il/b. (7) With an independent estimate of t, xs can be estimated from (7). Alternatively, xs could be estimated by setting total time for x5 units minus total time for xs – 1 units equal to ts and solving for xs. This procedure requires solving the following nonlinear difference equation: axs – a(xs -1)b+ = tS. (8) The approximate relationship between average time and marginal time shown in (6) becomes closer to the actual relationship as the units produced increases. When only one unit has been produced, marginal time is equal to 100% of average time; however, the ratio begins to converge rapidly to b + 1. To illustrate, consider an 80% learning curve which has a value of b + 1 equal to 0.678. The ratio of marginal time to average

COST-VOLUME-PROFIT ANALYSIS ADJUSTED FOR LEARNING 153 time reaches this value after approximately 240 units, but after 10 units the ratio is reduced to 0.689. Using either method of estimating tS, and (7) to estimate x5, cumulative profit on all units produced during and after the learning period can be estimated. If there are n simultaneous production lines, total production when each production line reaches the steady state is nxS. Cumulative profit for x > nxs can now be expressed as 7e =px-cnTs-cts(x-nxs)-f, x > nxs. (9) The second term in the right-hand side of (9) is the total labor cost of the first nxS units (n production lines producing xs units each), and the third term is the total labor cost of all units in excess of the first nxS units. If the initial break-even point on cumulative production occurs at some level XB > nxs (as shown by computations from (3) above), (9) can be used to solve for this quantity directly. The general solution for break-even under these conditions is presented in (10) below. x cn (T – tsxs) + f X > nxs (10) An additional computation of importance for planning purposes is the periodic break-even point for periods after steady-state conditions have been achieved. The equation to determine this quantity is the same as the conventional break-even equation, except that labor costs per unit are expressed in terms of the marginal time per unit for steady-state production. The periodic break-even point under steady-state production thus becomes XB = f/(p – Cts) (11) (11) assumes that variations in total employee production time occur within a pool of trained employees who produce, on average, at steady-state time. Alternative assumptions may be needed for wide fluctuations in production. Sensitivity Analysis CVP analysis is traditionally used to show the effects of changes in prices, costs, product mix, and output on profits and break-even quantities. These kinds of analyses are a form of sensitivity analyses. A natural extension of this use of CVP models when learning is an important factor is to perform additional sensitivity analyses to determine the effect of estimation errors in the learning rate, R, and estimated steady-state marginal time, ts, on projected profits and break-even quantities. These extensions are discussed in the two sections that follow. Errors in the Learning Rate The effect of errors in estimating the learning rate, R, on break-even and profit levels can be determined by substituting a new R (and thus a new value for b) in the equations of the previous sections. To avoid recomputing the cumulative break-even point by the trial and error procedure which would be necessary to solve (3), the differential, dxB, can be used to estimate the changes in the break-even quantity for small changes (errors) in R. Rewriting (3) to show the break-even condition, and writing b in terms of R, we obtain PXB – f = nca(xB/n)(1n R/ln 2)+ 1. In the preceding equation, XB is an implicit function of R and the differentiation process is facilitated by taking the natural logarithms of both sides of the equation. After taking logarithms, differentiating to find dxB/dR, and simplifying, we obtain the differential,

154 E. V. McINTYRE dxB, which may be used to approximate the change in XB for a small change in R, AR, given the original values of R, b, and XB. xB (ln XB – In n)(PXB f R (12) xB =[f(b + 1) – bPXBlR In 2 (12) is appropriate for estimating changes in the initial cumulative break-even point, when this point occurs before steady-state production is reached. The effect of a change in R on x5 and XB when the cumulative break-even point occurs after steady-state conditions are reached (xB > nx,) can easily be determined by substitut- ing the corresponding new value of b into (7) and (5) and using the new estimates of x5 and T7 in (10).6 Similar substitutions into (11) will show the effect of changes in the learning