M
A S I
M A N A G E R I A L R A T I O S
by
H A N S J E S S E N
The Technical University
of Denmark
November 1982
AMT Publication
DI.82.85A
.
ABSTRACT
In order to determine
managerial ratios as mathematical analytical functions of time there has
been developed a graphical model of a firm. This model shows the physical relationship
between fundamental principles of bookkee ping, operating statements and
managerial economics. The model is the structural basis of the
determination of the mathematical analytical functions for management.
The analytical
background of traditional ratio techniqu es, including, Bela Gold lit. 40
and the Dupont pyramid, is described by means of a new developed general
managerial ratio funktion.
.
 I 
CONTENTS
Page
Sumary V
Preface VI
Part A:
CHAPTER A
1. An analytical business model 1
1.1. Introduction 1
1.1.1. S. Eilon's model 1
1.1.1.1. Functional relationships and assumptions 3
1.1.1.1.1. Change in the cost structure 7
1.1.1.1.2. Change in the earnings structure 13
1.1.2. Assessment of S. Eilon's
model
18
Part B:
CHAPTER_B
2. An analytical graphical business model 20
2.1. Activity parameters 20
2.1.1. Sales
20
2.1.2. Purchases
20
2.1.3. Inventories 22
2.2. Payment parameters, operations
2.2.1. Sales
22
2.2.2. Purchases
23
2.3. Market parameters, sales 23
2.3.1. Cash sales ratio q 23
2.3.2.
The price p 24
2.3.3. Debit time d_{D} 24
2.4. Market parameters, purchases 25
2.4.l. Cash purchases ratio
e 25
2.4.2. The price q_{1} of raw materials 26
2.4.3.
The price q_{2}
of labor hours 26
2.4.4 Credit time d_{K} 27
 II 
3.1. Income statement 28
3.1.1.
Sales of goods 28
3.1.2.
Costs 29
3.1.2.1.
Inventories, additions (with signs) 30
3.1.3. Resource consumption (incl. F'_{i,1}) 33
3.1.4.
Operating profit (before interest and deprec.) 33
3.1.5.
Operating profit incl. inventory deprec. 33
4.1. Chanqe in liquidity (operations) 35
5.1. Cash balance 36
5.2. Bank loans 36
5.3. Loans (longterm)
37
6.1. Investment (in fixed capital)
38
7.1. Depreciation (for tax purposes) 39
8.1. Interest (for tax purposes)
40
9.1. Tax pavments 4O
10.1. Principal ratios 41
10.1.1.
Operating profit 0^{'}(t) 41
10.1.2.
Change in liquidity l^{'}(t) 42
10.1.3.
Working capital K(t)
43
10.1.4. Contribution ratio DG(t) 43
10.1.5.
Depreciation 44
10.1.6. Interest r^{'}_{BL}(t) 44
.
 III 
CHAPTER C
11. An analytical mathematical businessmodel 45
11.1. Physical and financial functions i
the operating
system
45
11.1.1. Sales
45
11.1.2. Inventories 46
11.1.3. Output
50
l1.1.4. Sales, ingoing payments 50
11.1.5. Purchases, outgoing payments
51
11.1.6. Change in liquidity 52
11.2. Capital tied up in the operating
system 52
11.2.1. Trade accounts receivable 52
11.2.2. Trade accounts payable 53
11.2.3. Raw materials invefitory 53
11.2.4. Finished goods inventory 54
11.2.5. Working capital (tied up in the operating system
54
12.1. Operatinq profit (for accountinq purposes) 55
12.2. Operating profit (computed on the
basis of Fig. 2.1.) 56
12.2.1. Operating profit incl. inventory depreciation
58
12.3.1. Bank loans
59
12.3.2. Loans (long term) 60
12.3.3.
Investments 60
12.4.1. Interest payments 61
12.4.2. Depreciation 61
12.4.3. Tax payments 61
12.4.4. Cash flow released 62
12.4.4.1.
Interest relative 63
12.4.4.2.
Depreciation relative 63
.
 IV 
13.1. Traditional ratios 64
13.1.1 Contribution ratio 64
13.1.2. Profit ratio 64
13.1.3. Breakeven sales 64
13.1.4. Margin of safety 65
13.1.5. Applications, examples 65
13.2. Dupont pyramid 66
13.2.1 Ratio mathematics, general
68
CONCLUSION 7O
BIBLIOGRAPHY 72
.
 V 
SUMMARY
For the determination of ratios as
analytical mathematical functions of time a graphical model of a firm has been
developed. This model is a graphical representation of the relationships
between fundamental aspects of the firm relating to bookkeeping (records),
accounting and managerial economics. The model forms the basis of the
following deve lopment of analytical mathematical functions. The
mathematical background of traditional ratio techniques, including Bela
Gold lit. 40 and the Dupont pyramid, is shown through the development of a
general ratio function.
Lyngby, November 1982
.
 VI 
PREFACE
The existing literature on accountancy
and managerial economics has ma de several attempts to improve the
theoretical basis in order to provide management with a better
understanding of business management possibili ties.
In lit. 20, Albert Danielsson is dealing
solely with purely analytical
aspects in relation to costs of
production, and he has in that connec tion developed symbolic flow charts
for analysis purposes. This work seems to be of a very special character and
not suited for overall ma nagement purposes where the firm is to be seen
as a whole. Links to inventories and the market are, for instance, missing.
Bela Gold, lit. 40, attempts to
generalise accounting ratios in a technical structure which includes
managerial ratios. This technique seems to be very practicable but only for
partial global business analyses. In this thesis a theoretical analysis of
general ratios will be made, including the Dupont pyramid and including,
in particular, Bela Gold's ratio technique.
J. W. Forrester, lit. 37, provides with
his special representation
technique based on computer technology an
excellent basis for analy sing company behavlour. It gives, in a certain
degree, a good insight into the behavlour of a firm in situations with
different external and internal influences. Also here a fundamental
mathematical model for purely analytical purposes is missing.
Dan Ahlmark, lit. 1, stresses the
necessity of developing an analysis
model of the business which makes it
possible to consider the current
integrated process, production,
investment, financing activities of the business. To illustrate this need,
an extensive empirical business ana lysis is made, using generally known
simulation techniques.
.
 VII 
Finn C. Sørensen, lit. 97, finds in his
review of traditional accoun
ting methods that a model should be
developed for man agement which is
suitable for illustrating general matters
in the firm, i.e. form the
basis of an actual managerial audit. By this
is meant an examination of activities and matters underlying the
financial/accounting report.
Samuel Eilon, lit. 30, attempts with his
mathematical model to compute
the rate of return as a function of
general business parameters, using, among other things, a symbolic
graphical representation technique to describe the inter relationships of
the equations. This work seems to be the most interesting work in the
literature seen in relation to the de velopment of a generalised business
analysis model.
Using the literature reviewed as a
starting point with special impor
tance being attached to the above
authors, the structure and field of
applications of Eilon's model in lit. 30
will be analysed in detail.
After this analysis, a graphical
analytical business model is developed in Chapter B including book keeping,
accounting and financial concepts to be employed by the business
management. Using this model it is possible to carry out an actual
managerial audit as described by Finn C. Sø rensen, among others.
Chapter C defines an analytical
mathematical business model based on the general graphical structure shown
in Chapter B. As a special starting point is taken the fact that any sales
curve may be composed of a piece wise linear function. The basic element
of the sales function is thus chosen as a linear function of time.
Based on the developed mathematical
functions the most common accounting ratios are computed as a function of
time.
.
 VIII 
Bela Gold's ratio technique is examined
more ciosely, using general mathematical ratio functions developed in this
report, and an attempt is made to explain it by means of these functions,
which are also used to illustrate
the technical background of the computation of the rate of return in the
Dupont pyramid.
.
C
H A P T E R A
.
 1 
1.
An analytical business model
1.1.
