M  A  S  I          The response of a linear sales curve imposed to a   micro economical model of a company   Hans Jessen   Management Simulator  M A S I ,  P.O. Box 171, DK-2630 Taastrup   and   Department of Mathematical Modelling  The Technical University of Denmark, DK-2800 Lyngby, Denmark   (November 11, 1998)   Abstract   A model of a company based on common accounting practice for tactical planning is developed containing physical flow of materials, manhours and deposits of materials, value flow and deposits of value and financial flow and deposits as functions of time. In the first place a graphical model is described naming each part by a mathematical function. Thereafter the functions of time are determined with respect to accountancy and their solutions are found imposing a linear sales curve. These solutions describe fundamental functions in time of basic theory of accountancy with reference to the flow of resources. E.g. profit and loss account, cash flow, working capital and main key figures of the Dupont Pyramide are determined as functions of time.    Key words: Flow of resources, accountancy, cash flow, working              capital, key figures, Dupont Pyramide.       1. INTRODUCTION   This paper is concerned with a model of a company containing common accoun- ting practice. Such models have been presented by Bela Gold,7 with a keynumber technique, which were based on a very simple ratio technique. Jay W. Forre- ster,6 developed models based on signal-graph   - 2 -   techniques, but these models of system dynamics are difficult to apply in practice because of the data to be found and to be interpreted. Models more applicable for management analysis and decisions were developed by Albert Danielsson,2-3 in the form of flow-graphs but containing no mathematical func- tions for evaluation. Samuel Eilon,4-5 made some mathematical approach to describe the primary problem of this article, the equations as functions of time between the working capital and the working system of profit and loss account, without also considering the derived cash flow. His model as well as others on this very aggregated data level are not able simultaneously to measure values from the basic theory of accountancy as functions of time. In the litterature of accountancy and management e.g. C. J. Malmborg,8 Alfred Rappaport,9 and R.S. Segal,10 one will find no functions of time describing and being consistent with accounting practice. Among all these efforts to describe the processes of products and finance in a company one will find Dan Ahlmark,1 as a primary source for this study. Dan Ahlmark only made a general desciption without mathematical modelling of ac- countancy with functions of time.                                                 - 3 –   2.  An analytical graphical business model   This Chapter describes an analytical graphical business model (see Fig. 2.1.). This model will form the basis of a mathematical analytical descrip- tion of the business which can be used by the business management in their principal planning activities. The model will integrate principal elements of managerial economics and the accounting theory, under the assumption that the business comprises an activity/ cash flow and related principal assets (accounts payable, accounts receivable, inventories). It is the management's task to achieve the best possible composition of this general structure by using some of the ratios defined in the model.     2.1.     Activity parameters   2.1.1.  Sales   The volume of goods sold by the firm per unit time is denoted S'u, where S'u = S'u(t). The dot denotes the physial dimension of “current”.   Sales are here divided into two main components of which one is the reference sales S'u,kon, which refers to the share of sales which is paid for in cash. The other component of sales is denoted with S'u,deb, which refers to the share of sales which is paid for by the trade accounts receivable the debit time dD after delivery from the firm. Here the following eguation applies:   S'u(t) = S'u,deb(t) + S'u,kon(t)                                       (1)     2.1.2.  Purchases   The firm is supplied with a number of labor hours per time unit denoted by a'i and with the volume of goods per time unit denoted by V'i. The flow of goods consists of two main components of which one is the reference purchase V'i,kon, and the other one in the goods purchased on credit V'i,kre, which are paid for by the firm after the credit time dK.     - 4 – - 5 -   The following equation applies:   V'i(t) = V'i,kre(t) + V'i,kon(t)                                       (2)   The firm is supplied with the fixed volume of resources per unit time F'i. This flow of resources may, for example, include electricity, administration, heating, rent, etc.     2.1.3.  Inventories   The volume Q'i of raw materials supplied per unit time is added to the raw materials inventory consisting of the volume RL. From the raw materials in- ventory is deduced the raw materials volume Q'u. The following equation appli- es here:          t RL =  ò (Q'i(t) - Q'u(t))dt                                          (3)       0   The volume of finished goods per unit time Z'i is added to the finished goods inventory consisting of the volume FL. From the finished goods invento- ry is deduced the finished goods volume Z'u. The following equation applies here:           t FL =   ò (Z'i(t) - Z'u(t))dt                                         (4)        0   2.2.     Payment parameters, operations   2.2.1.  Sales   The total volume of means of payment per time unit from the customers is denoted with S'i. This payments flow consists of two components. One component is the payments flow S'i,kon caused by the cash sales flow  S'u,kon. The other component S'i,deb is the payment flow caused by the credit sales flow S'u,deb. Here the following equation applies:   S'i(t) = S'i,kon(t) + S'i,deb(t)                                       (5)   - 6 -     2.2.2.  Purchases   The total volume of payment per unit time for operations is denoted by U'b. This payment flow is composed of three components, a'b and V'b and F'b. a'b is the payment flow corresponding to the flow of labour hours consumed a'i, V'b is the payment flow corresponding to the flow of raw material purchases V'i, F'b is the payment flow corresponding to the flow of fixed resources consumed F'i. The following equation applies:   U'b(t) = a'b(t) + V'b(t) + F'b(t)                                     (6)   The payments flow V'b is made up of two components. One component is the pay- ments flow V'b,kon corresponding to the cash purchases of rawmaterials V'i,kon; the other component is the payments flow V'b,kre corresponding to the credit purchase of raw materials V'i,kre. The following equation applies;   V'b(t) = V'b,kon(t) + V'b,kre(t)                                       (7)     2.3.  Market parameters, sales   In order to depict the fundamental financial effects of the market on the firm and its effects on earnings, the market is characterized by three basic components q , p and dD. They also describe the fundamental link between the firm's sales of goods and the related payment flows.     2.3.1.  Cash sales ratio q   The cash sales ratio is defined by the equation:   S'u,kon(t) = q S'u(t)                                                  (8)   where 0 £  q  £  1   In a manufacturing business q will typically have a value in the interval 0 £  q  £ 0.2. In a supermarket q  will typically be in the interval 0.8 £ q  £  1.           - 7 -     2.3.2.  The price p   The price of the firm's product(s) is defined by the eguations   S'u,kon,1(t) = p S'u,kon(t)                                            (9)   S'i,kon(t) = S'u,kon,1(t)                                             (10)   where S'u,kon,1(t) is the flow of debts corresponding to the sales flow   S'u,kon(t) (i.e. the current invoice flow stating the amount of debt; see equation (9)). Equation (10) expresses the fact that the flow of debts S'u,kon,1(t) is equal to the payments flow from the customers (cash payment).   In practice, it should be noted that there is normally a time lag between invoicing and sales. However, it has a temporary negative effect on liquidity and the computation of results. Management will therefore have in view that the invoicing is done without the mentioned delays.     2.3.3.  Debit time dD   This model defines the debit time dD as the time of delivery of the goods from the firm until the time of payment by the customer for the goods. In practi- ce, dD is spread over the individual customers but with well defined terms of payment the mean value can be detemined.   The definition of dD can be expressed by the equations   S'u,deb,1(t) = p S'u,deb(t)                                           (11)   V'deb,dD(t) = S'u,deb,1(t - dD)                                         (12)   S'i,deb(t) = V'deb,dD(t)                                              (13)                 - 8 -     S'u,deb,1 refers here to the invoice flow corresponding to the credit sales flow S'u,deb cf. equation (11). Equation (12) gives a functional description of a function V'deb,dD(t), which can be defined as the payments flow (documents) corresponding to the actual receipt of payments S'i,deb(t) cf. equation (13). In practice, no time lag is found between the two last mentioned functions.   In pratice, attention should be paid to the fact that there may be a time lag in the business between invoicing and sales, the result being changes in li- quidity and the computation of earnings. Management usually aims at applying equation (11) in practice, i.e. no time lag.     2.4.     Market parameters, purchases   With a view to depicting the fundamental financial effects of the purchasing market on the firm as well as its effects on costs, it is characterized by four basic components e, q1, q2 and dK. They describe the fundamental link between the firm's purchases of resources and the related payment flows.     2.4.1.  Cash purchases ratio e   The cash purchases ratio is defined by the equation:   V'i,kon(t) = e V'i(t)                                                (14)   where 0 £ e £ 1   In, say, a manufacturing business e will typically have a main value in the interval 0 £  e  £ 0.2. This is also a typical feature in a trading firm.   2.4.2.  The price q1 of raw materials   The price of the firms raw materials is defined by the equation:               - 9 -         V'i,kon,1(t) = q1 V'i,kon(t)                                          (15)   V'b,kon(t) = V'i,kon,1(t)                                             (16)   where V'i,kon,1(t) is the flow of debts corresponding to the raw materials flow V'i,kon(t) (i.e. the current receipt of invoices stating the amounts of debts); see equation (15). Equation (16) expresses the fact that the flow of debts V'i,kon,1(t) is equal to the payments flow to suppliers (cash payment).   In practice, attention should be paid to the fact that the time lag between the supplier's invoicing and the supplies of raw materials is usually a temporary feature which has a temporary positive effect on liquidity and the computation of results.     2.4.3.  The price q2 of labor hours   The price of the firm's labor hours is defined by the equations   a'i,1(t) = q2 a'i(t)                                                 (17)   a'b(t) = a'i,1(t)                                                    (18)   where a'i,1(t) is the time ticket flow corresponding to the flow of labor hours used a'i(t) (i.e. the current issuing of time tickets stating wages earned); see equation (17). Equation (18) expresses the fact that the time ticket flow a'i,1(t) is equal to the time rate flow a'b(t).   In practice there is a certain time lag between functions on the right hand side and the left hand side of the equal sign in equation (17). This time lag is ignored here. There is usually no time lag between the functions of equa- tion (18), or the time lag is relatively small and of no importance here.     - 10 -       2.4.4.  Credit time dK   This model defines the credit time dK as the time from the time of delivery of the raw materials to the firm until the time of payment by the firm for the raw materials. In practice, dK is spread over the individual suppliers but with well defined terms of payment the mean value can be used. The definition of dK can be expressed by the equations:   V'i,kre,1(t) = q1 V'i,kre(t)                                          (19)   V'kre,dK(t) = V'i,kre,1(t - dK)                                        (20)   V'b,kre(t) = V'kre,dK(t)                                              (21)   where V'i,kre,1(t) refers here to the invoice flow corresponding to the credit purchases flow V'i,kre(t), cf. equation (19). Equation (20) gives a functional description of a function V'kre,dK(t) which can be defined as the payment order flow (documents) corresponding to the actual effecting of payments V'b,kre(t), cf. equation (21). In practice, there is no time lag between the two last mentioned functions. In practice, attention should be paid to the fact that the time lag between the supplier's invoicing and the supplies of raw materials is usually a temporary feature which has a temporary positive affect on liquidity and the computation of results. The following equations are defined in relation to the fixed resources consumed F'i and the related fixed costs F'b.   F'i,1(t) = k F'i(t)                                                  (22)   F'b(t) = F'i,1(t)                                                    (23)                 - 11 -     where F'i,1(t) in equation (22) refers to the flow of debts in the form of in- voices (stating amounts) corresponding to the fixed resources flow F'i(t). k denotes a symbolic operator in the form of an average price of the fixed re- sources unit. In practice, there is some time lag between the functions in eguation (23). As, however, the fixed costs by definition are constant in ti- me, such a time lag is not important in this context.     3.1  Income statement   In this Chapter an income statement for operations is presented (before depre- ciation, etc.) using the general main principles of accounting theory.     3.1.1    Sales of goods   Sales of goods are defined on the basis of the following equations:   S'u,kon,2(t) = S'u,kon,1(t)                                            (24)   S'u,deb,2(t) = S'u,deb,1(t)                                            (25)   S'u,1(t) = S'u,kon,2(t) + S'u,deb,2(t)                                  (26)   Eguation (24) expresses the fact that the flow of debts (in the form of in- voices with statement of amounts) S'u,kon,1(t) gives rise to an equally large information flow S'u,kon,2(t). This quantity is identical with the current crediting to the cash sales account.   