UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN BOOKSTACKS Digitized by the Internet Archive in 2011 with funding from University of Illinois Urbana-Champaign http://www.archive.org/details/similaritiesunde144mors Faculty Working Papers SIMILARITIES UNDERLYING ACCOUNTING APPLICATIONS OF MATRIX ALGEBRA Wayne J. Morse 0144 College of Commerce and Business Administration University of Illinois at Urbana-Champaign FACULTY WORKING PAPERS College of Commerce and Business Administration University of Illinois at Urbana-Champaign December 31, 1973 SIMILARITIES UNDERLYING ACCOUNTING APPLICATIONS OF MATRIX ALGEBRA Wayne J. Morse //144 In recent years matrix algebra has been applied to a large number of accounting problems. Among -plications most widely discussed are cost allocation and estimating the allowance for doubtful accounts. Unfortunately, because of the way these and other applications of matrix algebra are formulated in the literat hey appear fundamentally different and unrelated. As a consequence, students may memorize in- dividual applications of matrix algebra in a time consuming manner and never understand the underlying concepts well enough to apply them to new situations. The purpose of this paper is to demonstrate how the applications of matrix algebra mentioned above can be formulated and examined in a manner that brings out their similarities. When there is a conflict between clarity of presentation and computational efficiency, clarity of presentation is emphasized. The author has found from experience that this approach reduces the c required to cover these and other applications of matrix a] The cost a! g service department costs to production departme . ervice departments provide reciprocal services to each other. When this condition exists the direct and step allocation methods are found wa because they do not recognize the reciprocal relations between service departments. Matrix algebra provides at least two methods for solving this preMen. First, the direct costs of each production and service department can be formulated as a series of linear equations and the total costs of the production departments found in a manner similar to that presented in finite mathematics courses.. Second, the relationship rtment can be formulated as a transition probability matrix and the portion of each service department's costs that are ultimo. llocats production department can be found by appropriate matrix operations. Linear Algebra : In linear algebra, the solution to a system of n linear equations with n unknowns can be found by first placing the equations in matrix notation and then postmultiplying the inverse of the coefficient matrix by the vector of knowns. The solution to: AX = B (1) is: X * A-lB (2) where: X * vector of unknowns; A » coefficient matrix; and B a vector of k This approach can be s situation presented by i Churchill in an ear There are three service departments, 5, of $2,000; $2,000; and $5,000 respectively, and four production departments, A, B, C, and D with direct costs of $10,000; $12,000; $14,000; and $8,000. The service departments allocate their costs as follows: From \ B C D S l S 2 S 3 .2 .4 .1 .1 .2 .1 .2 .2 .2 .3 .1 .1 .3 .4 C .1 The direct costs cf each department car. be expressed with the following series of linear equations: lA-OB-OC-OD-.2Sj-.IS2~.lS2 * 10,000 -0A+1B-0C-0D-.4SJ-.2S2-.1S3 ■ 12,000 -0A-OB+lC-OD-.lS r 0S 2 -.3S 3 = 14,000 -0A-0B-0OlD~.lSi-.2S 2 -.4S 3 - 8,000 (3) -OA-OB-OC-OD+lS r .2S 2 -0S 3 - 2,000 -OA-OB-OC-OD-OSj+IS^.lSs * 2,000 -0A-0B-0C-QD-.2S 1 -.3S 2 +1S 3 ■ 5,000 It should be noted that A, B, C, D f Sj, S 2 , and S- represent the total costs that flow to or through these departments , not the direct costs of these departments. The costs of department A are equal to the variable A less the cost: from, departments Sj., S 2 , and S 3 . Failure to understand atween the- direct costs of a department and th< through a department 2 can lead to confusi In matrix notation the above system of linear equations is expressed as follows: 1 -2 -.1 -.1 1 -.4 -.2 -.1 1 -.1 -.3 1 -.1 -.2 -.4 1 -.2 1 -.1 -.2 -.3 1 1*" 10,000 B 12,000 C 14,000 • D * 8,000 S l 2,000 2 »» -III (4) -4- The total costs of the production departments can now be determined by multiplying the vector of direct co the inverse of the coefficient matrix. "** "~1 - 1 A B C D s 1 s 2 J3 *i lO 1801 | I , 000 ' 8,000 2 , 000 2 e 000 S,000 ■■14 .; .02070 0352 . L99 i.o: fi o , ooo] 12,000 11,398 14,166 {16,141 = 11,296 2,526 2,629 6,294 I— (3) Hence, the total costs in production departments A, B, C, and D are $11,398; $14,166; $16,141; and $11,296 respectively. These amounts total to $53,001, the sum of the direct