Working Paper 153

An Unpublished Paper in 1988

This paper was rejected by two major journals and has not been revised since 1984. I considered this one of my better papers over the years, so now I am providing it for the world to judge. Referee comments that the paper is too hard to read are justified criticisms. However, I think that the analysis was timely in 1988 following up on the Manes, Park, and Jensen (1982) paper in The Accounting Review.

A PARAMETRIC PROGRAMMING
FORMULATION FOR RECIPROCAL
COST POST-OPTIMALITY ANALYSIS

Bob Jensen at Trinity University

Table of Contents

Abstract

Background of Reciprocal Cost Analysis

Background of Post-Optimality Analysis

Purposes of This Paper

A Well Known Illustration

Problem 1 Parametric Programming Solution

Problem 2 Parametric Programming Solutions

A Paradox: Reciprocal Cost Transfer Prices May Decline When Variable Costs Increase and Increase When Revenues Increase

The Relationship of Opportunity Costs to Fixed Costs and to Full Costing

Table 1

Table 2

Table 3

References

Appendix I - Linear Programming Tableaus for Problem 1

Appendix II - Final Linear Programming Tableaus for Problems 2-A, 2B, 2-C

Appendix III - Final Linear Programming Tableaus for Problems 3-A, 3-B, 3-C

Abstract

Simultaneous equation and linear programming analyses of the reciprocal cost problem have been proposed in a succession of papers beginning in the early 1960s and carrying through to the more recent mixed-integer formulation of Manes, Park, and Jensen [1982]. Although various papers have utilized shadow price (dual variate) analysis, none have extended this to post-optimality analysis.

The major purposes of this paper are to (i) formulate a post-optimality analysis and (ii) to point out some major dangers in traditional shadow price analysis. This paper argues that the costs of goods and services must reflect the costs of capacity as well as the variable costs of production. These costs are considered in a reciprocal costing setting in which the capacities of the interrelated departments are constrained. Unlike sensitivity analysis in linear programming, parametric programming accommodates simultaneous variations in parameters. A general parametric programming approach is formulated which provides formulas for rapid calculation of the impacts of capacity, cost, and price changes on reciprocal cost and output decisions. It is shown how cost increases may paradoxically cause reciprocal decreases and how revenue increases may paradoxically increase reciprocal costs.

Background of Reciprocal Cost Analysis

The problem of reciprocal costs and the need that such costs be determined by the solution of a system of simultaneous equations was understood long before the widespread use of computers and the application of quantitative methods in managerial accounting [Newlove & Garner, 1949, T. Lang, 1944]. A distributional simultaneous equation cost allocation technique using matrix algebra was proposed in Manes [1963]. Shortly thereafter, several papers introduced detailed explanations of the procedures involved in calculating reciprocal service center costs, notably Williams and Griffin [1964] and Churchill [1964].

The solutions suggested by these authors were based on matrices which expressed the related percentages of the total outputs of the interacting departments consumed by other service departments, and their approach shall be referred to as the "percentage of operations" method. Although this method calculates reciprocal service costs optimally, the firm must reformulate (and invert) the matrix of interdepartmental activities each and every time the relative proportions of the productive and service activities change; see Baker and Taylor [1979, p. 790].

Subsequently and separately Ijiri [1968], Livingstone [1969], and Farag [1967, 1968] showed that the reciprocal service cost problem of the firm is directly comparable to the macroeconomic, input-output models developed by Leontief [1951]. The submatrix of interacting service departments is like a subset of the national economy which produces only intermediate goods and services and nothing for final consumer demand. This development of the topic, by its reference to the technological coefficients of production (which we shall refer to as the "input-output" approach) focused its attention on variable costs. The culminating piece of the "input-output" articles was R. Kaplan's [1973] paper. See also Kaplan [1982]. Following an earlier and similar demonstration by Livingstone [1968], Kaplan settled the confusion regarding methods of mathematically formulating the matrix of interdepartmental relationships, a confusion introduced by Manes [1965] and compounded by Minch and Petri [1972]. In so doing, Kaplan showed how the input-output approach was related to the earlier efforts of Williams and Griffin [1964] and others. Kaplan presented a definitive explanation of the linear algebra operations required to calculate reciprocal (or simultaneously determined) variable service department costs. ¹