rate on the periodic break-even quantity under steady-state production. The effects of changes in R on projected profits at various levels of output can be determined in a similar fashion. Errors in Steady-State Estimates Errors in estimating x5 could occur because of errors in estimating R; i.e., a faster or slower learning rate would imply a lesser or greater number of units needed to reach steady-state conditions. The effect of errors in x5 resulting from errors in estimating R can be determined by the methods of the previous section. Errors in estimating x5 could also occur independent of errors in R. This type of estimation error may be viewed as a misestimation of x5 itself, or as a misestimation of steady-state marginal time, t, since x5 and t, are functionally related by (6) and (7). Since we assumed earlier that initial estimates were made in terms of t, it will be useful to examine the effect of misestimation of t, on profit and break-even levels. This can be accomplished by substituting the new estimate of t, into (6) and (7) to obtain new estimates of y, and x,. The error in estimated cumulative profit due to misestimation of t, will be equal to the error in total labor costs. Letting t,,y, and x5 denote the new estimates of t, y, and x5, the resulting error in estimated profit, Alg, is as follows: Ag = cn (y,x, -y yx,) + c [ ts (x – nxs) – t(s – nxs)] x > nxS. (13) In (13), the first term on the right is the change in the total labor costs of all units produced during the learning period, and the second term is the change in the total labor cost of units produced during steady-state production. A similar, but simpler, expression can be written for the error in periodic steady- state profit which results from a misestimation of ts. The resulting estimation error in steady-state profit is shown by (14). A7 = cx(tS – ts) (14) New estimates of related break-even quantities may be obtained by substituting appropriate new values of ts, xs and Ts into (10) and (11). The parameter b also can be used to estimate quickly the change in ys, xs and Ts due to estimation errors in ts. We will pursue this approach below because it provides an interesting interpretation of b as a coefficient of elasticity. Borrowing a term from economics, we can define the elasticity of the learning curve as the percentage change in y divided by the percentage change in x. The elasticity at any point on the curve, E, can be expressed as dy/ly_ dy x dx/x dx (15) 6 These computations assume that a change in the learning rate will cause a change in x5 rather than t8.

COST-VOLUME-PROFIT ANALYSIS ADJUSTED FOR LEARNING 155 To show that b = E, we rewrite (1) as y = ax b, then differentiate with respect to x dy/dx = baxb-l. (16) Substituting (1) and (16) into (15) we obtain E = baxb-l * (x/axb) = b. (17) (17) shows that for a given learning rate, the percentage change iny divided by the percentage change in x is constant and equal to b. Since by (6), marginal time, t, is a constant percentage of average time, the percentage change in t for a given percentage change in x is also constant and equal to b. Using these relationships the change in x5, Axx, and the change in y5, Ay5, due to an estimation error in t, LtS, can be approximated as follows: Axs/xs= I/bl (Atjt), and (18) Ys/ys = Ats/ts. (19) To compute the approximate change in the total time required to reach steady-state production, Ts we write Ts I S) ( Y I + )X (I Axs) (20) By substituting (18) and (19) into (20) we obtain ZT I S )( + S ) 1, (21) the desired approximation. Because b is a measure of point elasticity, it will provide the best approximations for relatively small changes in ts. The error caused by using b to estimate the change in xs is equal to the difference (I/b)(Atjlts) – Ax/xs. To obtain an expression for Axs/xs we utilize (7) to write ____ ~ I +At/lts) Il/b (+ x) [ (b + I)a J Solving the preceding equation for Axs/xs, we obtain AX5/X5 = (I + – )1- 1. The error term thus becomes Error – A(ts) [( I+ Ats ) I/_ (22) As an illustration, the preceding equation will show that with an 80% learning rate and a 5% estimation error in ts, the use of b will produce a 1.5% error in estimating the related percentage error in xs; e.g., a 15.5% error in xs would be estimated as a 14% error through the use of b. In a similar fashion it can be shown that the error caused by using b to estimate the change in Ts is equal to the following quantity:

156 E. V. MCINTYRE The Effect of Alternative Accounting Treatments of Production Costs The discussion to this point has assumed that in the profit calculation all labor costs incurred are charged to income as part of the cost of units sold. Under this procedure units produced early in the process will be reported at a much higher cost than units produced later in the process. For certain internal and external reporting purposes the firm may desire to adopt some averaging or smoothing procedure for reporting inventories and cost of sales. One way to accomplish this is to defer initial “learning costs” and charge those costs to income of later periods.7 If “learning costs” are interpreted to mean all labor costs in excess of cost per unit during steady-state production, and if such costs are initially deferred and written off on a “unit of production” basis, periodic profit may be expressed as 7T= px – ctsx – (cnT, – ctnxS)x/xT -f x XT, (23) where XT represents total estimated production or some specified number of units over which learning costs are to be amortized. All other symbols are as previously defined. (23) charges revenue of the period with the steady-state cost of each unit sold plus a write-off of deferred learning costs. The bracketed expression is the estimated total learning costs to be deferred; i.e., the excess of total labor cost of production during the learning period over the steady-state cost of these nx, units. After some algebraic manipulation, (23) can be rewritten as -7 =px – x[cnTI + cts(xT- nxs)1] – f, x < XT. (24) (24) shows that utilizing a unit of production amortization method for deferred learning costs is identical to expensing all units sold during the period at the estimated average labor cost of XT units. The first term inside the brackets of (24) is the labor cost of the first xs units produced by each of n production lines, and the second term is the labor cost of XT -nxs units produced under steady-state conditions. The sum of these two terms is the total labor cost of XT units, and this sum divided by XT is the estimated average of these units. If this average is represented by YT, (23) and (24) can be expressed more simply as ST =PX -YTXJ, x X XT, (25) which can be solved easily for the periodic break-even quantity.8 Firms also may wish to adopt a "normal costing" approach to fixed overhead allocations. If fixed manufacturing costs are allocated equally to all units produced during the period, units manufactured early in the learning process will have a much higher unit cost than later units, since fewer units will be produced in early periods. This can be avoided by allocating fixed manufacturing costs to units on the basis of estimated average fixed cost per unit over XT units. If this procedure were adopted, 7Deferral of "learning" costs was suggested by Morse [13] and is mentioned by the Securities and Exchange Commission in SEC Release No. 33-5492 [17]. 8 Viewing the unit-of-production amortization method as equivalent to costing all units at the average cost of XT units shows clearly that the deferral account will always have a debit balance, i.e., the cumulative write-offs will not exceed cumulative deferrals at any time. For write-offs to exceed cumulative deferrals, the average labor cost of XT units would have to be greater than the average labor cost of x units (x < xT). Given the nature of the learning curve employed, this is not possible. Other amortization methods do not guarantee this result. With straight-line amortization (with respect to time), for example, write-offs may exceed cumulative deferrals if a small number of units are produced in an early period, thus making current deferrals small but not affecting straight-line write-offs. COST-VOLUME-PROFIT ANALYSIS ADJUSTED FOR LEARNING 157 periodic profit becomes (assuming deferral of learning costs as previously discussed) 77 = PX – YTX – (Nf/xT)x, x < XT. (26) In (26), N represents the number of periods required for production of XT units. With this method of accounting for overhead costs, a periodic volume variance would be reported and deferred which would measure the deviation of actual production from average periodic production over N periods. Employee Turnover and Continuous Learning Learning curves may be relevant to CVP analysis even after steady-state production has been reached by a majority of employees. If turnover is high and new employees must learn complex work tasks, it may be worthwhile to consider explicitly the effect of continuous learning by a segment of the labor force on average cost and projected profit and break-even levels. Learning due to employee turnover may be incorporated into CVP models by adopting an approach similar to that used for multi-product firms. Although we are presently considering only one product, it may be viewed temporarily