costs of the production and service departments. An immediate advantage of this presentation is its similarity to the solution of any system of linear equations. It is strange that the solution to the cost allocation problem is not presented this way in 4 the literature. It should be noted that the costs that .flow through the service de- partments exceed the direct costs of those departments because costs are reallocated back and forth between them a number of times before they are finally allocated to, cr absorbed by , the production departments. In reality, an absorbing Markov process is taking place. Accordingly, the cost allocation problem can also be formulated as an absorbing Markov process. Markov Process : The transition probability matrix for an absorbing Markov process has the following standard from: 5 r o CO -5- where : I ■ probability of going from one absorbing state to another (identity matrix) ; = probability of going t> state to a nonabsorbing state (zero ) ; R = probability c .orbing state to an absorbing state; and Q = probability of going from one nonabsorbing state to another. Ultimately, the process will be absorbed. The important questions are how many transitions will it take for the process to be absorbed and in what absorbing state will values in the nonabsorbing states end up. The first question is answered by solving the following equation: N - (I-Q)" 1 (7) where : N * average number of times a value in a nonabsorbing state will be in various nonabsorbing states before it is absorbed. The second question is answered by solving: B * N-R (8) where : B » portion of ng state that will end up in various absorbing state Once B is determined, the uitim; > direct costs of various service de, ostrr.ultiplying the row vector of direct service department costs by B. Once again, consider the situation presented by Churchill. The transition probability matrix for production and service department costs is as follows: ■6- l .3 .1 .2 .4 2 .1 (9) The matrix indicates the disposition of costs that flow through various departments during one iteration of the allocation process. For example, of the costs that flow through service department S,, 20 percent go to production department A, 40 percent go to department B, etc. To find the average number of times a direct service department cost flows through various service departments before it is absorbed, solve for N. N .2 [0.1 J! 1-i 1,00414 0.02070 0.20704 S Q704 1.03520 0.35199 (10) [0.02070 0.10350 1.03520 For example, in the allocation process a dollar of direct xosts in Sj will flow through Sj an ave .00414 times, S average of 0.02070 times, and S-, an average of 0.20704 times before it is absorbed. The similarity of the numbers in (10) and those in the lower right hand corner of (5) should be noted. The total flow of dollars through the service departments is easily found by multiplying (10) and the vector of direct service departments costs: [2,000 2, COO 5,' [2. 526 2.629 6 ,294 J 1.00414 0.02070 0.20704 0.33199 0.02070 0. 10350 1.05520 cm • 7> As expected, the total flow of costs through the service departments exceeds the direct costs of the service departments. This answer can be compared with that obtained in (5) . To find the portion of t rs in various service departments that will ultimately end up in r production department, solve for B. B = 'l. 00414 0.02070 0.20704 L2 .4 .1 .11 0.20704 1.03520 C. 55199 . .1 .2 .0 .2 0.02070 0.10350 1.03520J [. 1 .1 .3 .4] 0.22360 0.42650 0.162S3 0.18737 0.18013 0.32506 0.12630 0.36854 0.11801 0.13250 0.31263 0.43685 (12) For example, a dollar of direct costs in S will ultimately be allocated l to A, B, C, and D in accordance with values in the first row of (12). The similarity of the numbers in (12) and those in the upper right had corner of (5) should be noted. The ultimate allocation of direct service department costs can be found by multiplying (12) and the row vector of direct service department costs: 5253 0.18737* [2,000 2,000 5,000] • 0.12630. .0.36854 t3 0.43685 As expected, when the direa snt cosl e added to those allocated from the >artments , Final solution is the same as • that obtained when the problem was formulated as a series of linear equations. An advantage of presenting the solution to the cost allocation problem as a series of linear equations and then as an absorbing Markov process is the progression from a procedure to winch the student has hid previous exposure to a less familiar cr.e. -8- UNCOLLECTABLE ACC ESTIMATION Once the cost allocatior s been solved by the use of an absorbing Markov process > estimating he a ce for doubtful accounts becomes merely another application of a previously used