Introduction
During the mentioned review of the
litterature only one source was found, which was suitable for forming the
basis of the development of
the general mathematical business model
in Chapters B and C. This sour ce was Samuel Eilon's article in OMEGA
1997, Vol. 5, No. 6: "A Profita bility Model for Tactical
Planning", lit. 30.
In the following the mentioned article
will therefore be discussed as
an introduction to the analytical
mathematical business model in Chap
ter C.
1.1.1
S. Eilon's model
The article starts by pointing out that
simple models reflecting aggre gate company behaviour in response to
changes imposed by management de cisions and/or outside factors provide
useful tools for management for tactical planning purposes.
As his starting point, Eilon takes the
rate of return r expressed as:
(p  c)V
r = ¾¾¾¾¾
(1)
I
or
earnings
r =
¾¾¾¾¾¾¾¾¾¾¾ (2)
total investment
where
.
 2 
p = unit price per unit of output
c = unit cost per unit of output
V = output units per time unit
I = total investment
Attention is drawn to the fact that for
macro economic purposes it is
possible to compute the ratio r from the
definition equation (2) and
thus obtain information on an industry's
"profitability".
Equation (1) is a micro economic
rewriting of (2) based on "logical"
considerations. The numerator in equation
(1) is fairly well defined,
also in a micro economic model, but the
denominator I, total assets,
which serves the main purpose, is
difficult to determine in practice.
Additions to and disposals of assets as
well as changes in the market
value of these assets take place currently.
The practical purpose of the computation
of the rate of return is a
desire to obtain an equivalent measure of
the return on investment. It
appears from the above that in practice
the com putation of r involves
great uncertainty so that r is a relatively
uncertain measure of profi tability. If the following definitions are now
introduced
dp dc dI
p^{*} = ¾¾¾ ;
c^{*}
= ¾¾¾ ; I^{*} = ¾¾¾ ,
p d I
where the changes dp, dc, dV and dI are
given, equation (1) can be
transformed into
1 1 + V^{*}
r^{*} = ¾¾¾¾¾¾ (¾¾¾¾¾ (p^{*}  (1  a) c^{*}) + V^{*}  I^{*}) (3)
1 + I^{*} a
.
 3 
p  c
given definition a = ¾¾¾¾¾ = the relative
profit margin.
p
As regards equation (3), S. Eilon
observes that it is an analytical tool for assessing the effects on r of
changes of the variables of the
right hand side.
In practice, a functional inter
relationship exists very often between
these variables; a later change in the
selling price or the cost price will, for instance, bring about changes not
only in investments (in
the working capital) but also in demand
and hence output.
1.1.1.1.
Functional relationships and assumptions
Eilon assigns to the cost c per output
unit the following conventional
functional expression:
F J
c = s + ¾¾ + ¾¾ (4)
V V
where
s represents direct
unit costs
F
¾¾ represents indiret unit costs excl. interest
V
J
¾¾ represents interest
charge per unit.
V
Equation (4) is a so called traditional
economic calculation of total
unit costs. It should, however, be noted
that from a general accoun
ting point of view there is no real
justification for equation (4). It
is simply an appropriate formula for the
unit cost function in relati
on to the traditional theory of
managerial economics, which makes it
possible to carry out simple partial
operations research computations
.
 4 
concerning, for instance, profit
maximization in relation to various
alter natives.
The stressing of the point that equation
(4) has no real physical ju
stification is due to the fact that
equation (4) is a simple transfor mation of equations (5) and (6).
TO = c V (5)
TO = s V + F + J (6)
Equation (5) is here a purely non
physical definition equation (addi
tion of simple "krone amounts")
for the total costs TO specified via
the definition equation (6). It will be
seen that these definition equa tions give rise to problems in connection
with the physical interpreta tion. What is, for instance, meant by fixed
costs, and how are they de fned in relation to, say, the interest charges
J ?
A transformation of equation (4) using
the definitions s = f_{1} c , F/V
= f_{2} c and J/V = f_{3} c gives
1
c^{*} = f_{1} s^{*} + ¾¾¾¾¾(f_{2} F^{*} + f_{3} J^{*}  (1  f_{1}) V^{*}) (7)
1 + V^{*}
This equation (7) shows that with a good
approximation we have:
c^{*} = f_{1} s^{*} + f_{2} F^{*} + f_{3} J^{*}  (1  f_{1}) V^{*} (8)
Interpretation of the contents of, for
example, (8) will show that the
percentage change of the unit costs is
equal to the weighted sum of the percentage change of s, F, J and V.
.
 5 
Concerning investments I, S. Eilon
assumes:
I_{W} = A + B (9)
I = I_{W} + I_{F} (10)
where
I_{W} = the working capital
I_{F} = investment in fixed assets
B
= bank loans + overdrafts
A
= other loans
S. Eilon also defines:
w = I_{W}/I (10a)
l = B/I_{W} (10b)
J = j B , where j is the interest rate (lOc)
Equation (10) can now be transformed
into:
I^{*} = w I^{*}_{W} + (1  w)I^{*}_{F}
(11)
Equation (9) can be transformed into:
I^{*}_{W} = (1  l) A^{*} + l B^{*} (12)
A combination of equation (12) and
equation (11) will take the form:
I^{*} = w ((1  l) A^{*} + l B^{*}) + (1  w) I^{*}_{F} (13)
For use in the actual planning process of
the business, S. Eilon assu
mes that
.
 6 
I^{*}_{F} = 0 (14)
and
A^{*} = 0
(15)
It is also assumed that w and l are constant (the artiticle does not
mention this explicitly).
Based on the mentioned assumptions
equations (12) and (13) are then
reduced to
I^{*}_{W} = l B^{*} (16)
and
I^{*} = w l B^{*} (17)
gives the conditions (14) and (15).
In connection with the determination of
changes in the working capital
I_{W}, S. Eilon writes:
"No single relationship between
working capital and the 1evel of ac
tivity in the firm is universally
accepted and we may proceed to ex
plore two possible assumptions."
These two assumptions are combined as a
linear combination
I^{*}_{W} = g (p v)^{*} + h(c V)^{*} (18)
which denotes that the working capital
(tied up in the operating sy
stem is changed as a linear combination
of the change in sales and the
change in cost. S. Eilon claims that no
controller has difficulty in
determining empirically the constants g
and h. It must therefore be
.
 7 
possible to find a physical model which
describes these empirical facts. A mathematical analytical solution to this
problem is described
in Chapter C.
S. Eilon proceeds to consider three cases
which are relevant for tac
tical planning purposes:
1. Change of s, F and j
2. Change of V
3. Change of p
In the first case changes in the cost
structure are considered. The
following two cases deal with changes in
sales and changes in the mar
ket price, i.e. two situations where the
earnings structure is chan
ged. However, as regards cases 2 and 3, it
is natural to describe them
together as will be seen later. The
things to be discussed are therefo re as follows:
1. Change in the cost structure caused by
changes in s, F and j.
2. Change in the earnings structure
caused by changes in V given the
market elasticity e.
1.1.1.1.1. Change in the cost structure
Attention is drawn to the fact that in
his case 1 S. Eilon discusses
an iterative process, physical and
mathematical, in connection with the final computation of c^{*}. From a physical point of view,
this is in full accordance with the accounting theory, as will be shown
later in the ge neral mathematical business model. S. Eilon attempts to
provide this "fact" of the expressions de scribed here through the mathematical convergence in the computation
of total unit costs as shown in the article. In this respect, however, it
does not seem to be a good idea to combine physical and mathematical facts
too much since, as has already been
 8 
mentioned, S. Eilon employs a non physical
definition of unit costs (see equation (4), which highly weakens the
foundation of Eilon's conclusi ons.
S. Eilon elaborates on this definiton of
the unit cost, one of the prin ciples of traditional theories of
managerial economics, in case 1. It is exactly these conflicts between the
physical conditions in the firm and the traditional theory of managerial
economics which have caused the de velopment of the mathematical business
model described in Chapter C.