Although not dealing with reciprocal cost problem per se (which may include direct labor, direct material, and overhead), a related literature on overhead allocation to products evolved using mathematical models. Kaplan and Thompson [1974] utilized linear programming based upon a firm's production function. Kaplan and Welam [1974] extended the linear analysis to a more general nonlinear production function (in their Appendix). A more general survey of overhead allocation is given in Atkinson [1987]. More recently, Balachandran, Li, and Magee [1987] discuss the economic theory of overhead allocation. The paper is discussed by Atkinson.

Capettini and Salamon [1977] extended reciprocal cost analysis by adding semi-variable costs and externally available services (resources) competing with services produced internally. Baker and Taylor [1979] provide an innovative linear programming formulation that utilizes the dual model for shadow price anlaysis of opportunity values coupled with externally available services. This paper was followed by the Manes, Park, and Jensen [1982] mixed-integer programming formulation that incorporated relevant (avoidable) fixed costs.

The most recent papers on reciprocal cost analysis are Mensah [1988] and Jensen [1988]. Mensah presents a matrix alegbra approach for reapportionment of budgeted variable costs. Jensen introduces capacity cost opportunity loss concepts that will be analyzed more formally in parametric programming analysis in this paper. The paper also extends to post-optimality analysis, the Balachandran, Li, and Magee [1987] theory for allocating fixed service department costs.

Background of Post-Optimality Analysis

A linear programming (LP) formulation of reciprocal cost analysis was introduced by Baker and Taylor [1979] and extended to a mixed integer LP model by Manes, Park, and Jensen [1982]. However, neither paper pursues one of the principal benefits of the LP approach, i.e., the benefits of post-optimality analysis. There are two major approaches for post-optimality analysis:

(1) The single-parameter variation approach terms "sensitivity analysis" in which one model parameter is varied ceteris paribus relative to all other model parameters. The purpose is to test the sensitivity of a solution (results) to a critical input parameter that is typically subject to estimation error and/or possible non-stationarity over time.

(2) The multliple-parameter variation approach termed "parametric programming" in which multiple model parameters are jointly varied to determine the sensitivities of results to parameters that typically move jointly, e.g., various input costs that move due to correlation with underlying factors in the economy.

Many years ago, sensitivity analysis in accounting was proposed in Rappaport [1967]. It was critically analyzed and discussed in an integer programming context in Jensen [1968]. Sensitivity analysis was also the focus in Glover [1969], Hartley [1970], Currin and Spivey [1972], Kaplan and Thompson [1974], and Kaplan and Welham [1974]. Parametric programming has seldom appeared in accounting literature. Jensen [1968] introduced parametric programming for joint cost allocation.

Purposes of This Paper

Jensen [1988] alludes to capacity opportunity costs for reciprocal costing. However nothing in detail appears in the literature to date regarding postoptimality analysis of the reciprocal cost problem. The major purpose of this paper is to extend the Jensen [1968] parametric programming analysis to the reciprocal cost problem and to formalize the Jensen [1988] discussion. Hillier and Lieberman [1986, pp. 282-283] describe parametric programming as follows:

Sensitivity analysis involves changing one parameter at a time in the original model to check its effect on the optimal solution. By contrast, parametric linear programming (or parametric programming for short) involves the systematic study of how the optimal solution changes as many of the parameters change simultaneously over some range. This study can provide a very useful extension of sensitivity analysis, e.g., to check the effect of "correlated" parameters that change together due to exogenous factors such as the state of the economy. However, a more important application is the investigation of tradeoffs in parameter values. For example, if the c_j represent the unit profits of the respective activities, it may be possible to increase some of the c_j at the expense of decreasing others by an appropriate shifting of personnel and equipment among activities. Similarly, if the b_i represent the amounts of the respective resources being made available, it may be possible to increase some by the b_i by agreeing to accept decreases in some of the others. In some applications, the main purpose of the study is to determine the most appropriate tradeoff between two basic factors, such as costs and benefits. The usual approach is to express one of these factors in the objective function (e.g., minimize total cost) and incorporate the other into the constraints (e.g., benefits > minimum acceptable level), as was done for the Nori & Leets Co. air pollution problem in Sec. 3.4. Parametric linear programming then enables systematic investigation of what happens when the initial tentative decision on the tradeoff (e.g., the minimum acceptable level for the benefits) is changed by improving one factor at the expense of the other. This approach is illustrated by the case study in Sec. 8.5., where the two basic factors are the distance travelled by high school students and the degree of racial balance achieved in their schools.

The algorithmic technique for parametric linear programming is a natural extension of that for sensitivity analysis, so it too is based on the simplex method.

The simplex algorithmic parametric programming approach mentioned above is illustrated in the appendices of this paper:

 

Appendix I - Linear Programming Tableaus for Problem 1

Appendix II - Final Linear Programming Tableaus for Problems 2-A, 2B, 2-C

Appendix III - Final Linear Programming Tableaus for Problems 3-A, 3-B, 3-C

Related purposes of this paper are to demonstrate that:

(1) Traditional shadow price analysis based upon assumed parameter stationarity not only is of limited value but may be misleading in the presence of parameter variations or tradeoffs.

(2) Reciprocal costs (and transfer prices) of most or all interactive service departments are jointly impacted "cost creep" at or near capacity operations. Parametric programming shows the impact of creeping costs due to economy-wide price increases.

Here, an important point must be made. Reciprocal costs and opportunity losses of all service departments rise sharply if any single one, or any combination of departments, are at capacity. Under normal conditions, we can expect that in the well managed, going concern at least some service departments would be operating near or at full capacity. As Dixon [1953] pointed out long ago:

If the company is in a position to make permanent additions to its side activities without adding to its fixed costs, something must be out of order. Fixed costs reflect capacity to operate, and they appear throughout the organization. Evidently if fixed costs can be ignored in such a calculation, the company is overequipped and overstaffed for its regular work [p. 49, emphasis added].

R. Anthony, in a discussion of a well known managerial accounting case, Martall Blanket, points out that in a practical sense management should not think of shadow prices only when capacity is already completely occupied but instead should think in such terms whenever any major addition to activities would utilize remaining slack. ²

A Well Known Illustration

In order to pursue the discussion and to illustrate points made, we shall utilize the same problem chosen by Kaplan [1973] and later by Capettini and Salamon (C&S) [1977] and Kaplan [1982] to demonstrate a linear algebraic solution of reciprocal service costs. The problem, which is attributed to David Green, is printed below as it appears in C. Horngren's Cost Accounting 5th Edition, p. 500:

The Prairie State Paper Company located a plant near one of its forests. At the time of construction, there were no utility companies equipped to provide this plant with water, power, or fuel. Therefore, included in the original facilities were (1) a water plant, which pumped water from a nearby lake and filtered it; (2) a coal-fired boiler room that produced steam, part of which was used for the manufacturing process and the balance for producing the electricity; and (3) an electric plant.

An analysis of these activities has revealed that 60 percent of the water is used for the production of steam and 40 percent is used in manufacturing. Half of the steam produced is used for the production of electric power and half for manufacturing. Twenty percent of the electric power is used by the water plant and 80 percent goes to manufacturing.