From equation (25) follows that the flow of debts S'u,deb,1(t) causes an equally large information flow S'u,deb,2(t). This quantity is identical to the current crediting to the credit sales account.   Total sales in the form of the information flow S'u,1(t) corresponding to the total crediting to the sales account are then obtained from equation (26).         - 12 -       3.1.2  Costs   The costs of the firm in connection with production and sales are defined by the following equations:   V'i,kon,2(t) = V'i,kon,1(t)                                            (27)   V'i,kre,2(t) = V'i,kre,1(t)                                            (28)   a'i,2(t) = a'i,1(t)                                                  (29)   F'i,2(t) = F'i,1(t)                                                  (30)   U'd(t) = V'i,kon,2(t) + V'i,kre,2(t) + a'i,2(t) + F'i,2(t)                (31)   Equation (27) expresses the fact that the invoice flow from the cash purchase V'i,kon,1(t) is currently debited to the cash purchases account to the extent of the cash flow V'i,kon,2(t).   Equation (28) expresses the fact that the invoice flow from the credit pur- chase V'i,kon,1(t) is currently debited to credit purchases account to the extent of the cash flow V'i,kre,2(t).   Equation (29) denotes the functional relationship between the time ticket flow a'i,1(t) and the current debiting to the time rate account of the wage payment flow a'i,2(t).   Equation (30) expresses the functional relationship between the invoice flow F'i,1(t) for fixed costs and the current debiting of the cash flow F'i,2(t) to the fixed costs account.   The total cost flow is defined by equation (31).                 - 13 -       3.1.2.1  Inventories, additions (with signs)   By way of introduction, it is mentioned that the signs relating to additions to inventories (as a mean time value) are assumed to be the same as those relating to additions to sales (as a mean time value). Against this background the ad- ditions to the individual inventories will for principal planning purposes ha-ve the same signs. The inventories only serve as "standby stores" in case of emergency events "i.e. in normal operation state" the materials and products go directly through the factory. Thus, the following systems of equations apply:   The increase of sales S'u is supplied directly by the production and the inventories are increased proportionally with S'u.                              Q'i(t) > 0                              Q'u(t) = 0 d S'u ¾¾¾¾  > 0      Þ                                                                        (32)  dt                              Z'i(t) > 0                              Z'u(t) = 0   Constant sales S'u is supplied directly by the production and the inventories remain constant.                                                                      Q'i(t) = 0                              Q'u(t) = 0   d S'u ¾¾¾¾  = 0      Þ                                                                        (33)  dt                              Z'i(t) = 0                              Z'u(t) = 0   The decrease of sales S'u is supplied directly by the production and the flow from inventories. The inventories are decreased proportionally with S'u.       - 14 -                                Q'i(t) = 0                              Q'u(t) > 0   d S'u ¾¾¾¾   < 0      Þ                                                                         (34)  dt                              Z'i(t) = 0                              Z'u(t) > 0   The system of equations (32) denotes that inventories rise when sales rise.   The system of equations (33) denotes that inventories are constant when sales remain unchanqed.   The system of equations (34) denotes that inventories fall when sales fall.   Based on these main principles for the model the following equations can be developed.   Q'i,1(t) = qR Q'i(t)                                                 (35)   Q'u,1(t) = qR Q'u(t)                                                 (36)   Z'i,1(t) = qF Q'i(t)                                                 (37)   Z'u,1(t) = qF Z'u(t)                                                 (38)   U'tl(t) = Q'i,1(t) + Z'i,l(t)                                         (39)   U'al(t) = Q'u,1(t) + Z'u,1(t)                                         (40)   where    Q'i,1(t)   is the flow of additions to raw materials invento-                     ries corresponding to the additions to rawmateri-                     als inventory records with statement of amounts.         - 15 -            Q'u,1(t)   is the flow of deductions to raw materials inven-                       tories corresponding to the deductions to raw                     materials inventory records with statement of                      amounts.            Z'i,1(t)   is the flow of additions to finished goods inven-                     tories corresponding to the additions to finished                     goods inventory records with statement of amounts.            Z'u,1(t)   is the flow of deductions to finished goods inven-                     tories corresponding to the deductions to finished                     goods inventory records with statement of amounts.             qR           denotes the calculated rav material price per unit                      of finished goods.             qF           denotes the calculated direct cost price per unit                      of finished goods.             U'tl(t)   is total additions to inventories.             U'al(t)   is total deductions from inventories.   The system of equations (32), (33) and (34) can now be given the form:   d S'u ¾¾¾¾  > 0      Þ       U'tl(t) > 0  and  U'al(t) = 0              (41)  dt       d S'u ¾¾¾¾  = 0      Þ       U'tl(t) = 0  and  U'al(t) = 0              (42)  dt   d S'u ¾¾¾¾  < 0      Þ       U'tl(t) = 0  and  U'al(t) > 0              (43)  dt   Attention is drawn to the fact that the physical model based on the FIFO principle can be desribed mathematically only by               - 16 -               d S'u sign ( ¾¾¾¾ ) = sign (U'tl(t))                                   (44)         d t   given U'al(t) = 0                                                   (45)     and U'tl(t) is computed with signs.       3.1.3.  Resourceconsumption (incl. F'i,1)   Resources consumed U'd,1,1(t) can be defined by the following equations:    d S'u ¾¾¾¾  > 0      Þ        U'd,1,1(t) = U'd(t) - U'tl(t)              (46)   dt                                                  given U'al(t) = 0    d S'u ¾¾¾¾  = 0      Þ        U'd,1,1(t) = U'd(t)                       (47)   dt        d S'u ¾¾¾¾  < 0      Þ        U'd,1,1(t) = U'd(t) + U'al(t)              (48)   dt                                       given U'tl(t) = 0     3.1.4.  Operation profit (before interest and depreciation)   The operating profit (before interest and depreciation etc.) is defined by the equation:   O'(t) = S'u,1(t) - U'd,1,1(t)                                         (49)       3.1.5  Operating profit incl. inventory depreciation   If a tax year of the length T is considered in a period of time t1 £ t £ t1 + T where t1 is a time selected at random, the following functions can be defined:           t1+T Vkøb =   ò  q1 V'i(t) dt                                             (50)         t1   w = w(t1)                                                          (51)   an = an(t)                                                         (52) - 17 -       In equation (50) Vkøb represents the purchases of goods in the period          t1 £ t £ t1 + T.   