With a view to solving the existing
mathematical problem, equation
(1Oc) is transformed into:
J^{*} = j^{*} + (1 + j^{*}) B^{*} (2o)
The mathematical problem can now be
solved by means of the followinq
previously shown equations (7), (16) and
(18) together with the reated conditions:
Equations:
1
c^{*} = f_{1} s^{*} + ¾¾¾¾ (f_{2} F^{*} + f_{3} J^{*}  (1  f_{1}) V^{*}) (21)
1 + V^{*}
J^{*} = j^{*} + (1 + j^{*}) B^{*}
(22)
I^{*}_{W} = l B^{*} (23)
I^{*}_{W} = g(p v)^{*} + h(c v)^{*} (24)
.
 9 
given the conditions
V^{*} = 0
(25)
p^{*} = 0 (26)
A solution is obtained as follows:
From equation (24) with the conditions
(25) and (26) it follows that
I^{*}_{W} = h c^{*} (27)
Equation (27) and (23) give
h
B^{*} = ¾¾
c^{* } (28)
l
A combination of equation (28) and (22)
gives
h
J^{*} = j^{*} + (1 + j^{*}) ¾¾ c^{*} (29)
l
A combination of equation (29) and (21)
gives
f_{1}
s^{*} + f_{2} F^{*} + f_{3} j^{*}
c^{*} = ¾¾¾¾¾¾¾¾¾¾¾¾¾¾ (30)
h
1  (1 + j^{*}) ¾¾ f_{3}
l
In equation (30) the question is raised whether
h
(1 + j^{*}) ¾¾ f_{3} < 1
l
has been satisfied as, in practice,
equation (lOb) shows that
1 I_{W}
¾¾ = ¾¾
l B
.
 l0 
Normally will I_{W} < B, hence
1
¾¾ < 1
l
As typically in practice h < 1, f_{3} < 0.5 and (1 + j^{*}) < 1.5, inequa1ity
(31) gives
h 1
(1 + j^{*}) ¾¾ f_{3} < 1,2 ¾¾ 0.5
l 1
or
h
(1 + j^{*}) ¾¾ f_{3} < 0.6
l
From this will be seen that, in practice,
inequality (31) has been sa
tisfied.
S. Eilon introduces a new ratio H = I_{W} /(c V) for the purpose
showing
that inequality (31) has been satisfied
in practice. This seems to be
a purely mathematical exercise without
any relevant justification phy
sically. It is once more pointed out that
S. Eilon overinterprets the
mathematical consequences of the use of
the equation (4) defined on the basis of managerial economics.
For the purpose of computing the rate of
return the equations (3),
(lOa) and (18) are used to solve the
equations:
1 1 + V^{*}
r^{*} = ¾¾¾¾¾ (¾¾¾¾¾(p^{*}  (1  a) c^{*}) + V^{*}  I^{*}) (32)
1 + I^{*} a
1
I =
¾¾ I_{W} (33)
w
.
 11 
I^{*}_{W} = g(p V)^{*} + h(c v)^{*} (34)
with the conditions:
f_{1}
s^{*} + f_{2} F^{*} + f_{3} j^{*}
c^{*} = ¾¾¾¾¾¾¾¾¾¾¾¾¾¾ (35)
h
1  (1 + j^{*}) ¾¾ f_{3}
l
V^{*} = 0 (36)
p^{*} = 0
(37)
Equation (34) gives, cf. equation (27)
I^{*}_{W} = h c^{*} (38)
Equation (33) is transformed into
I^{*} = w I^{*}_{W} (39)
Equatioin (38) combined with equation
(39) gives
I^{*} = w h c^{*} (40)
Equation (32) gives with equation (40)
and the conditions (35),
(36) and (37) the following expression
c*
1  a
r^{*} =  ¾¾¾¾¾¾¾¾ (¾¾¾¾¾ h w ) (41)
1 + w h c^{*} a
given
f_{1}
s^{*} + f_{2} F^{*} + f_{3} j^{*}
c^{*} = ¾¾¾¾¾¾¾¾¾¾¾¾¾¾ (30)
h
1  (1 + j^{*}) ¾¾ f_{3}
l
.
 12 
As regards equation (30) it should be
noted that S. Eilon finds it
"justified" to define a
quantity
c^{*}_{0} = f_{1} s^{*} + f_{2} F^{*} + f_{3} j^{*} (42)
as unit costs, if h = 0, i.e. if no
changes occur in the working ca
pital with the given conditions V^{*} = O and p^{*} = 0 , cf. equa tion (38).
Equation (35) now takes the form
c^{*}_{0}
c^{*} = ¾¾¾¾¾¾¾¾¾¾¾¾¾¾ (30)
h
1  (1 + j^{*}) ¾¾ f_{3}
l
given
c^{*}_{0} = f_{1} s^{*} + f_{2} F^{*} + f_{3} j^{*}
Further, on the basis of the denominator
in equation (43), S. Eilon
defines a ratio u as he seems to find it desirabie
that all ratios oc
cur in product form. For instance, as
mentioned previously in this
connection, he also defines the ratio H =
I_{W}/(c V), which from the
point of view of accounting theory is a
very specific concept.
A look at equation (43) will show that it
takes the form of "ratios",
i.e. it contains dimensionless
quantities, which are all ratios in the
firm. Therefore, it does not seem to be a
very desirable measure to
introduce further ratios to give the equa tion a changed algebraic
structure.
However, for analytical purposes in
connection with an analysis of the
numerical "behaviour" of
equation (43) it may be useful to define a
parameter x given by
.
 13 
h
x =
(1 + j^{*}) ¾¾ f_{3 } (44)
l
so that equation (43) is transformed into
c^{*}_{0}
c^{*} = ¾¾¾¾¾ (45)
1  x
given
h
x =
(1 + j^{*}) ¾¾ f_{3} (46)
l
and
c^{*}_{0} = f_{1} s^{*} + f_{2} F^{*} + f_{3} j^{*} (47)
Thus, by a preliminary nurnerical
analysis of (45), x may be a11owed
to vary in the interval 0 < x < 1.
It should be noted that x is here a
parameter. In the second phase of such an
anal ysis a numerical ana lysis of
equation (46) can be carried out, given certain selected va lues of x.
1.1.2.1.2. Change in the earnings structure
In this case where management wishes to
consider the influence of the
market on the rate of return, etc., the
following expression is assu
med to apply
V^{*} =  e p^{*} (48)
given p^{*}.
.
 14 
The problem is thus given by the
equations
1
c^{*} = f_{1} s^{*} + ¾¾¾¾¾ (f_{2} F^{*} + f_{3} J^{*}  (1  f_{1}) V^{*}) (49)
1 + V^{*}
J^{*} = j^{*} + (1 + j^{*}) l B^{*}
(50)
I^{*}_{W} = l B^{*} (51)
I^{*} = g(p V)^{*} + h(c V)^{*}
(52)
with the conditions:
s^{*} = 0
(53)
F^{*} = 0
(54)
j^{*} = 0 (55)
V^{*} =  e p^{*} (56)
Here equation (52) is transformed into
I^{*}_{W} = (g + h)V^{*} + (g p^{*} + h c^{*})(1 + V^{*}) (56a)
After the transformation of the above
equations and with the above
conditions the following equations are
developed:
1
c* = ¾¾¾¾¾ (f_{3} B^{*}  (1  f_{1}) V^{*}) (57)
1 + V^{*}
1
B* =
¾¾((g + h) V^{*} + (g p^{*} + h c^{*})(1 + V^{*})) (58)
l
given the condition V^{*} =  e p^{*} (59)
.