For the year 19_9, the costs charged to these departments were:

Kaplan and C&S assume demand for the final product of b₄ = 100 thousand units, and from this demand they generate a production schedule as follows:

The "true" or simultaneously determined costs of water, steam and electricity respectively proceed from the solution of the following system:

X = C + AX,

X - AX = C

X = (1-A) -¹C

where j^th term of (m x 1) vector X, represents the increase in total service departmental cost for production of one additional unit of service from the j^th department, C is an (m x 1) vector, c_j is the traceable variable cost per unit of service department j, and A is an (m x m) matrix, a_ij being the number of units of service department j required for each unit of output for service department i.

In this problem

		0	0	.8				$	13.33
A	=	.5	0	0	,	and	C =	$	100.00	.
		0	.15	0				$	10.00

As solved correctly by Kaplan and C&S, per units costs of water, steam and electricity are p₁ = $35.46, p₂ = $117.73, and p₃ = $27.66 respectively for this system.

As elegant and as useful as prior literature is in the development of the cost allocation methodology, it fails to consider a major part of the above problem and of service department cost calculations in general, namely that of costs of capacity. A central consideration, after all, in the creation and operation of many (if not most service departments) is in the acquisition of a large piece of equipment, such as a power generating unit, a pumping system, a printing press, a photocopy machine or a computer. The cost of computing services, for example, certainly should include the cost of buying or leasing the equipment and the expense of compensating the computer center supervision.

Earlier writers, such as Horngren, Kaplan, and C&S, implicitly assume that product demand (e.g., b₄ = 100 thousand units of paper) can always be satisfied by internal resource capacity or external purchase of resource inputs. We will now extend this to alternate problems (cases) where product demand cannot be fully satisfied.

The well-known problem or case in which x₄ product demand (x₄ < b₄) is the only binding (active) restraint will be termed "Problem 1." In Problem 1, the Water Plant, Steam Room, and Electric Plant are all assumed to have enough or more than enough (slack) capacity to meet forecasted product demand. In Problem 2-A, the Water Plant does not have adequate capacity, i.e., x₁ < b₁ replaces x₄ < b₄ as an active constraint. There is no way product demand of b₄ units of paper can be satisfied such that there must be b₄ - x₄ thousand units of demand for paper that remains unsatisfied. In Problem 3-A, the Water Plant has the same active constraint, but unlike in Problem 2-A, it is possible to circumvent this problem by buying added steam externally such that it becomes feasible (with higher costs) to fully satisfy demand of b₄ units of paper.

The problems (cases) described above plus several other problems are summarized in Table 1. These problems are analyzed in subsequent portions of this paper.

See Table 1

It might be noted that in these problems the c_j parameters are as follows:

c1	=	$ 2,000/150	=	$13.33 variable cost per thousand gallons of water produced in the Water Plant
c2	=	$18,000/180	=	$100.00 variable cost per thousand cubic feet of steam produced in the Steam Room
c3	=	$ 6,000/600	=	$10.00 variable cost per thousand k.w.h. of electricity generated by the Electric Plant
c4	=	$70,000/100	=	$700.00 price per thousand units of paper produced by the Prairie State Paper Company

The LP solutions to the x_j primal variables, y_j dual variables, and p_j reciprocal cost transfer prices are shown in Table 2. Only the Problem 1 solution conforms to the Kaplan [1973, 1982], Horngren [1982], and C&S [1977] solutions. Solutions to Problems 2-A, 2-B, 2-C, 3-A, 3-B, and 3-C with active resource capacity constraints have not been previously analyzed.

See Table 2

Problem 1 Parametric Programming Solution

The algorithmic approach to solving parametric programming problems is illustrated in the various tableaus of Appendices I, II, and III.