Equation (51) defines w(t1) as the total inventory value at time t1. an(t) in the equation defines the inventory depreciation rate.   Materials consumed computed for tax purposes is then derived from the follow- ing equation (53):   Vtax = Vkøb + w(t1) - (w(t1)/(1 - an(t1))                t1+T             +  ò (U'tl(t) - U'al(t)) dt) (1 - an(t1 + T))                 (53)           t1   For principal planning purposes the mean time value of an(t) for a given business will be a constant an and limited i.e.   0 < an < 0.3 . Based on this assumption equation (53) gives                                                                             t1+T Vtax = Vkøb - (1 - an) ò (U'tl(t) - U'al(t)) dt                       (54)                       t1   Materials consumed for operations is defined by the following equation:                                                  t1+T Vdrift = Vkøb + w(t1) - (w(t1) +  ò (U'tl(t) - U'al(t)) dt)           (54a)                                 t1 or                     t1+T Vdrift = Vkøb -  ò (U'tl(t) - U'al(t)) dt)                             (55)                t1   If equation (55) and equation (54) are combined, the following equations are developed:                       t1+T Vtax = Vdrift + an ò (U'tl(t) - U'al(t)) dt                            (56)                  t1   - 18 -                         t1+T Vtax = Vdrift +  ò an(U'tl(t) - U'al(t)) dt                            (57)                t1   On the basis of equation (57) the following functions can be defined:   U'tl,1(t) = U'tl(t)                                                  (58)   U'al,1(t) = U'al(t)                                                  (59)   In equation (58) U'tl,1(t) denotes total additions to inventories from a taxation point of view. U'al,1(t) denotes in equation (59) total deductions from invento-ries from a taxation point of view.   With the following definition equation:     B'ln(t) = an (U'tl,1(t) - U'al,1(t))                                  (60)   equation (57) can be transformed into                   t1+T Vtax = Vdrift +  ò B'ln(t) dt                                        (61)                t1   On the basis of equation (61) the following equation (62) can be defined:   O'DS = O' - B'ln                                                     (62)   where O'DS is the operating profit adjusted for inventory depreciation.       4.1.     Change in liquidity (operations)   The cash flow released by operations, the change in liquidity, is defined by the following equation (63):   l'(t) = S'i(t) - U'b(t)                                              (63)         - 19 -     5.1.     Cash balance   The cash balance of the firm is designated by M, which, in relation to the present principal planning model, is very small in practice, i.e. M(t) = 0. The folloving equation can now be developed:   i'e = l' + i'K - y'B - y'L - H'S,1                                     (64)   where   i'e      is the self financing flow   y'B      is the service of bank loans   y'L      is the service of other loans   i'K      is current raise of loans for operations   H'S,1    is tax payments   5.2.  Bank loans.   The firm is financed currently by trading credits in the form of the cash flow i'B. The equation is defined as follows:   i'B,1(t) = i'B(t)                                                   (65)   where i'B,1(t) is the information flow in the form of loan documents with statement of amounts corresponding to the cash flow i'B(t). The bank charges currently interest r'B(t) on the amount outstanding   B = B(t) where r'B(t) is the document flow with statement of interest. The following equation applies:   n'B(t) = i'B,1 + r'B                                                 (66)   where n'B(t) is the firm's current crediting to the bank account.       - 20 -     The current service payments y'B(t) to the bank give rise to a payment order flow with statement of amounts y'B,1(t). We have:   y'B,1(t) = y'B(t)                                                   (67)   The payment order flow y'B,1(t) involves a corresponding current debiting to the bank account in the form of y'B,2(t). The following equation therefore ap- plies:   y'B,2(t) = y'B,1(t)                                                  (68)     5.3.  Loans (long term)   The long term financing of the business is represented by the cash flow i'L. The following equation applies:   i'L,1(t) = i'L(t)                                                   (69)   where i'L,1(t) is the information flow in the form of loan documents with statement of amounts corresponding to the cash flow i'L(t). On the loan L current interest r'L(t) is charged where r'L(t) is the document flow with statement of interest. The following equation applies:   n'L(t) = i'L,1(t) + r'L(t)                                           (70)   where n'L(t) is the firm's total current crediting to the loan account.   The following equation applies:   i'L(t) = i'L,1(t) + i'D(t)                                           (71)   where i'L,D(t) denotes the long term financing flow to the working capital, and i'L,1(t) is the long term financing flow to the fixed capital.             - 21 -     The folloving equation applies:   i'K(t) = i'B(t) + i'L,D(t)                                           (72)   The current service payments y'L(t) to lender give rise to a payment order flow with statement of amounts y'L,1(t). We have   y'L,1(t) = y'B(t)                                                    (73)   The payment order flow y'L,1(t) involves a corresponding current debiting to the loan account in the form of y'L,2(t). The following equation therefore applies:   y'L,2(t) = y'L,1(t)                                                  (74)       6.1.     Investment (in fixed capital)   The firm's current investment in fixed capital is denoted i'(t). The following equation applies:   i'(t) = i'L,1(t) + i'e(t)                                            (75)   It is pointed out that, in practice, i'L,D(t) currently converts short term liabilities into long term liabilities, which means that at a strategic level alone i'L,D = 0. As to i'e(t), there is no unique definition of i'e(t) as it de- pends on the financing and market situation. Roughly speaking, i'e(t) is the average cash flow which can be withdrawn from the business without changing the existing product, investnent and financing structure and the necessary financial reserves set aside for an appropriate future development of the bu- sinees.                           - 22 -       7.1.     Depreciation (for tax purposes)   It is normal to distinguish between depreciation for tax purposes and depre- ciation for accounting purposes. Depreciation for accounting purposes is used with the object of comparing alternative projects on the basis of special cost principles. These principles are purely OR mathematical models and do not reflect the physical business situation.   Here we shall only take an overall view of the financial flow of the firm for which reason depreciation for tax purposes will be used. Such depreciation will only reflect the actual effects on liquidity (after tax).   The following equations apply:   i'1(t) = i'(t)                                                      (76)             t D(t) =   ò  (i'1(t) - d'1(t))dt                                     (77)          0   where i'1(t) represents the current debiting to the tax depreciation account corresponding to the investment flow i'(t). d'1(t) is the current crediting to the same account (i.e. current "depreciation").   