as two products with separate costs. Units produced by new employees will require more time and thus have a higher unit cost than units produced by seasoned employees. For a given turnover rate, we can compute an average labor cost per unit based on the implied mix of units produced by "turned-over" employee positions and "stable" employee positions. To construct such a model, the following additional terms are employed: d =working days per period per employee. r = employee turnover rate per period. L = number of employees in labor force. h = time units per working day (measured in the same units as the parameter a). Using these terms, and ts as previously defined, we can write the following relationships. rL = number of employee positions "turned-over" or replaced per period. h / ts = number of units produced per working-man day under steady-state condi- tions. To obtain an expression for the periodic production of "turned-over" employee positions a number of additional assumptions are necessary. These assumptions are listed below. As with any model, to the extent that violations of the assumptions affect the model's predictive ability, the assumptions may be replaced with alternative ones which more closely describe the underlying process being modeled. 1. We assume that each employee independently produces units of finished prod- uct.9 2. Employees who are replaced during the period are assumed to have completed the learning process.10 3. The learning period (Ts) is assumed to be not longer than the period under consideration; i.e., Ts < dh. 4. A certain regularity in turnover is assumed such that employees who are hired during the last Ts working days of the period, and thus have not completed learning, 9 If new employees work with other, seasoned employees to produce units jointly, it may be possible to identify a learning curve applicable to separate tasks which are performed by new employees. If so, this learning curve could be used to estimate the resulting increase in average time and average labor cost per unit. 10 Alternatively, one may assume that replaced employees leave, on average, after Q days, where Q is less than the average learning period. With this assumption, Qh replaces T, in (27), and x, is replaced by (Qh/a)l/(b+'), the total estimated production of an individual during the first Q days of the learning period. 158 E. V. MCINTYRE are approximately complemented by employees hired during the last T, working days of the preceding period, and are completing production during the current period. The result of this assumption is that each “turned-over” position may be viewed as producing for one complete learning cycle each period, plus d – T7 days of steady- state production. With these assumptions, the periodic production by employee positions “turned- over” during the period, denoted P1, can be expressed as follows: P1 = rL[xs + (dh – Ts)/ts]. (27) In (27), xs is the number of units produced during the learning period by each employee. The second term within the brackets is the units produced by each employee position during the days of steady-state production. Production by the stable or nonturnover segment of the labor force, denoted P2, is P2 = (1- r)(L)(h/ts)d. (28) Total production equals P1 + P2, and, after some rearranging, may be written in the following form. P1 + P2 = dL(h/ts) + rL[xS – TS(1/)]. (29) (29) shows that total production is equal to the units that would have been produced if the entire labor force had produced at steady-state levels (the first term in the right-hand side of the equation) less a correction factor equal to the difference between the number of units rL employees produced during the learning period and the number they would have produced during this period if they were seasoned employees (the second term on the right in (29), which will always be negative). The above expression for total production may also be used to compute the effect on production of a change in the turnover rate, r. For example, if the turnover rate doubled, periodic production would decrease by the quantity L[xs – Tsts] with a corresponding increase in average labor cost per unit (which can be viewed as one cost of turnover). To obtain average labor cost per unit (denoted by c below), total labor cost is divided by total production;” i.e., c = Ldhc/(PI + P2) (30) The profit function of a firm with turnover of employees subject to a learning period (and the implied assumptions) can now be expressed as S7- =(p – OX -f- (31) (31) may be used to estimate periodic profit and break-even levels, as well as to estimate the effect of a change in the turnover rate on those quantities.12 Conclusion The objective of this paper is to make CVP analysis applicable to more situations by incorporating alternative assumptions concerning the nature of labor costs. Specifi- 1 (30) assumes a constant labor rate for all employees. This could be modified easily to reflect a, lower wage rate for new employees. 