 15 
Now equations (57), (58) and (59) give by
simple reduction
e
p^{*
}^{ }1 f_{3}
c^{*} = (1  f_{1}) ¾¾¾¾¾ ¾¾¾¾¾¾¾¾¾¾ (1  ¾¾¾¾¾¾¾
1  e
p^{*} f_{3 } (1  f_{1}) l_{}
_{ }1  ¾¾¾ h
l
(h + g(1 + p^{*}  e^{1}))) (60)
given the conditions
s^{*} = 0
(61)
F^{*} = 0
(62)
j^{*} = 0 (63)
V^{*} =  e p^{*} (64)
In connection with the practical use of
equation (60) it might be de
sirable to define a change in the unit
cost cx* given by
e
p^{*}
c^{*}_{x} = (1  f_{1}) ¾¾¾¾¾¾ (65)
1  e
p^{*}
which has been obtained by putting s^{*} = 0, F^{*} = 0 and J^{*} = 0 in equati on (7). c^{*}_{x} can here be interpreted as the
change in the unit cost if the only thing to be considered is a change in
the price p.
Moreover, from equation (60) can be
defined
g
y = ¾¾¾
h
which may be interpreted as the need of
investment in the working ca
pital caused by sales in relation to
caused by costs (see equation (18)).
.
 16 
The system of equations (60) .... (64) is
now given the form
c^{*}_{x} x
c* = ¾¾¾¾ (1  ¾¾¾¾ (1 + y(1
+ p^{*}  e^{1}))) (66)
1  x 1  f_{1}
given the conditions
h
x =
(1 + j^{*}) ¾¾ f_{3 } (67)
l
g
y = ¾¾¾
h
Using equations (3), (17), (57) and (60)
.... (64) the following sy 
stem of equations can now be defined for
the determination of the rate
of return.
Equation:
1 1 
e p^{*}
r^{*} = ¾¾¾¾¾¾¾ (¾¾¾¾¾¾ (p^{*}  (1  a) c^{*})  e p^{*}  w I^{*}_{W}) (69)
1 + w I^{*}_{W} a
given the conditions
s^{*} = 0 (70)
F^{*} = 0
(71)
j^{*} = 0
(72)
V^{*} =  e p^{*} (73)
I^{*}_{W} = (g + h)V^{*} + (g p^{*} + h c^{*})(1 + V^{*}) (74)
e
p^{*}^{ }1 f_{3}
c^{*} = (1  f_{1}) ¾¾¾¾¾ ¾¾¾¾¾¾¾¾¾¾ (1  ¾¾¾¾¾¾¾
1  e
p^{*} f_{3} (1  f_{1}) l_{}
1  _{ }¾¾¾ h
l
(h + g(1 + p^{*}  e^{1}))) (75)
.
 17 
If changes are recorded only in V, the
following system of equations
is obtained by replacing p^{*} with V^{*} and putting p^{*} = 0 and e^{1} = 0 in
the above equations:
V^{*} 1 f_{3}
c^{*} =  (1  f_{1}) ¾¾¾¾¾ ¾¾¾¾¾¾¾¾¾¾ (1  ¾¾¾¾¾¾¾
1  V^{*} f_{3} (1  f_{1}) l_{}
1  _{ }¾¾¾ h
l
(h + g)) (76)
given the conditions
s^{*} = 0
(77)
F^{*} = 0
(78)
j^{*} = 0 (79)
e^{1} = 0
(80)
p^{*} = 0
(81)
and the system of equations:
1 1  a
r^{*} = ¾¾¾¾¾¾¾ (V^{*}  ¾¾¾¾¾ (1 + V^{*}) c^{*} + w I^{*}_{W}) (82)
1 + w I^{*}_{W} a
given the conditions
s^{*} = 0
(83)
F^{*} = 0 (84)
j^{*} = 0
(85)
I^{*}_{W} = (g + h(1 + c^{*})) V^{*} + h c^{*} (86)
V^{*} 1 f_{3}
c^{*} =  (1  f_{1}) ¾¾¾¾¾ ¾¾¾¾¾¾¾¾¾¾ (1  ¾¾¾¾¾¾¾
1  V^{*} f_{3} (1  f_{1}) l_{}
1  _{ }¾¾¾ h
l
(h + g)) (87)
Attention is called to the fact that the
resuits in this Chapter dif
fer from S. Eilon's results in cases 2
and 3. The following Chapter
will indude a general discussion of S.
Eilon's results and models in
the light of the results achieved here.
 18 
1.1.2. Assesment of S. Eilon's model
It has already been pointed out that the
basis of S. Eilon's model gi ves rise to the question as to whether it
serves any purpose to carry out these computations and at the same time
attach such fundamental importance to the models shown in the article in
relation to the phy sical business situation.
Thus, S. Eilon assumes that eguation
(4) is fundamental, i.e. a funda mental starting point for
considerations based on managerial economies. With reference to eguations
(5) and (6) it was stated that this is a point of view which should be
examined more closely. This examination leads to the point that eguation
(4) is a purely mathematical definition equation, i.e. an equation which
is not founded on real physical facts (equation (6)'s right hand side
consists of a sum of elements of widely differing physical origin with only
one thing in
common: the value "DKK").
Owing to the mathematical structure of
equation (4) it will mathema
d J(V)
tically be convergent as where U £ ¾¾¾ (¾¾¾)
£ K , in practice U
dV V
and K are constants. The physical
convergence also exists in connec tion with the changes in the tactical
planning process (transients) under consideration. It should be noted that
the mathematical model shows the relationships between changes in
states" (i.e. time is not included explicitly) with the related
mathematical characteristics of the manner of converging. The physical
activity/cash flow model of the business is knovn also in practice to
possess convergent characteri stics as a function of time. See chapter C.
Against this background it is important
not to attach too great impor
tance here to the applicability of S.
Eilon's model to an interpreta
tion of the dynamics of the firm (for
tactical planning purposes).
Thus, the mathematical business model,
Chapter C, is not to take state
functions as its starting point but only
use a time description of the
functions.
.
 19 
The graphical description used by S.
Eilon can only be regarded as a
dear description of the equations between
the individual variable.
being studied.
In Chapter B a physical model description
of the business will the
refore be given first, the greatest
importance being attached to ma
king the physical/financial description
as realistic as possible. Af
ter this the mathematical déscription is
developed in Chapter C.
The results achieved in the present
Chapter A differ from S. Eilon's
results as far as computations of the
effects of changes in the ear
nings structure are concerned. It is
pointed out that S. Eilon's un
structuralized consideration of the
mathematical methods of solution
may be the reason for the deviating
results in the article.
The central equation (18), which
estimates the relationship between the working capital tied up in the
operating system, will be analysed in detail in Chapter C.
.
C H A P T E R
B
.
 20 
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 description of the business so that this
description can be used by the business management for their principal
planning activities. The model will integrate principal elements of
managerial economics and the ac counting theory, it being assumed that the
business comprises an acti vity/cash flow and related principal assets
(accounts payable, accounts receivable, inventories). It is management's
task to achieve the best possible composition of this general structure by
using some of the ra tios defined in the model.
2.1. Activity
parameters
2.1.1. Sales
The volume of goods sold by the firm per
unit is denoted with S^{'}_{u}. 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
deltaD
after delivery
from the firm. Here the following eguation applies:
S^{'}_{u}(t) = S^{'}_{u,deb}(t) + S^{'}_{u,kon}(t)
(88)
2.1.2.
Purchases
The firm is supplied with a number of
labor hours per time unit a^{'}_{i}.
The firm is supplied with the volume of
goods per time unit V^{'}_{i}. This flow of goods consists
of two main components of which one is the refe rence purchase V^{'}_{i,kon}, and the other the goods
purchased on credit V^{'}_{i,kre}, which are paid for by the
firm after the credit time d_{K}.
 21 
Figure
2.1
(Click
on the figure for 200%)
.
 22 
The following equation applies:
V^{'}_{i}(t) = V^{'}_{i,kre}(t) + V^{'}_{i,kon}(t)
(89)
The firm is supplied with the fixed
volume of resources per time unit F^{'}_{i}. This flow of resources may,
for example, include electricity, administration, heating, rent, etc.