D(t) represents the balance of the tax depreciation account. The depreciation charges d'(t) are calculated on the basis of this account, and the following expressions apply:   d'1(t) = d'(t)                                                     (76a)   d'2(t) = d'(t)                                                     (76b)   where d'2(t) is the depreciation flow which is included on the basis of compu- tation of the taxable income.             - 23 -       8.1.  Interest (for tax puroses)   Interest is usually computed for two main purposes. One concerns the income statement for tax purposes, the other concerns internal computation purposes such as the effect of interest on the income statement as a whole or in con- nection with special computations.   No distinction will be made here between the two purposes. The interest charges will be placed in this model with the sole aim of depicting the fundamental fi-nancial characteristics.   The following equations are defined:   r'B,1(t) = r'B(t)                                                    (78)   r'L,1(t) = r'L(t)                                                    (79)   r'BL(t) = r'B,1(t) + r'L,1(t)                                         (80)   where r'B,1(t) denotes the current recording of interest payment to the bank. r'L,1(t) denotes the current recording of interest payments to other lenders. The recording of total interest payments is designated r'BL(t).     9.1.  Tax payments   According to the principles governing computation of the taxable income the following equations apply:   f'u(t) = d'2(t) + r'BL(t)                                            (81)   H'S(t) = s (O'DS(t) - f'u(t))                                        (82)   H'S,1(t) = H'S(t)                                                   (82a)                 - 24 -       where f'u(t) is a state function for the computation of tax, cf. equation  (81), s is the tax rate, H'S(t) is the computed tax payment and H'S,1(t) is the tax payment flow.       10.1.    Principal ratios As appears from Fig. 2.1, the following principal ratios in the firm are im-portant to the understanding of the dynamic (tactical) characteristics of the firm.            Operating profit                             O'(t)            Change in liquidity                          l'(t)            Working capital (net)                        K'(t)            Contribution ratio                           DG(t)            Depreciation                                 d'2(t)            Interest                                      r'BL(t)   These ratios will be discussed in detail in the following.       10.1.1.  Operating profit O'(t)   Using different assumptions concerning prices and changes in principal assets (accounts payable, accounts receivable, inventories) it is possible via Fig. 2.1 to assess the effects on the operating profit. A reduction of the raw materials inventories in a situation with raw materials prices which are higher than the prices of the raw materials inventories but otherwise constant will increase the profit temporarily in the period concerned.   One of the things that will be seen is that the profit O'(t) is independent of the volume of trade accounts payable and the volume of trade accounts recei- vable.   - 25 -       10.1.2.  Change in liquidity l'(t)   Other things being equal, the following expression, cf. Fig. 2.1., applies:    d S'u ¾¾¾¾  > 0      Þ     l'(t) < O'(t)                               (83)  d t   Equation (83) shows that the profit O'(t) is larger than the change in liqui- dity in the case of growing sales in the firm, the reason being the funds ti- ed up, calculated with signs, in principal assets (accounts receivable and inventories),   d S'u ¾¾¾¾  = 0      Þ     l'(t) = O'(t)                               (84) d t   Equation (84) shows that the change in liquidity is equal to the profit in the case of constant sales, the reason being an unchanged volume of principal as-sets (accounts payable, accounts receivable and inventories).   d S'u ¾¾¾¾  < 0      Þ     l'(t) > O'(t)                               (85) d t   From equation (85) appears that in the case of falling sales the change in liquidity becomes greater than the operating profit owing to a reduced volume of principal assets (accounts payable, accounts receivable and inventories).   The above shows how important it is for the business to keep the cash budget currently up to date as the profit and the financial circumstances of the business may differ substantially from each other. It should be noted that if the net principal assets are negative, the inequality signs in (83) and (85) must be reversed.             - 26 -       10.1.3.  Working capital K(t)   If the working capital is denoted K(t), the definition eguation for net capi- tal tied up in the operating system will apply:   K(t) = Vdeb(t) + FL(t) + RL(t) - Vkre(t)                            (86)   The following definition equation will also apply:   d K(t) ¾¾¾¾  + l'(t) = O'(t)                                            (87)  d t   Equation (87) shows that the profit is equal to the change in liquidity + the increment of the net working capital tied up.   If equation (87) is transformed, the following equation is derived:   d K(t) ¾¾¾¾  = O'(t) - l'(t)                                            (88)  d t   Equation (88) denotes that the difference between the operating profit and the change in liquidity is equal to the financing requirements for operations in the period under review.       10.1.4.  Contribution ratio DG(t)   The contribution ratio is defined by equation (89):   DG(t) = (O'(t) + F'b(t))/S'u,1(t)                                    (89)   Equation (89) shows that DG is independent of the amount of sales and defines the share of sales which will cover fixed costs, etc. The point is stressed here that a high contribution ratio does not imply that there is "money" to cover the fixed costs. For further details see section 10.1.2. as the size of l'(t) gives only an indication of the ability of the firm to pay fixed costs, etc.         - 27 -       10.1.5.  Depreciation   Depreciation contributes to influencing the firm's liquidity, cf. equation (82). Assuming that the investments are made as individual projects at time intervals, it is shown that depre­ciation in the periods between investments causes liquidity to rise owing to the reduction in tax payments.   However, it should be noted that of the cash flow released after tax there must be funds to cover repayment commitments in connection with loans raised. The effect of the cash flow released after tax described above is therefore partial and must be seen in relation to the repayment commitments.   Later there will be shown that for practical reasons the division described here is desirable for the understanding of the financial components of the cash flow released.       10.1.6.  Interest r'BL   From Fig. 2.1 and from equations (81) and (82) is apparent that interest pay-ments reduce the cash flow released after tax. Thus, the net effect on cash flow released (to be defined later) stems partly from the computation of income for tax purposes, partly from the payment of interest on total loans.   The computation of interest on total loans seen in relation to a given level of activity will be defined later.                                     - 28 -           11.  An analytical mathematical business model   This Chapter presents a new analytical mathematical model de­scription of the business. This model has been developed for use in the tactical planning pro- cess. No reference can be made to a similar model in existing literature. The theoretical literature which gets nearest is S. Eilon's article discussed in Chapter A in “Economical Keynumbers”.     11.1.  Physical and financial functions in the operating system   In the following further definitions of mathematical functions and their rela- tionships will be established. The sole justification of these definitions is that they provide the basis of a clear and generally coherent system of equa-tions between ratios.     11.1.1.  Sales   A basic sales volume is defined:   S'u0 = S'u(0)                                                       (90)   where S'u0 is the volume of sales S'u at time t = 0, i.e. at the beginning of the simulation period.   The development of sales during the time period is defined by equation (91):   d S'u(t) ¾¾¾¾   = as S'u0                                                   (91)  d t   where as is constant.                     - 29 -       The following equation now applies:   S'u(t) = S'u0(1 + as t)                                              (92)   where t ³ 0       11.1.2.  Inventories   Let a ratio hF be defined so that equation (93) applies:   hF = FL(t)/S'u(t)                                                   (93)   for t ³ 0, hF being a positive constant which is designated "finished goods inventory time". Another ratio hR is defined so that equation (94) applies:   hR = RL(t)/S'u(t)                                                   (94)   for t ³ 0  being a positive constant which is designated "raw materials inventory time".   From equation (93) follows:   FL(t) = hF S'u(t)                                                   (95)   The definition equation applies:                     t FL(t) = FF(0) +  ò  Z'i(t) dt                                       (96)                  0   which substituted into equation (95) gives: - 30 -                 t FL(0) +  ò Z'i(t) dt = hF S'u(t)                                     (97)          0   or     t  ò Z'i(t) dt = hF S'u(t) - FL(0)                                     (98)  0   If equation (92) is used in equation (98), the following expression is derived:     t  ò Z'i(t) dt = hF S'u0 as t + (hF S'u0 - FL(0))                       (99)  0   For t = 0 equation (93) gives the following expression:   hF = FL(0)/S'u(0)                                                  (100)   Using equation (100) together with equation (99) we have:     t  ò  Z'i(t) dt = hF S'u0 as t                                        (101)  0   The solution to the integral equation (101) is:   Z'i(t) = hF S'u0 as                                                 (102)   The flow of goods Z'i(t) to the finished goods inventory may then be defined by equations (103) and (104):   Z'i(t) = hF S'u0 as                                                 (103)   for as ³ 0   and         - 31 -       Z'u(t) = - hF S'u0 as                                               (104)   for as < 0   Mathematically the physical equations (103) and (104) may be described by equation (105) for all values of as, i.e.   Z'i(t) = hF S'u0 as                                                 (105)   for  - ¥ < as < ¥   With equation (105) the physical inventory system has been converted to a mathematical model where Z'i(t) can change sign and where Z'u(t) = 0 for all t, cf. equation (102).   From equation (94) follows:   RL(t) = hR S'u(t)                                                  (106)   The definition equation applies:                     t RL(t) = RL(0) +  ò  Q'i(t) dt                                      (107)                  0   which combined with equation (106) gives:             t RL(0) +  ò  Q'i(t) dt = hR S'u(t)                                   (108)          0   or    t ò  Q'i(t) dt = hR S'u(t) - RL(0)                                    (109) 0   If equation (92) is used in equation (109), the following equation is derived:               - 32 -      t ò  Q'i(t) dt = hR S'u0 as t + (hR S'u0 - RL(0))                       (110) 0   For t = 0 equation (94) gives:   hR = RL(0)/S'u(0)                                                  (111)   Using equation (110) together with equation (111) we have:     t  ò  Q'i(t) dt = hR S'u0 as t                                        (112)  0   The solution to the integral equation (112) is:   Q'i(t) = hR S'u0 as                                                 (113)   The flow of goods Q'i(t) to the raw materials inventory can now be defined by equations (114) and (115):   Q'i(t) = hR S'u0 as                                                 (114)   for as ³ 0   Q'u(t) = - hR S'u0 as                                               (115)   for as < 0   Mathematically the physically equations (114) and (115) can be described by equation (116) for all values of as, i.e.   Q'i(t) = hR S'u0 as                                                 (116)   for  - ¥ < as < ¥   With equation (116) the physical inventory system has been converted to a ma-thematica1 model where Q'i(t) can change sign and where Q'u(t) = 0 for all t.           - 33 -       11.1.3.  Output   Total output T'p(t) is given by:   T'p(t) = S'u(t) + Z'i(t)                                            (117)   If the ratio ba is here defined as the number of labor hours used per unit of output and the ratio bR as the raw materials consumption per unit of finished goods, the equations, resource balance equations, will apply:   a'i(t) = ba T'p(t)                                                  (118)   V'i(t) = bR T'p(t) + Q'i(t) bR                                       (119)   If equation (117) and equation (119) are combined, the following equation is obtained:   V'i(t) = bR S'u(t) + bR Z'i(t) + Q'i(t) bR                            (120)   If equations (92), (105) and (116) are substituted into equation (120), the following equation is obtained:   V'i(t) = bR S'u0 (1 + (hF + hR + t) as)                              (121)   Using equations (117), (92) and (105), equation (118) gives:   a'i(t) = ba S'u0 (1 + as(t + hF))                                  (121a)       11.1.4.  Sales, ingoing payments   Using equations (8), (9) and (10) we obtain payments derived from cash sales:   S'i,kon(t) = p q S'u0 (1 + as t)                                     (122)             - 34 -       Using equations (1), (8), (11), (12) and (13) we obtain payments derived from debit sales:   S'i,deb(t) = p (1 - q) S'u0 (1 + as(t - dD))                         (123)   Equations (1), (122) and (123) give:   S'i(t) = p q S'u0 (1 + as t) + p (1 - q) S'u0          (1 + as(t - dD))                                         (124)   or   S'i(t) = p S'u0 (1 + as(t - dD(1 - q)))                             (125)     11.1.5.  