12 Under the assumption of continuous turnover, each period will incur some learning costs, and thus deferral of these costs is probably not needed. Use of average costing, coupled with a predetermined fixed overhead application rate which takes into account the effect of turnover on periodic production, will smooth out fluctuating unit costs. (29), as with other equations presented in this article, retains the usual assumption of equality between production and sales; otherwise, to obtain conventional income figures, an adjustment would have to be made for the change in fixed manufacturing overhead in inventory (see Solomons [18] or McIntyre [12]. COST-VOLUME-PROFIT ANALYSIS ADJUSTED FOR LEARNING 159 cally, we have relaxed the assumption that all labor is composed of homogeneous units which can be varied at constant marginal costs. Other articles have dealt briefly or in a very general way with the effect of learning on profit (and more specifically with various aspects of planning that would affect profit -e.g., learning and capital budgeting). This paper attempts to deal more explicitly with the influence of learning on CVP analysis by formally including appropriate factors into the CVP model. The usefulness of the results to a particluar situation is necessarily circumscribed by the applicability of the implied assumptions. However, the procedures employed are intended to be illustrative of a more general class of computations which could be adapted to alternative sets of assumptions. Even with these restrictions, the set of circumstances to which CVP can be applied has been expanded. Hopefully, this will increase the utility of CVP models as useful tools for planning, analysis and control. Appendix Derivation of Average Learning Curve Equation The average learning curve model used in this article states that each time cumulative production doubles, average time per unit is R % of the previous average. This process can be described in a straightforward fashion by the following equation, where m represents the number of times production doubles and all other notation is as previously defined. Y = aRtm. (1) Cumulative production, x, can be expressed as x = 2m. (2) Solving (2) for m and substituting the results into (1) yields y = aR log x/log 2 (3) (3) expressed in logarithmic form becomes logy = log a + (log x/log 2)- log R and substituting b for log R/log 2 we obtain logy = log a + b log x. Finally, taking antilogs provides the learning curve in the form used in the body of the article y = axb (4) Range of R For an average learning curve model, the maximum range of R is 0.50 < R < 1. If R = 1, no learning occurs; if R = 0.50, total time remains constant as production continues to double, an obvious impossibil- ity. Marginal Time Model The expression used for the average learning curve can also be used for a marginal time model. In the marginal model, a, x, and b have the same interpretation as in the average model; however, y represents the marginal time to produce the xth unit. In other words, the marginal model states that the time required to produce the xth unit is reduced by 1 - R every time production doubles. Other things being equal, a marginal time model implies a slower rate of learning than an average time model. For comparison purposes, marginal time, average time, and total time functions for the marginal model are shown below. Marginal time = y = ax Total time = x Y = ax b + I/(b + 1), Average time = axb/(b + 1) =y/(b + 1). In the marginal model, the theoretical limits on R are: 0 < R < 1. References 1. ABERNATHY, W. J. AND BALOFF, N., "A Methodology for Planning New Product Start-ups," Decision Sciences (January 1973), pp. 1-20. 2. ANDRESS, F. J., "The Learning Curve as a Production Tool," Harvard Business Review (January/February 1954), pp. 87-97. 3. BALOFF, N., "Estimating the Parameters of the Startup Model-An Empirical Approach," The Journal of Industrial Engineering (April 1967), pp. 248-53. 160 E. V. McINTYRE 4. AND KENNELLY, J., “Accounting Implications of Product and Process Startups,” Journal of Accounting Research (Autumn 1967), pp. 131-43. 5. AND McKERSIE, R. B., “MOtivating Startups,” The Journal of Business (October 1966), pp. 473-84. 6. BHADA, Y. K., “Dynamic Cost Analysis,” Management Accounting (July 1970), pp. 11-14. 