Appendix I - Linear Programming Tableaus for Problem 1

Appendix II - Final Linear Programming Tableaus for Problems 2-A, 2B, 2-C

Appendix III - Final Linear Programming Tableaus for Problems 3-A, 3-B, 3-C

To avoid both complexity and lengthy discussion, details of this algorithm will not be provided here. Readers are referred to Hillier and Lieberman [1986, pp. 280-285] or other operations research literature on parametric programming. Suppose the standard LP objective function (for the Table 1 problems)

Max x₀ = å⁴_{j = 1} c_jx_j

is revised to parametric programming form using conventional notation to read

Max Z (q) = å⁴_{j = 1} ( c_j + a_jq ) x_j,

where in the words of Hillier and Lieberman [1986, p. 280]:

the a_j are given input constants representing the relative rates at which the coefficients are being changed. Therefore, gradually increasing q from zero changes the coefficients at these relative rates. The values assigned to the a_j may represent interesting simultaneous changes of the c_j for systematic sensitivity analysis of the effect of increasing the magnitude of these changes. They may also be based on how the coefficients (e.g., unit profits) would change together with respect to some factor measured by q. This factor might be uncontrollable, e.g., the state of the economy. However, it may also be under the control of the decision maker, e.g., the amount of personnel and equipment to shift from some of the activities to others. For any given value of q, the optimal solution of the corresponding linear programming problem can be obtained by the simplex method. This solution may have been obtained already for the original problem where q = 0. However, the objective is to find the optimal solution of the modified linear programming problems (maximize Z(q) subject to the original constraints) as a function of q.

By way of illustration, suppose the relative joint rates of coefficient change are as follows:

a1	=	0.5	º	rate of change of water variable cost
a2	=	1.0	º	rate of change of steam variable cost
a3	=	3.0	º	rate of change of electricity variable cost
a4	=	-0.5	º	rate of change of paper sales price

The variable costs/prices are as follows:

c1	+	a1q	=	$ 13.33	+	0.5(q)	per thousand gallons of water.
c2	+	a2q	=	$100.00	+	1.0(q)	per thousand cubic feet of steam.
c3	+	a3q	=	$ 10.00	+	3.0(q)	per thousand k.w.h. of electricity.
c4	+	a4q	=	-$700.00	-	0.5(q)	per thousand units of paper sold.

The parametric solutions to the dual model shadow prices for Problem 1 production of 100 thousand units of paper are:

y1	=	$	0	opportunity value of added Water Plant capacity.
y2	=	$	0	opportunity value of added Steam Room capacity.
y3	=	$	0	opportunity value of added Electric Plant capacity.
y4	=	$	400	- 3.0(q) - 1.8(q) - 2.0(q) + 2.0(q)
	=	$	440	- 4.8(q) º opportunity value of added paper demand.

Recall that in Problem 1, there is slack capacity in production of water, steam, and electricity. Hence, there is zero value in adding capacity of production. This slack capacity can be used in producing more paper. Each unit demand of paper has an opportunity value of y₄ shown above. When q = 0, this opportunity value is comprised of the following:

For q > 0 increases, however, paper prices and all variable costs jointly increase. Costs increase faster than do paper prices such that the y₄ opportunity value of paper declines according to $440 - 4.8q. This opportunity value hits zero when q = 440 / 4.8 » 91.67.

Let p_j (q) depict the Department j reciprocal cost transfer price as a function of q. This can be shown to be

p_j (q) = p_j (q) + å⁴_{i = 1} ¡_ija_iq,

where ¡_ij is the reciprocal augmentation coefficient of the Department i cost change on Department j output. Each ¡_ij value is a negation of the value found in the final t_j1 column (and the row corresponding to the b_i capacity constraint) in the final tableau of Appendix I for Problem 1. For example, the reciprocal cost transfer price of water becomes

In an analogous fashion we derive reciprocal costs (transfer prices) for all departments compared below with variable cost changes:

Hence, if q depicts a state of rising price levels in the economy, the economy impacts both absolutely and relatively harder on electricity variable and reciprocal costs vis-a-vis water and steam variable and reciprocal costs.