Purchases, outgoing payments   The outgoing payments flow corresponding to cash purchases of raw materials is expressed by means of equations (14), (15), (16) and (121) as   V'b,kon(t) = q1 e bR S'u0 (1 + (hF + hR + t)as)                      (126)   Credit purchases of raw materials cause an outgoing payments flow which by means of equations (7) (14), (19), (20) and (21) is computed at:   V'b(t) = e q1 V'i(t) + (1 - e)q1 V'i (t - dK)                         (127)   Equation (127) is transformed by means of equation (121) into:     V'b(t) = e q1 bR S'u0(1 + (hF + hR + t) as) +         (1 - e)q1 bR S'u0(1 + (hF + hR + t - dK)as)                  (128)   Equation (128) is reduced to:   V'b(t) = q1 bR S'u0(1 + as(hF + hR + t - dK (1 - e)))                 (129)         - 35 -   The total payments flow to purchases of resources is then obtained by using equations (6), (17), (18) and (129):   U'b(t) = q2 a'i(t) + q1 bR S'u0(1 + as (hF + hR + t - dK           (1 - e))) + F'b(t)                                       (130)   By combining equation (121a) and equation (130) the total outgoing payments flow is then given by:   U'b(t) = S'u0(q2 ba (1 + as (t + hF)) + q1 bR(1 + as           (hF + hR + t - dK(1 - e))) + F'b(t)                       (131)     11.1.6.  Change in liquidity   The accounting concept, change in liquidity l'(t), here also called cash flow, can then by the use of equations (63), (125) and (131) be given the following form:   l'(t) = S'u0(p(1 + as(t - dD(1 - q)))              - q2 ba (1 + as(t + hF)) - q1 bR(1 + as                (hF + hR + t - dK(1 - e)))) - F'b(t)                 (132)     11.2.    Capital tied up in the operating system   Depending on the firm's level of activity capital will be tied up in the ope- rating system. Capital will be tied up in trade accounts payable, raw materi- als inventories and finished goods inventories as well as accounts receivable (the amounts are indicated with signs).       11.2.1.  Trade accounts receivable   The volume of trade accounts receivable is defined by the following equation, equations (1), (8), (11) and (12) being used:             dD Vdeb(t) = ò  p(1 - q)S'u(t - x) dx                                  (133)          0         - 36 -       In this model it is assumed that equation (92) applies. From this equation combined with (133) follows:                           dD Vdeb(t) = p(1 - q) S'u0  ò (1 - as(t - x)) dx                       (134)                        0   The computation of the integral in equation (134) allows equation (134) to be reduced to:       Vdeb(t) = p(1 - q) S'u0 dD (1 + as(t - 0.5 dD))                      (135)     11.2.2.  Trade accounts payable   The volume of trade accounts payable is defined by the following equation, equations (2), (14), (19) and (20) being used:              dK Vkre(t) =  ò q1 bR(1 - e) V'i(t - x)) dx                            (136)           0   Assuming that sales satisfy equation (92) and that equation (121) applies, equation (136) develops the following expression:                              dK Vkre(t) = q1 bR(1 - e) S'u0  ò (1 + as(hF + hR + t - x)) dx           (137)                           0   By computing the integral in equation (137) this equation is reduced to:   Vkre(t) = q1 bR(1 - e) S'u0 dK (1 + as(hF + hR + t - 0.5 dK))         (138)     l1.2.3  Raw materials inventory   The volume of the raw materials inventory is given by equation (106). The value of the raw materials inventory RL,1(t) satisfies the equation:   RL,1(t) = q1 bR RL(t)                                              (139)       - 37 -       If equations (106) and (92) are substituted into equation (139), we have:   RL,1(t) = q1 bR hR S'u0(1 + as t)                                   (140)     10.2.4.  Finished goods inventory   The volume of the finished goods inventory is given by equation (95). The calculated consumption of materials and labor hours per unit of finished goods is given by qF, cf. equation (37). The definition equation applies:   qF = bR q1 + ba q2                                                (141)   The value of the finished goods inventory FL,1(t) satisfies the equa-ion:   FL,1(t) = qF FL(t)                                                (142)   If equations (95), (92) and (141) are substituted into equa­tion (142), the following expression is obtained:   FL,1(t) = (bR q1 + ba q2) hF S'u0(1 + as t)                         (143)     11.2.5.  Working capital (tied up in the operating system)   The total capital tied up in the operating system, i.e. the working capital K(t), is through the use of equations (135), (138), (140) and (143) given by:   K(t) = Vdeb(t) - Vkre(t) + RL,1(t) + FL,1(t)                        (144)   or by substituting into the relevant places   K(t) = p(1 - q)S'u0 dD (1 + as(t - 0.5 dD))        - q1 bR (1 + e)S'u0 dK (1 + as(hF + hR + t - 0.5 dK))        + q1 bR hR S'u0(1 + as t)        + (bR q1 + ba q2) hF S'u0(1 + as t)   or       - 38 -       K(t) = S'u0(1 + as t)(hF (bR q1 + ba q2) + q1 bR hR)        + p(1 - q) dD S'u0(1 + as(t - 0.5 dD))        - q1 bR (1 - e) dK S'u0(1 + as(hF + hR + t - 0.5 dK))         (145)     12.1.  Operating profit (for accounting purposes)   In the following, functions are established for the computation of operating profit based on accounting theory.   The turnover of the firm is obtained by using equations (9), (11), (24), (25) and (26) and is expressed as:   S'u,1(t) = p S'u(t)                                                (146)   Using equation (92) and equation (136) gives:   S'u,1(t) = p S'u0(1 + as t)                                         (147)   Raw materials consumed corresponding to sales S'u(t) are given by the equation:   V'for(t) = q1 bR S'u(t)                                             (148)   or by using equation (92):   V'for(t) = q1 bR S'u0(1 + as t)                                      (149)   The wages paid, time rates, corresponding to sales S'u(t) are given by the equation:   a'for(t) = q2 ba S'u(t)                                             (150)   or by using equation (92)   a'for(t) = q2 ba S'u0(1 + as t)                                      (151)         - 39 -       By using equations (147), (149) and (151) the operating profit O'(t) can now be given the form:   O'(t) = S'u,1(t) - V'for(t) - a'for(t) - F'b(t)                       (152)   or   O'(t) = S'u0(1 + as t)(p - (q1 bR + q2 ba)) - F'b                    (153)       12.2.  Operating profit (computed on the basis of Fig. 2.1)   In this section the operation profit will as an alternative be computed directly on the basis of Fig. 2.1.   The costs U'd(t) in connection with sales S'u(t) are given by equation (31). If equations (2), (15), (17), (19), (22), (23), (27), (28), (29), (30), (121) and (121a) are substituted into equation (31), the following equation is de­ veloped:   U'd(t) = q1 bR S'u0(1 + (hF + hR + t) as)          + q2 ba S'u0(1 + (hF + t)as) + F'b                         (154)   Computed with a plus or minus sign (positive for inventory) the following va- lue is added to the raw materials inventory, cf. equation (35):   Q'i,1(t) = q1 bR Q'i(t)                                             (155)   or equation (116) may be used:   Q'i,1(t) = q1 bR hR S'u0 as                                          (156)   Here the definition equation for cost prices of raw materials per unit of finished goods has been used:   qR = q1 bR                                                        (157)             - 40 -       The following value is added to the finished goods inventory, cf. equation (37):   Z'i,1(t) = qF Z'i(t)                                                (158)   or equation (102) may  be used:   Z'i,1(t) = (q1 bR + q2 ba) hF S'u0 as                                (159)   The total value flow to inventories now amounts to, cf. equations (44) and (45):   U'tl(t) = q1 bR hR S'u0 as + (q1 bR + q2 ba) hF S'u0 as                (160)   or by reduction   U'tl(t) = S'u0 as(q1 bR hR + (q1 bR + q2 ba) hF)                      (161)   The total operating profit is obtained by using equations (147), (154) and (161) and is expressed as:   O'(t) = S'u(t) - (U'd(t) - U'tl))                                    (162)   or by substituting into the right hand side:   O'(t) = p S'u0(1 + as t)         - (q1 bR S'u0(1 + (hF + hR + t) as)         + q2 ba S'u0(1 + (hF + t) as) + F'b          - S'u0 as (q1 bR hR + (q1 bR + q2 ba) hF))                   (163)                 - 41 -       or by reduction:   O'(t) = S'u0(1 + as t)(p -(q1 bR + q2 ba)) - F'b                     (164)   It will be seen that equations (153) and (164) are identical, i.e. a systema- tic use of Fig. 2.1. gives here the same result as the use of a simple "logi- cal" accounting method.     