2.1.3.
Inventories
The
volume of raw materials per time unit Q^{'}_{i} is added to the raw mate
rials inventory consisting of the volume
R_{L}. From the raw materials in
ventory is deduced the raw materials volume Q^{'}_{u}. The following equation
applies here:
t
R_{L} = ò (Q^{'}_{i}(t)  Q^{'}_{u}(t))dt (90)
0
The
volume of finished goods per time unit Z^{'}_{i} is added to the finished
goods
inventory consisting of the volume F_{L}. From the finished goods
inventory is deduced the finished goods
volume Z^{'}_{u}. The following equa tion applies here:
t
F_{L} = ò(Z^{'}_{i}(t)  Z^{'}_{u}(t))dt
(91)
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} stemming from the cash sales
flow
S^{'}_{u,kon}. The other component S^{'}_{i,deb} is the payments flow stemming
from the credit sales flow S^{'}_{u,deb}. Here the following equation
applies:
S^{'}_{i}(t) = S^{'}_{i,kon}(t) + S^{'}_{i,deb}(t)
(92)
.
 23 
2.2.2. Purchases
The total volume of means of payment per
time unit for operations is denoted with U^{'}_{b}. This payments flow is
composed of three components, a^{'}_{b} and V^{'}_{b} and F^{'}_{b}. a^{'}_{b} is the payments flow
corresponding to the flow of hours consumed a^{'}_{i}, V^{'}_{b} is the payments flow
corresponding to the flow of raw material purchases V^{'}_{i}, F^{'}_{b} is the payments 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) (93)
The payments flow V^{'}_{b} is made up of two components.
One component is the payments 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)
(94)
2.3.
Market parameters, sales
With a viev to depicting the fundamental
financial effects of the mar ket on the firm as well as its effects on
earnings the market is cha racterized by three basic components q , p and d_{D}. They also describe the
fundamental link between the firm's sales of goods and the related payments
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) (95)
where 0 £ q £
1
In a manufacturing business q will typically be placed in the in
terval 0 £ q
£ 0.2.
In a supermarket q will typically be in the interval 0.8 £ q £ 1.
.
 24 
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) (96)
S^{'}_{i,kon}(t) = S^{'}_{u,kon,1}(t)
(97)
where S^{'}_{u,kon,1}(t) is the flow of debts
corresponding to the sales flow S^{'}_{u,kon}(t) (i.e. the current sending
out of invoices stating the amount of debt; see equation (96)). Equation
(97) 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 only a tempora ry time lag between invoicing and sales.
However, it has a temporary negative effect on liquidity and the
computation of results. Manage ment will therefore as far as possible make
sure that invoicing is done without the mentioned delays.
2.3.3.
Debit time d_{D}
This model defines the debit time d_{D} as the time from the time of
de livery of the goods from the firm until the time of payment by the cu
stomer for the goods. In practice, d_{D} is spread over the individual cu stomers but with well defined
terms of payment the mean value can be adopted.
The definition of d_{D} can be expressed by the
equations
S^{'}_{u,deb,1}(t) = p S^{'}_{u,deb}(t)
(98)
V^{'}_{deb,}_{d}_{D}(t) = S^{'}_{u,deb,1}(t  d_{D}) (99)
S^{'}_{i,deb}(t) = V^{'}_{deb,}_{d}_{D}(t) (100)
.
 25 
S^{'}_{u,deb,1} refers here to the invoice
flow corresponding to the credit sales flow S^{'}_{u,deb} cf. equation (98). Equation (99)
gives a funational description of a function V^{'}_{deb,}_{d}_{D}(t), which can be defined as
the pay ments flow (documents) corresponding to the actual receipt of
payments S^{'}_{i,deb}(t) cf. equation (100). 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 liquidity and the computation of
earnings. Management usu ally aims at applying equation (98) in practice,
i.e. no time lag.
2.4. Market parameters, purchases
With a view to depicting the fundamental
financial effects of the pur chasing market on the firm as well as its
effects on costs, it is cha racterized by four basic components epsilon, q_{1}, q_{2} and d_{K}. They describe the
fundamental link between the firm's purchases of resources and the related
payments 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) (101)
where 0 £ e £
1
In, say, a manufacturing business e will typically be placed in the interval
0 £ e £ 0.2.
This is also a typical feature in a trading firm.
2.4.2.
The price q_{1} of raw materials
The price of the firms raw materials is
defined by the equation:
.
 26 
V^{'}_{i,kon,1}(t) = q_{1} V^{'}_{i,kon}(t) (102)
V^{'}_{b,kon}(t) = V^{'}_{i,kon,1}(t)
(103)
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 (102). Equation
(103) 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 bepaid 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.
2.4.3. The price q_{2} of labor hours
The price of the firm's labor hours is
defined by the equations
a^{'}_{i,1}(t) = q_{2} a^{'}_{i}(t) (104)
a^{'}_{b}(t) = a^{'}_{i,1}(t) (105)
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 (104). This time lag is ignored here. There is
usually no time lag between
the functions of equation (105), or the
time lag is relatively small and of no importance here.
.
 27 
2.4.4. Credit time d_{K}
This model defines the credit time d_{K} as the time from the time of de
livery of the raw materials to the firm until the time of payment by the
firm for the raw materials. In practice, d_{K} is spread over the in
dividual suppliers but with well defined terms of payment the mean va lue
can be used. The definition of d_{K} can be expressed by the equati ons:
V^{'}_{i,kre,1}(t) = q_{1} V^{'}_{i,kre}(t) (106)
V^{'}_{kre,}_{d}_{K}(t) = V^{'}_{i,kre,1}(t  d_{K})
(107)
V^{'}_{b,kre}(t) = V^{'}_{kre,}_{d}_{K}(t) (108)
where V^{'}_{i,kre,1}(t) refers here to the invoice
flow corresponding to the
credit purchases flow V^{'}_{i,kre}(t), cf. equation (106).
Equation (107) gives a functional description of a function V^{'}_{kre,}_{d}_{K}(t) which can be defined as
the payment order flow (documents) corresponding to the actual effecting of
payments V^{'}_{b,kre}(t), cf. equation (108). 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)
(109)
F^{'}_{b}(t) = F^{'}_{i,1}(t)
(110)
.
 28 
where F^{'}_{i,1}(t) in equation (109) refers to
the flow of debts in the form of invoices (stating amounts) corresponding
to the fixed resoures flow F^{'}_{i}(t). k denotes a symbolic operator in the form
of an average price of the fixed resources unit. In practice, there is some
time lag between the functions in eguation (110). As, however, the fixed
costs by definition are constant in time, such a time lag is not important
in this context.
3.1 Income statement
In this Chapter an income statement for
operations is presented (be fore depreciation, etc.) using the general
main principles of accoun ting 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)
(111)
S^{'}_{u,deb,2}(t) = S^{'}_{u,deb,1}(t)
(112)
S^{'}_{u,1}(t) = S'_{u,kon,2}(t) + S^{'}_{u,deb,2}(t) (113)
Eguation (111) expresses the fact that
the flow of debts (in the form of invoices with statement of amounts) S^{'}_{u,kon,1}(t) gives rise to an e qually
large information flow S^{'}_{u,kon,2}(t). This quantity is
identital with the current crediting to the cash sales account.
From equation (112) 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 iden
tical to the current crediting to the credit sales account.
Total uales in the form of the
information flow S^{'}_{u,1}(t) corresponding to the total
crediting to the sales account are then obtained from e quation (113).
.