Problem 2 Parametric Programming Solutions

Problems 2-A, 2-B, and 2-C in Table 1 have active resource constraints such that it is impossible to fully satisfy customer demand for paper produced by Prairie State Paper Company. Suppose variable costs have the same parametric functions as illustrated previously in Problem 1:

Parametric programming solutions to opportunity values and reciprocal costs are shown in Table 3.

See Table 3

Note the tremendous impact that active service center constraints have upon their reciprocal costs (transfer prices). For example, in Problem 1 the Water Plant had excess capacity, and the reciprocal cost of water was only $35.46 + 3.210q per thousand gallons. In Problem 2-A, the water capacity shortage (coupled with inability to buy water, steam, or electricity externally) raises the reciprocal cost of water to a huge $357.51 - 10.995q per thousand gallons. When q = 0, this is nearly a tenfold increase caused by the enormous opportunity value of increasing Water Plant capacity.

Also note how the reciprocal costs get passed along. For example, in Problem 2-A, the increase in steam from $117 + 2.615q to $273.75 - 4.495q results from the increased price of water purchased from the Water Plant by the Steam Room. Similarly, the rise in the reciprocal cost of electricity is due mainly to the opportunity value of water passed along in water and steam prices to the Electric Plant.

The tremendous increase in the reciprocal cost of water is mitigated by general price rises in the economy. Because variable costs of water, steam, and electricity jointly rise more than the sales price of paper as q increases, increasing q values decrease the opportunity value of water capacity (in Problem 2-A) or steam capacity (in Problem 2-B) or electric capacity (in Problem 2-C). For example, when q = 0, the Problem 2-A reciprocal cost of water is $347.51 per thousand gallons, the bulk of which is due to the opportunity value of Water Plant capacity. For q > 0 price rises in the general economy, however, the reciprocal cost of water is reduced by $10.945q according to the parametric programming solution (in Problem 2-A):

p₁(q) = $347.51 - 10.945(q).

For q = 347.51 / 10.945 » 31.75, the reciprocal cost of water is reduced to zero, because variable costs (of water, steam and electricity jointly) have risen to exactly equal the opportunity value of water capacity.

A Paradox: Reciprocal Cost Transfer Prices May Decline
When Variable Costs Increase and Increase When Revenues Increase

It is evident in Table 3 that p_j reciprocal cost transfer prices often decrease when variable costs increase and increase when revenues increase. Such paradoxical phenomena may occur when a service Department J has a variable cost increase and a capacity opportunity value (shadow price) y_j > 0. The increase in variable cost may be more than offset by a decline in the opportunity value such that the reciprocal cost p_j actually declines. Increases in other service department costs may also have decreasing effects because of reciprocal impacts on opportunity value.

The opposite situation exists if the price increases on a firm's final output product such that more revenue is generated. In any Department j having a capacity opportunity value (shadow price) y_j > 0, the reciprocal cost transfer price p_j will increase, because the increase in revenue increases the reciprocal costs of other departments.

Recall that the reciprocal costs of water in Problem 1 versus Problem 2-A are as follows:

Problem 1:	p1(q)	=	$	35.46	+	3.210(q)
Problem 2-A:	p1(q)	=	$	347.51	-	10.995(q)

The paradoxical reason that reciprocal costs increase by +3.210(q) in Problem 1 versus decrease by -10.995(q) in Problem 2-A can now be explained. Recall that a_j parameters are such that, as q increases to reflect economy-wide price increases, variable costs of water, steam, and electricity increase faster than paper prices. In Problem 1, p₁(q) is not affected by paper revenues, becaue p₁(q) contains (rising) variable costs only, i.e., when the Water Plant has slack capacity the reciprocal cost transfer price of water includes zero opportunity value for added production capacity.