7. , “Dynamic Relationships for Accounting Analysis,” Management Accounting (April 1972), pp. 53-57. 8. BRENNECK, R., “B-E Charts Reflecting Learning,” NA.A. Bulletin (June 1959), p. 34. 9. BUMP, E. A., “Effects of Learning on Cost Projections,” Management Accounting (May 1974), pp. 19-24. 10. HARVEY D. W., “Financial Planning Information for Production Start-ups,” The Accounting Review (October 1976), pp. 838-45. 11. KEACHIE, E. C. AND FONTANA, R. J., “Effects of Learning on Optimal Lot Size,” Management Science (October 1966), pp. 102-8. 12. MCINTYRE, E. V., “An Algebraic Aid in Teaching the Differences Between Direct Costing and Full-Absorption Costing Models: An Extension,” The Accounting Review (October 1974), pp. 839-40. 13. MORSE, W. J. “Reporting Production Costs That Follow the Learning Curve Phenomenon,” The Accounting Review (October 1972), pp. 761-73. 14. , “The Use of the Learning Curve in Financial Accounting,” The CPA (January 1974), pp. 51-57. 15. , “Learning Curve Cost Projections With Constant Unit Costs,” Managerial Planning (March- /April 1974), pp. 15-21. 16. PEGELS, C. C., “Start Up or Learning Curves-Some New Approaches,” Decision Sciences (October 1976), pp. 705-13. 17. SECURITIES AND EXCHANGE COMMISSION, Accounting Series Release No. 33-5492, Washington, D. C., 1974. 18. SOLOMONS, D., “Break-even Analysis Under Absorption Costing,” The Accounting Review (July 1968), pp. 447-52. 19. SUMMERS, E. L. AND WELSCH, G. A., “How Learning Curve Mod pROFESSORS Lau and Lau [1981] have made good use of the formulas for partial moments provided by Winkler et al. [1972] in presenting an alternative derivations for E(T) and Var (T) of the profit distribution which I developed in my paper [1979]. While the Laus’ result for E(T) is virtually no different from mine except for the need to consult different Normal tables, their derivation of Var (T) represents a sig- nificant simplification over my long for- mula. The authors of the Comment also briefly questioned the use of P1=Prob [GT<(R-V)Q-F] instead of P2= Prob [G?< T] as an objective function for optimization. In what follows, I would like to respond briefly to that first, and then to point out a few implications of my model formula- tion for the stochastic break-even analysis as a final wrap-up of my two Replies. Before addressing P2, let's define Q1 and Q2 as the production quantity that maximizes P1 and P2, respectively. It is true that the use of P2 would lead to a simple Q2. However, there is a major drawback in using P2, which is: Q2 will actually reduce P2 to P(T= G). As such, when P2 is used, there will be no chance of making a profit larger than G. There is no such limit with Q1, and therefore the upper bound on profit associated with Q1 and P1 will be higher than that associ- ated with Q2 and P2. As a result, P1 would appear to be more practical to the management than P2. Finally, I wish to point out that the model formulation developed in my paper could be extended and applied to the break-even analysis under uncer- tainty in which the production quantity would play a significant role. For exam- ple, the management may want to sug- gest that a product warrants further consideration for marketing only if: (a) P(T?U)?ca, or (b) P(T ? G) ?x. It can be easily shown that (a) implies that a product is worth marketing if -F + VQ YD > R Za aD; (1) and, similarly, (b) implies that a product should be marketed if F + G + VG YD > R -Za D. (2) The value of Q, of course, can be decided in a variety of ways depending on the goals and objectives of the top man- The author wishes to thank Dr. Joyce Chen of the University of Illinois at Chicago Circle for her sugges- tions on the extension of the break-even analysis. Wei Shih is Professor of Operations Research, Bowling Green State University. Manuscript received and accepted February 1981. 984

Shih 985 agement. It is clear from (1) and (2) that the production quantity Q now enters the stochastic break-even analysis as a factor in the determination of the lower bound of the mean demand. This is a significant departure from the traditional break-even analysis, and thus may be viewed as a potential for future research. REFERENCES Lau, Amy Hing-Ling, and Hon-Shiang Lau, “A Comment on Shih’s General Decision Model for CVP Analysis,” THE ACCOUNTING REVIEW (October 1981), pp. 980-983. Shih, Wei, A General Decision Model for Cost-Volume-Profit Analysis under Uncertainty,” THL Ac- COUNTING REVIEW (October 1979), pp. 687-706. Winkler, R. L., G. M. Roodman, and R. R. Britney, “The Determination of Partial Moments,” Manage- ment Science (November 1972), pp. 290-296.