 29 
3.1.2
Costs
The costs of the firm in connection with
production and sales are de fined by the following equations:
V^{'}_{i,kon,2}(t) = V^{'}_{i,kon,1}(t)
(114)
V^{'}_{i,kre,2}(t) = V^{'}_{i,kre,1}(t) (115)
a^{'}_{i,2}(t) = a^{'}_{i,1}(t)
(116)
F^{'}_{i,2}(t) = F^{'}_{i,1}(t)
(117)
U^{'}_{d}(t) = V^{'}_{i,kon,2}(t) + V^{'}_{i,kre,2}(t) + a^{'}_{i,2}(t) + F^{'}_{i,2}(t) (118)
Equation (114) expresses the fact that
the invoice flow from the cash purchase V^{'}_{i,kon,1}(t) is corrently debited to
the cash purchases account
to
the extent of the cash flow V^{'}_{i,kon,2}(t).
Equation (115) expresses the fact that
the invoice flow from the cre dit purchase 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 (116) 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 (117) expresses the functional
relationship between the in voice 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 (118).
.
 30 
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 additions to the individual
inventories will for principal planning purposes have the same signs. The
inventories only serve as "standby stores" in case of emergancy events"
i.e. in normal operation state "the materials and products go directly
through the factory. Thus, the following systems of equations apply:
Q^{'}_{i}(t) > 0
Q^{'}_{u}(t) = 0
d S^{'}_{u}
¾¾¾¾ > 0 Þ (119)
dt
Z^{'}_{i}(t) > 0
Z^{'}_{u}(t) = 0
Q^{'}_{i}(t) = 0
Q^{'}_{u}(t) = 0
d S^{'}_{u}
¾¾¾¾ = 0 Þ (120)
dt
Z^{'}_{i}(t) = 0
Z^{'}_{u}(t) = 0
Q^{'}_{i}(t) = 0
Q^{'}_{u}(t) > 0
d S^{'}_{u}
¾¾¾¾ < 0 Þ (121)
dt
Z^{'}_{i}(t) = 0
Z^{'}_{u}(t) > 0
.
 31 
The system of equations (119) denotes
that inventories rise when sales rise.
The system of equations (120) denotes that
inventories are constant when sales remain unchanqed.
The system of equations (121) 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) = q_{R} Q^{'}_{i}(t) (122)
Q^{'}_{u,1}(t) = q_{R} Q^{'}_{u}(t) (123)
Z^{'}_{i,1}(t) = q_{F} Q^{'}_{i}(t) (124)
Z^{'}_{u,1}(t) = q_{F} Z^{'}_{u}(t) (125)
U^{'}_{tl}(t) = Q^{'}_{i,1}(t) + Z^{'}_{i,l}(t)
(126)
U^{'}_{al}(t) = Q^{'}_{u,1}(t) + Z^{'}_{u,1}(t)
(127)
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.
Q^{'}_{u,1}(t)
is the flow of deductions to raw materials invento
ries corresponding to the deductions to raw
mate
rials inventory records with statement of
amounts.
Z^{'}_{i,1}(t)
is the flow of additions to finished goods invento
ries corresponding to the additions to
finished
goods inventory records with statement of
amounts.
.
 32 
Z^{'}_{u,1}(t)
is the flow of deductions to finished goods inven
inventories corresponding to the deductions
to fi
nished goods inventory records with
statement of
amounts.
q_{R } denotes the caiculated rav material price
per unit
of finished goods.
q_{F} denotes the caiculated 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 (119), (120) and
(121) can nov be given the form:
d S^{'}_{u}
¾¾¾¾
> 0 Þ U^{'}_{tl}(t) > 0
and U^{'}_{al}(t) = 0 (128)
d t
d S^{'}_{u}
¾¾¾¾
= 0 Þ U^{'}_{tl}(t) = 0
and U^{'}_{al}(t) = 0 (129)
d t
d S^{'}_{u}
¾¾¾¾
< 0 Þ U^{'}_{tl}(t) = 0
and U^{'}_{al}(t) > 0 (130)
d t
Attention is dravn to the fact that the
physical model based on the FIFO principle can be deseribed mathematically
only by
d S^{'}_{u}
sign ( ¾¾¾¾ ) = sign (U^{'}_{tl}(t)) (131)
d t
given U^{'}_{al}(t) = 0 (132)
and U^{'}_{tl}(t) is computed with signs.
.
 33 
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)
(133)
d t given U^{'}_{al}(t) = 0
d S^{'}_{u}
¾¾¾¾
= 0 Þ U^{'}_{d,1,1}(t) = U^{'}_{d}(t) (134)
d t
d S^{'}_{u}
¾¾¾¾
< 0 Þ U^{'}_{d,1,1}(t) = U^{'}_{d}(t)  U^{'}_{al}(t)
(135)
d t
given U^{'}_{tl}(t) = 0
3.1.4.
Operation profit (before interest and depreciation)
The operating profit (before interest and
depreciation etc.) is defi ned by the equation:
O^{'}(t) = S^{'}_{u,1}(t)  U^{'}_{d,1,1}(t) (136)
3.1.5
Operating profit incl. inventory depreciation
If a tax year of the length T is
considered in a period of time
t_{1} £ t £
t_{1} + T where t_{1} is a time selected at random,
the following
functions can be defined:
t_{1}+T
V_{køb} = ò q_{1} V^{'}_{i}(t) dt
(137)
t_{1}
w = w(t_{1}) (138)
a_{n} = a_{n}(t) (139)
In equation (137) V_{køb} represents the purchases of goods
in the period t_{1} £ t £ t_{1} + T.
Equation (138) defines w(t_{1}) as the total inventory value
at time t_{1}.
a_{n}(t) in the equation defines
the inventory depreciation rate.
.
 34 
Materials consumed computed for tax
purposes is then derived from the following equation (140):
V_{skat} = V_{køb} + w(t_{1})  (w(t_{1})/(1  a_{n}(t_{1}))
^{ }
t_{1}+T^{}
+ ò (U^{'}_{tl}(t)  U^{'}_{al}(t)) dt) (1  a_{n}(t_{1}  T)) (140)
t_{1}
For principal planning purposes the mean
time value of a_{n}(t) for a given business will
be a constant a_{n} and limited i.e.
0 < a_{n} < 0.3 . Based on this
assumption equation (140) gives
t_{1}+T^{}
V_{skat} = V_{køb}  (1  a_{n}) ò (U^{'}_{tl}(t)  U^{'}_{al}(t)) dt (141)
t_{1}
Materials consumed for operations is
defined by the following equation
(54a):
t_{1}+T^{}
V_{drift} = V_{køb} + w(t_{1})  (w(t_{1}) + ò (U^{'}_{tl}(t)  U^{'}_{al}(t)) dt) (141a)
t_{1}
or
t_{1}+T^{}
V_{drift} = V_{køb}  ò (U^{'}_{tl}(t)  U^{'}_{al}(t)) dt) (142)
t_{1}
If equation (142) and equation (141) are
combined, the following equations are developed:
t_{1}+T^{}
V_{skat} = V_{drift} + a_{n} ò (U^{'}_{tl}(t)  U^{'}_{al}(t)) dt (143)
t_{1}
t_{1}+T^{}
V_{skat} = V_{drift} + ò a_{n}(U^{'}_{tl}(t)  U^{'}_{al}(t)) dt (144)
t_{1}
.
 35 
On the basis of equation (144) the
following functions can be defined:
U^{'}_{tl,1}(t) = U^{'}_{tl}(t) (145)
U^{'}_{al,1}(t) = U^{'}_{al}(t)
(146)
In equation (145) U^{'}_{tl,1}(t) denotes total additions to
inventories from a taxation point of view. U^{'}_{al,1}(t) denotes in equation (146)
total deductions from inventories from a taxation point of view.
With the following definition equation:
B^{'}_{ln}(t) = a_{n} (U^{'}_{tl,1}(t)  U^{'}_{al,1}(t)) (147)
equation (144) can be transformed into
t_{1}+T
V_{skat} = V_{drift} + ò B^{'}_{ln}(t) dt
(148)
t_{1}
On the basis of equation (148) the
following equation (149) can be defined:
O^{'}_{DS} = O^{'}  B^{'}_{ln} (149)
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 de fined by the following equation (150):
l^{'}(t) = S^{'}_{i}(t)  U^{'}_{b}(t) (150)
.