Conversely, in Problem 2-A, the p₁(q) reciprocal cost transfer price includes non-zero opportunity value for added Water Plant capacity. This opportunity value increases as the price of paper increases. However, when q increases (to reflect economy-wide price increases) paper price increases lag behind increases in the variable costs of water, steam, and electricity. Hence, opportunity value of added paper production (facilitated by adding water production capacity in Problem 2-A) declines as q increases. This accounts for the minus sign in p₁(q) = $347 - 10.995(q). If paper prices increased faster than variable costs the minus sign would change into a plus sign.

The Relationship of Opportunity Costs
to Fixed Costs and to Full Costing

Referring back to the example problem, we know that the profit that has been maximized is actually the direct costing, gross margin of the firm, i.e., that c₄ is the price less direct variable costs per unit of x₄ and that c₁, c₂, and c₃ are the traceable direct variable costs of service departments 1, 2, and 3 respectively. The price of paper is set high enough to cover all out-of-pocket fixed costs per period and some allocation of sunk fixed costs and also to yield a profit. The shadow prices of a capacity constraint (which when multiplied by a positive-valued right hand side constraint value b₁, equals profit) thus include fixed costs, but they do so only in a very situation-specific way in that the fixed costs are reflected through the price of the paper and all fixed costs are charged to the service department whose capacity is binding. Supplemented by post-optimality analysis (i.e., sensitivity analysis or parametric programming), opportunity value shadow prices provide the necessary relevant data for the immediate short run, but they do not provide us with very useful calculations of reciprocal service costs for longer run planning. The analysis thus suffers from myopia in a sense analogous to what Arrow and Lind [1970] term a myopic property. However, Baumol [1971, p. 654] points out "myopic decision rules can be helpful in practice because, where they apply, they can simplify the calculation of marginal costs and depreciation charges."

There are three reasons for which the shadow prices related to the services department technological constraints may prove inadequate for furnishing useful costs for planning. The first, referred to in the previous paragraph is that they include an amount equal to the profit of the firm which uses the services. This is not the most serious difficulty and it can be adjusted for by recalculating the optimal solutions of Problem 2-A, 2-B, and 2-C at a break even price for paper. On this point, Professor Dixon [1953, p. 53] argued that:

to justify the addition of a permanent new activity on the grounds of cost savings, the best available supplier's price must be shown not only to exceed the full cost of production, with no apportionable costs omitted, but it should be higher by an amount at least equal to the rate of profit which the company is able to make through its principal operation (emphasis added).

If one agrees with Dixon, no adjustment for profit would be necessary.

The second reason for being cautious in the use of shadow prices as costs is related to the mathematical nature of their calculation. For every constraint in the LP solution there is a basic variable and a shadow price; and for every non-slack, basic variable there is a positive shadow price. In the problem examined above, there are three kinds of constraints: (1) there are technological constraints, m in all, one for each service department; (2) there are m or less constraints corresponding to the capacities of the service departments; and (3) there are n or less market constraints for the n final products, the demand vector of final output. However, there can only be m + n or less non-slack basic variables with positive values and given the reciprocal, input-output nature of the service departments, for any positive final output at all there will necessarily exist m technological constraint shadow prices. That means there remain only n shadow prices to be calculated for both the n(or less) marketing constraints and for the m(or less) service department capacity constraints.

As a result of these limits, under certain conditions it is possible for a service department capacity to be binding but to have a zero shadow price. In fact for alternate case Problems 2-A, 2-B, and 2-C, should two or more service departments ever operate at full capacity simultaneously in the production of paper, a degenerate solution results. Specifically, one or more of the slack variables for constrained service department capacity remain in the basis at a zero value and, consequently their related shadow prices have zero values, all of the profit being imputed to the other service department via capacity constraints and/or market constraints. There being no shadow price for these service departments, no cost is attributed to the department in the solution of X* = [I - A]^-1C*. The resulting amounts calculated for reciprocal costs of service departments augmented for capacity costs can thus vary, depending on a tie-breaking rule in the LP algorithm rather than on the intrinsic costs of the service departments. In general, the problem of degeneracy and the resulting unsatisfactory conditions are more likely to result when m is larger than n.