12.2.1.  Operating profit incl. inventory depreciation   If equations (161) and (58) are substituted into equation (60), U'tl,1(t) being computed with a plus or minus sign, the following equation is obtained:   B'ln(t) = an S'u0 as (q1 bR hR + (q1 bR + q2 ba) hF)                  (165)   The operating profit incl. inventory depreciation is given by equation (62). If equations (164) and (165) are substituted into this equation, the following expression is derived:   O'DS(t) = S'u0(1 + as t)(p -(q1 bR + q2 ba)) - F'b             - an S'u0 as (q1 bR hR + (q1 bR + q2 ba) hF)             (166)   or by reduction:   O'DS(t) = S'u0((1 + as t)(p -(q1 bR + q2 ba))           - an as (q1 bR hR + (q1 bR + q2 ba) hF)           - F'b                                                   (166a)                               - 42 -       12.3.1.  Bank loans   This model takes as its starting point that the net working capital tied up K(t) can be given the form:   K(t) = K0 + Kinc(t)                                               (167)   where K0 is the net working capital tied up at time t = 0, and Kinc(t) is the change in the working capital tied up at time t. It is assumed that equation (168) applies:   d Kinc(t) ¾¾¾¾¾  = i'B(t)                                                (168)  d t   This means that the increase in the capital tied up in the operating system is financed by the bank overdraft.   If equation (168) is used together with equation (167), the following equation will also apply:   d K(t) ¾¾¾¾¾  = i'B(t)                                               (168a)  d t   It is assumed that:   B(0) = 0                                                         (169)   This means that the overdraft amounts to DKK B(0) = 0 at time t = 0.   As regards the mathematical model it is pointed out that in equation (168) i'B(t) may be both positive and negative as it is also assumed here that, besides equations such as (65), (66), (67) and (68), the following equation applies:   y'B(t) = r'B(t)                                                    (170)       - 43 -       12.3.2.  Loans (long term)   It is assumed that i'L(t) is discreet, i.e. that   i'L(t) = 0 and i'D(t) = 0                                          (171)   for all t > 0, apart from certain selected times tq  where, in practice, chan-ges take place in financing conditions, and new investments are made. Subject to these assumptions equation (72) may be reduced to   i'K(t) = i'B(t)                                                    (172)   with the condition i'L,D(t) = 0   In close connection with the operational financial possibilities of equations (171) and (172) this model also assumes that equation (173) applies:   y'L(t) = r'L(t)                                                    (173)       12.3.3.  Investments   Investments are defined by i'(t). It is here assumed that i'(t) = 0 apart from certain times tp corresponding to the forms of investment seen in practice.   In this mathematical model equation (75) is changed into:   i'(t) = i'L,i(t)                                                   (174)   where i'e(t) thereafter becomes the quantity, cash flow released, for the following purposes:                           - 44 -                                New investments                              Instalments on loans                              Etc.       This change of equation (75) is desirable seen in relation to the possibili- ties of implementing this mathematical model on a computer.       12.4.1.  Interest payments   From equations (78), (79), (80), (170) and (173) the total interest payment is derived:   y'B(t) + y'L(t) = rB B(t) + rL L(t)                                (175)   where rB is interest rate bank and rL is interest rate lender.       12.4.2. Depreciation   Depreciation to tax computation is obtained from equation (77) and is expressed as:   d'2(t) = aD D(t)                                                   (176)   where    aD is the depreciation rate per time period.       12.4.3.  Tax payments   From equations (81), (175) and (176) the following equation is derived:   f'u(t) = aD D(t) + rB B(t) + rL L(t)                               (177)   By using equations (82) and (177) total tax payments are expressed as:   H'S,1(t) = s(O'DS(t) - aD D(t) - (rB B(t) + rL L(t)))                (178)         - 45 -       12.4.4.  Cashflow released   With the special definition of i'e(t) given in 12.3.3. cash flow released is defined by:   i'e(t) = O'(t) - H'S,1(t) - (y'B(t) + y'L(t))   which together with equation (88) gives:   i'e(t) = l'(t) + dK(t)/dt - H'S,1(t) - (y'B(t) + y'L(t))   If equation (168a) including the related assumption is used here, the follow- ing equation is obtained:   i'e(t) = l'(t) - H'S,1(t) - (y'B(t) + y'L(t)) + i'B(t)                (179)   or if equations (175) and (178) are used:   i'e(t) = l'(t) + i'B(t) - s O'DS(t) + s aD D(t)          - (1 - s)(rB B(t) + rL L(t))                             (180)   By using equations (62) and (87), the following equation is derived from equation (180):   i'e(t) = O'(t) - s O'DS(t) + s aD D(t)          - (1 - s)(rB B(t) + rL L(t))                             (181)   If the function O'L(t) is defined by the equation:   O'L(t) = O'(t) - s O'DS(t)                                          (182)   O'L(t) may be designated as the profit after tax from the operating system.   Equation (181) is now transformed into:   i'e(t) = O'L(t)(1 + s aD D(t)/O'L(t)          - (1 - s)(rB B(t) + rL L(t))/O'L(t)                       (183)   It appears from equation (183) that it may be appropriate to define the fol- lowing managerial ratios:   - 46 -       12.4.4.1.  Interest relative   Interest relative is defined by equation (183):                     rB B(t) + rL L(t) rrel = - (1 - s)   ¾¾¾¾¾¾¾¾¾¾¾¾                              (184)                        O'L(t)   rrel can be interpreted as interest payments after tax in relation to profit after tax from the operating system.       11.4.4.2.  Depreciation relative   Depreciation relative is defined by equation (183):               aD D(t) arel =  s ¾¾¾¾¾¾¾                                              (185)              O'L(t)   arel may be interpreted as the improvement in cash flows after tax as a result of depreciation in relation to profit after tax from the operating system.                                                                 - 47 -       13.1.  Traditional ratios   In this Chapter some traditional ratios will be computed on the basis of the functional expressions derived in section 12.     13.1.1.  Contribution ratio   The contribution ratio DG(t) is defined by the following equation (186):            S'u,1(t) - (U'd,1,1(t) - F'