 36 
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 following equation can
now be developed:
i^{'}_{e} = l^{'} + i^{'}_{K}  y^{'}_{B}  y^{'}_{L}  H^{'}_{S,1} (152)
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)
(152) ^{}
^{ }
where i^{'}_{B,1}(t) is the information flow in
the form of loan documents with statement of amounts corresponding to the
cash flov 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 appliess:
n^{'}_{B}(t) = i^{'}_{B,1} + r^{'}_{B}
(153)
where n^{'}_{B}(t) is the firm's current
crediting to the bank account.
.
 37 
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) (154)
The payRent order flov y^{'}_{B,1}(t) involves a corresponding current
debi ting to the bank account in the form of y^{'}_{B,2}(t). The following equati on
therefore applies:
y^{'}_{B,2}(t) = y^{'}_{B,1}(t)
(155)
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)
(156)
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)
(157)
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)
(158)
where i^{'}_{L,D}(t) denotes the long term
financing flow to the working ca pital, and i^{'}_{L,1}(t) is the long term financing
flow to the fixed capi tal.
.
 38 
The following equation applies:
i'_{K}(t) = i^{'}_{B}(t) + i^{'}_{L,D}(t) (159)
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) (160)
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 equati on
therefore applies:
y^{'}_{L,2}(t) = y^{'}_{L,1}(t)
(161)
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) (162)
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 defi
nition of i^{'}_{e}(t) as it depends on the financing and market situation.
Roughly speaking, i^{'}_{e}(t) is the average cash flow
which can be with drawn from the business without changing the existing
product, invest nent and financing structure and the necessary financial
reserves set aside for an appropriate future development of the businees.
.
 39 
7.1. Depreciation (for tax purposes)
It is normal to distinguish between
depreciation for tax purposes and depreciation for accounting purposes.
Depreciation for accounting pur poses is used with the object of comparing
alternative projects on the basis of special cost principles, which, for
example, are mentioned in connection with equation (4). These principles
are pure ly OR mathe matical 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) (163)
t
D(t) = ò (i^{'}_{1}(t)  d^{'}_{1}(t))dt (164)
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 cur rent crediting
to the same account (i.e. current "depreciation").
D(t) represents the balance of the tax
depreciation account. The de preciation charges d^{'}(t) are calculated on the
basis of this account, and the following expressions apply:
d^{'}_{1}(t) = d^{'}(t) (163a)
d^{'}_{2}(t) = d^{'}(t) (164a)
where d^{'}_{2}(t) is the depreciation flow
which is inciuded in the basis of computation of the taxable income.
.
 40 
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 compu tation purposes such as the effect of interest on
the income statement as a whole or in connection with special computations.
No distinction will be made here between
the two purposes. The inter est charges will be placed in this model with
the sole aim of depic ting the fundamental financial characteristics.
The following equations are defined:
r^{'}_{B,1}(t) = r^{'}_{B}(t)
(165)
r^{'}_{L,1}(t) = r^{'}_{L}(t)
(166)
r^{'}_{BL}(t) = r^{'}_{B,1}(t) + r^{'}_{L,1}(t)
(167)
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)
(168)
H^{'}_{S}(t) = s (O^{'}_{DS}(t)  f^{'}_{u}(t))
(169)
H^{'}_{S,1}(t) = H^{'}_{S}(t)
(169a)
.
 41 
where f^{'}_{u}(t) is a state function for
the computation of tax, cf. equa tion (168), 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 important to the understanding of the
dynamic (tactical) characte ristics 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 pos sible via Fig. 2.1 to assess the
effects on the operating profit. A reduction of the raw materials
inventories in a situation with raw ma terials prices which are higher
than the prices of the raw materials inventories but othervise constant
will increase the profit temporari ly in the period concerned.
One of the things that will be seen is
that the profit O^{'}(t) is inde pendent of the volume of trade accounts payable
and the volume of tra de accounts receivable.
 42 
10.1.2.
Change in liquidity l^{'}(t)
Other things being equal, the following
expression, cf. Fig. 2.1., ap plies:
d S^{'}_{u}
¾¾¾¾ > 0 Þ l^{'}(t) < O^{'}(t) (170)
d t
Equation (170) shows that the profit O^{'}(t) is larger than the change
in
liquidity in the case of growing sales in
the firm, the reason being the funds tied up, calculated with signs, in principal
assets (ac
counts receivable and inventories),
d S^{'}_{u}
¾¾¾¾
= 0 Þ l^{'}(t) = O^{'}(t) (171)
d t
Equation (171) shows that the change in
liquidity is equal to the profit in the case of constant sales, the reason
being an unchanged volu me of principal assets (accounts payable, accounts
receivable and in ventories).
d S^{'}_{u}
¾¾¾¾
< 0 Þ l^{'}(t) > O^{'}(t) (172)
d t
From equation (172) 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 circum stances 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 (170) and (172) must be reversed.
.
 43 
10.1.3.
Working capital K(t).
If the working capital is denoted K(t),
the definition eguation for net capital tied up in the operating system
will apply:
K(t) = V_{deb}(t) + F_{L}(t) + R_{L}(t)  V_{kre}(t) (173)
The following definition equation will
also apply:
d K(t)
¾¾¾¾
+ l^{'}(t) = O^{'}(t) (174)
d t
Equation (174) shows that the profit is
equal to the change in liqui dity + the increment of the net working capital
tied up.
If equation (174) is transformed, the
following equation is derived:
d K(t)
¾¾¾¾
= O^{'}(t)  l^{'}(t) (175)
d t
Equation (175) denotes that the
difference between the operating pro fit 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 (176):
DG(t) = (O^{'}(t) + F^{'}_{b}(t))/S^{'}_{u,1}(t) (176)
Equation (176) 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.
.
 44 
10.1.5. Depreciation
Depreciation contributes to influencing the
firm's liquidity, cf. e quation (169). Assuming that the investments are
made as individual projects at time intervals, it is shown that
depreciation in the peri ods between investments causes liquidity to rise
owing to the reducti on 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 repay ment commitments.
In chapter C 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 (168)
and (169) is apparent that in terest payments reduce the cash flow
released after tax. Thus, the net effect on cash flow released (to be defined in
charter C) stems partly from the computation of income for tax purposes,
partly from the pay ment 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.
.
C H A P T E R
C
.
 45 
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 process. No reference can be
made to a similar model in exist ing literature. The theoretical
literature which gets nearest is S. Eilon's article discussed in Chapter A
in the thesis.
11.1.
Physical and financial functions in the operating system
In the following further definitions of
mathematical functions and their relationships will be established. The
sole justification of these defi nitions is that they provide the basis of
a clear and generally coherent system of equations between ratios.
11.1.1.
Sales
A basic sales volume is defined:
S^{'}_{u0} = S^{'}_{u}(0) (177)
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 = a_{s} S^{'}_{u0} (178)
d t
where a_{s} is constant.
.
 46 
The following equation now applies:
S^{'}_{u}(t) = S^{'}_{u0}(1 + a_{s} t) (179)
where t ³ 0
11.1.2.
Inventories.
Let a ratio h_{F} be defined so that equation
(180) applies:
h_{F} = F_{L}(t)/S^{'}_{u}(t)
(180)
for t ³ 0, h_{F} being a positive constant
which is designated "finished goods inventory time". Another ratio
h_{R} is defined so that equation
(181) applies:
h_{R} = R_{L}(t)/S^{'}_{u}(t)
(181)
for t ³ 0 being a positive constant which is designated "raw
materi als inventory time".
From equation (180) follows:
F_{L}(t) = h_{F} S^{'}_{u}(t)
(182)
The definition equation applies:
t
F_{L}(t) = F_{F}(0) + ò Z^{'}_{i}(t) dt (183)
0
which substituted into equation (182) gives:
.