The third reason for rejecting shadow prices for more than short run decision making purposes (e.g., for calculating standards or long run costs) is also related to the nature of the mathematical solution of the LP problem. Shadow prices only appear when capacity is reached; the shadow prices act like a cattleprod or an electrified fence or an alarm at point of contact and not like a radar system which signals the approach of full capacity. Unlike the results of marginal analysis, in which solution values shift gradually for continuous functions, shadow prices change abruptly as solutions move from one extreme point on the convex set to another. For this same reason, shadow prices have not proven practical for transfer price determination [Manes, 1970] and, as Dopuch and Drake [1964] have pointed out, the LP algorithm does not discriminate between marketing or capacity constraint.

And so, although we concur with Professor Dixon's [1953, p. 50] position, "that (capacity) fixed costs simply cannot be ignored in making the produce-or-purchase decision unless one is satisfied with a very short sighted analysis," for the above reasons shadow prices will often fail to make an appropriate provision for fixed costs.

Since we are not satisfied with shadow prices as service department costs, we are reduced to examining the full costs reciprocal cost. Resorting again to the paper company problem, we adjust the traceable cost vector C to include a "per unit" capacity cost. As Abel [1978] argues, capacity is created in anticipation of a certain volume of production and therefore the capacity costs should be allocated on the basis of planned utilization rather than on the basis of actual utilization. ³

Kaplan points out the production schedule of service center output will flow from the firm's planned needs for final output (Q - UP(I - A)^-1 where U is the (1 X n) vector of demand for final product, P is the (n X M) matrix of input requirements of final production for service, and Q is the (1 X m) vector of service department output required). Thus for the problem at hand, suppose Q = [150, 180, 600] and

Different C*s and X*s could be calculated for every change in the scheduled utilization of services but would probably only be calculated for major proposed changes in the use of capacity.

Solving for X*, we obtain:

These full costs fall within the range of costs for 2-A, 2-B, and 2-C adjusted to omit profit. For the going concern, which sizes its service operations according to its needs, such costs are far more representative of the opportunity costs of the firm than are those costs derived from a variable-cost-only computation.

One can argue then that, because of the inadequacies of shadow prices for any planning beyond the next immediate decisions, full costing is the best proxy of an opportunity cost system. Note that in a competitive economic system the outside supplier of services, in the long run, must recover fixed or capacity costs as well. Therefore, the price demanded for services will include a provision for the capacity costs. For the purpose of long run planning, management must consider the efficiency of capacity utilization vis-a-vis others in the market. Often, the use of excess capacity to support new services leads to busying facilities in activities for which they were not originally planned. This view is also supportive of Zimmerman's [1979] paper which provides additional defense of traditional cost accounting techniques by showing how cost allocation (1) control the overconsumption of agent perquisites, and (2) proxy the costs of degraded service, delays and future expansion, costs that arise when a common resource (or service) is shared by several decision makers. In that sense our paper is a formal expression of Zimmerman's second argument.

________________
¹ Kaplan [1977] in his Beyer lecture at the University of Wisconsin states, "Perhaps the biggest triumph of mathematical modeling to cost accounting has been the matrix approach to allocating costs from interacting service departments to revenue producing departments."

² Anthony, R. A. and J. S. Reece, Managerial Accounting: Test and Cases, 5th Ed., [1975]. In the solution to the Martall Blanket case, p. 605, Anthony points out, "that a company does not have to be at full capacity before it switches from a contribution-analysis approach to a normal-pricing approach.... (I know of a steel company that sold its last increment of "excess" capacity to an auto producer at less than normal margin, only to have a nation-wide steel shortage occur months later. The result was a loss of tens of thousands of dollars...").

³ We are mainly concerned with determination of long run costs for planning. Should the actual production deviate from the planned utilization volumes, variances will arise. Examination of the variances is also a managerial responsibility which is not discussed in this paper.