concept. Consider the situation presented b md Thompson in the appendix to their article on doubtful account nation. There are two absorbing states, 0", an account is collected, and 2, an account is declared bad, and two nonabsorbing states, 0, an account is current, and I, an account is one period old. The transition probability matrix for movement between these various states is as follows. To From IT 2 1 2 1 h 1 ,a ,s .2 .5 .] .1 \ (14) To find the average number of times an account is in various non- absorbing states before it is absorbed, solve for N. N « 1 |l- : 1 i.S .31 .51 .3 1.28 L For example, an account Ln an average of .31 periods in state and .51 periods in state 1 before it is absorbed. To find the portion of the accounts in a nonabsorbing state that will ultimately end up in a particular absorbing state solve for B. (15) B = 2.31 .51 .77 1.28 .3 .5 .1 .95 .05 .87 .13 (16) For example, 95 percent of the accounts in state will ultimately end up in absorbing state 0" and 5 percent of the accounts in state will ultimately end up in absorbing state 2. If all of the accoun mabsorbing states are of approximately equal size an estimate of the dollar amount of the accounts that will ultimately be collected or go bad \ssume there are $10,000 in state and $5,000 in state 1. Then, the expected final disposition of these dollars is as follows: m .95 .05 [10,000 5,000] «[ 13,850 900 ] (17) .87 .13 The allowance for uncollectable accounts should be $900. SIMILARITY Many additional accounting applications of matrix algebra, such as inventory valuation in process costing and consolidated income deter- mination with intercorporate stockholdings may be formulated as either a system of linear equations or a Markov process. Yet, regardless of the *ay they are formulated the notions of "flow" and "absorption" underly all applications mentioned in this paper. In cost allocation problems, costs flow through service departments and are absorbed by production departments. In accounts receivable problems, revenues flow through various age categories and are absorbed by being :ollected or written off. In process costing problems, costs flow through 5roduction departments and are absorbed by inventories or the scrap heap. In consolidated income problems with intercorporate stockholdings, income flows through the affiliated corporations and is absorbed by the majority 2nd r.inoritv interests. 10- Once the significance of the notions of "flow" and "abscrbtion" is grasped, the student is able erstand the similarities underlying many accounting application gebra and solve these problems without the aid of the instruc spe example. CONl 3NS The student's understanding of various accounting applications of matrix algebra can be enhanced if these applications are related to pre- viously learned concepts „ such as the solution of a system of linear equations, and if the similarities underlying these applications are emphasised. In so doing a certain amount of computational efficiency may be lost. However, this loss of computational efficiency is not critical in an undergraduate accounting class because of the widespread availability of computers and the size of the problems considered. Just as important as the increased clarity of presentation is the reduction in class time that must be devoted to matrix algebra. The vast increase in the number of issues that should be considered in the classroom requires that each of them be presented mot a less time consuming manner. 11- FOOTNOTES *N. Churchill, "Linear Algebra and Cost Allocations: Some Examples," The Account ing Review (October, 1964), pp. 894-904. 2 R. Manes, "Comment on Matrix Theory and Cost Allocation," The Accou nting Review (July, 1965), pp. 640-643; J. Livingstone s . "Matrix Algebra and Cost Al location ,".The Accounting Review (July, 1968), pp. 503-508. There is a one dollar rounding error in the calculations. 4 Dispite the fact that a system of linear equations is normally solved in matrix algebra by multiplying the vector of constants by the inverse of the coefficient matrix, in cost allocation models, most authors break the coefficient matrix down into a number of smaller matricies and perform additional operations on them, While such a procedure may have computational advantages, it lacks clarity of presentation. 5 J. Kemeny, A. Schleifer, Jr., J. Snell, and G. Thompson, Finite Mathematics , (Prentice Hall, 1972), pp. 224-229. ^R. M. Cyert, H. J. Davidson, and G. L„ Thompson, "Estimation of the Allowance for Doubtful Accounts by Markov Chains," Management Science (April, 1962), pp. 287-303. 7 Churchili, pp. 89 °C. II. Griffin, T. V rson, Advanced Accounting , (Irwin, 1971), pp. 516-519.