 47 
t
F_{L}(0) + ò Z^{'}_{i}(t) dt = h_{F} S^{'}_{u}(t) (184)
0
or
t
ò Z^{'}_{i}(t) dt = h_{F} S^{'}_{u}(t)  F_{L}(0) (185)
0
If equation (179) is used in equation (185),
the following expression is derived:
t
ò Z^{'}_{i}(t) dt = h_{F} S^{'}_{u0} a_{s} t + (h_{F} S^{'}_{u0}  F_{L}(0)) (186)
0
For t = 0 equation (180) gives the
following expression:
h_{F} = F_{L}(0)/S^{'}_{u}(0) (187)
Using equation (187) together with
equation (186) we have:
t
ò Z^{'}_{i}(t) dt = h_{F} S^{'}_{u0} a_{s} t (188)
0
The solution to the integral equation
(188) is:
Z^{'}_{i}(t) = h_{F} S^{'}_{u0} a_{s} (189)
The flow of goods Z^{'}_{i}(t) to the finished goods
inventory may then be defined by equations (190) and (191):
Z^{'}_{i}(t) = h_{F} S^{'}_{u0} a_{s} (190)
for a_{s} ³ 0
and
Z^{'}_{u}(t) =  h_{F} S^{'}_{u0} a_{s} (191)
for a_{s} < 0
.
 48 
Mathematically the physical equations
(190) and (197) may be described by equation (192) for all values of a_{s}, i.e.
Z^{'}_{i}(t) = h_{F} S^{'}_{u0}^{ }a_{s} (192)
for
 ¥ < a_{s} < ¥
With equation (192) 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 (189).
From equation (181) follows:
R_{L}(t) = h_{R} S^{'}_{u}(t) (193)
The definition equation applies:
t
R_{L}(t) = R_{L}(0) + ò Q^{'}_{i}(t) dt (194)
0
which combined with equation (193) gives:
t
R_{L}(0) + ò Q^{'}_{i}(t) dt = h_{R} S^{'}_{u}(t) (195)
0
or
t
ò Q^{'}_{i}(t) dt = h_{R} S^{'}_{u}(t)  R_{L}(0) (196)
0
If equation (179) is used in equation
(196), the following equation is
derived:
t
ò Q^{'}_{i}(t) dt = h_{R} S^{'}_{u0} a_{s} t + (h_{R} S^{'}_{u0}  R_{L}(0)) (197)
0
For t = 0 equation (181) gives:
.
 49 
h_{R} = R_{L}(0)/S^{'}_{u}(0) (198)
Using equation (197) together with
equation (198) we have:
t
ò Q^{'}_{i}(t) dt = h_{R} S^{'}_{u0} a_{s} t (199)
0
The solution to the integral equation
(199) is:
Q^{'}_{i}(t) = h_{R} S^{'}_{u0} a_{s}
(200)
The flow of goods Q^{'}_{i}(t) to the raw materials inventory
can now be de fined by equations (201) and (202):
Q^{'}_{i}(t) = h_{R} S^{'}_{u0} a_{s}
(201)
for a_{s} ³ 0
Q^{'}_{u}(t) =  h_{R} S^{'}_{u0} a_{s} (202)
for a_{s} < 0
Mathematically the physically equations
(201) and (202) can be descri
bed by equation (203) for all values of a_{s}, i.e.
Q^{'}_{i}(t) = h_{R} S^{'}_{u0} a_{s}
(203)
for
 ¥ < a_{s} < ¥
With equation (203) the physical
inventory system has been converted to a mathematica1 model where Q^{'}_{i}(t) can change sign and where
Q^{'}_{u}(t) = 0 for all t.
.
 50 
11.1.3.
Output
Total output T^{'}_{p}(t) is given by:
T^{'}_{p}(t) = S^{'}_{u}(t) + Z^{'}_{i}(t) (204)
If the ratio b_{a} is here defined as the number
of labor hours used per unit of output and the ratio b_{R} as the raw materials
consumption per unit of finished goods, the equations, resource balance
equations, will apply:
a^{'}_{i}(t) = b_{a} T^{'}_{p}(t) (205)
V^{'}_{i}(t) = b_{R} T^{'}_{p}(t) + Q^{'}_{i}(t) b_{R}
(206)
If equation (204) and equation (206) are
combined, the following equa tion is obtained:
V^{'}_{i}(t) = b_{R} S^{'}_{u}(t) + b_{R} Z^{'}_{i}(t) + Q^{'}_{i}(t) b_{R} (207)
If equations (179), (192) and (203) are
substituted into equation (207), the following equation is obtained:
V^{'}_{i}(t) = b_{R} S^{'}_{u0} (1 + (h_{F} + h_{R} + t) a_{s}) (208)
Using equations (204), (179) and (192),
equation (205) gives:
a^{'}_{i}(t) = b_{a} S^{'}_{u0} (1 + a_{s}(t + h_{F})) (208a)
11.1.4. Sales, ingoing payments
Using equations (95), (96) and (97) we
obtain payments derived from cash sales:
S^{'}_{i,kon}^{'}(t) = p q S^{'}_{u0} (1 + a_{s} t) (209)
.
 51 
Using equations (88), (95), (98), (99)
and (100) we obtain payments derived from debit sales:
S^{'}_{i,deb}(t) = p (1  q) S^{'}_{u0} (1 + a_{s}(t  d_{D})) (210)
Equations (88), (209) and (210) give:
S^{'}_{i}(t) = p q S^{'}_{u0} (1 + a_{s} t) + p (1  q) S^{'}_{u0}
(1 + a_{s}(t  d_{D})) (211)
or
S^{'}_{i}(t) = p S^{'}_{u0} (1 + a_{s}(t  d_{D}(1  q))) (212)
11.1.5.
Purchases, outgoing payments
The outgoing payments flow corresponding
to cash purchases of raw ma terials is expressed by means of equations
(101), (102), (103) and (208) as
V^{'}_{b,kon}(t) = q_{1} e b_{R} S^{'}_{u0} (1 + (h_{F} + h_{R} + t)a_{s}) (213)
Credit purchases of raw materials cause
an outgoing payments flow which by means of equations (94) (101), (106),
(107) and (108) is computed at:
V^{'}_{b}(t) = e q_{1} V^{'}_{i}(t) + (1  e)q_{1} V^{'}_{i} (t  d_{K}) (214)
Equation (214) is transformed by means of
equation (208) into:
V^{'}_{b}(t) = e q_{1} b_{R} S^{'}_{u0}(1 + (h_{F} + h_{R} + t) a_{s}) +
(1  e)q_{1} b_{R} S^{'}_{u0}(1 + (h_{F} + h_{R} + t  d_{K})a_{s}) (215)
Equation (215) is reduced to:
V^{'}_{b}(t) = q_{1} b_{R} S^{'}_{u0}(1 + a_{s}(h_{F} + h_{R} + t  d_{K} (1  e))) (216)
.
 52 
The total payments flow to purchases of
resources is then obtained by using equations (93), (104), (105) and (216):
U^{'}_{b}(t) = q_{2} a^{'}_{i}(t) + q_{1} b_{R} S^{'}_{u0}(1 + a_{s} (h_{F} + h_{R} + t  d_{K}
(1  e))) + F^{'}_{b}(t) (217)
By substituting equation (208a) into
equation (217) the total outgoing payments flow is then given by:
U^{'}_{b}(t) = S^{'}_{u0}(q_{2} b_{a} (1 + a_{s} (t + h_{F})) + q_{1} b_{R}(1 + a_{s}
(h_{F} + h_{R} + t  d_{K}(1  e))) + F^{'}_{b}(t) (218)
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 (150), (212)
and (218) be given the following form:
l^{'}(t) = S^{'}_{u0}(p(1 + a_{s}(t  d_{D}(1  q)))
 q_{2} b_{a} (1 + a_{s}(t + h_{F}))  q_{1} b_{R}(1 + a_{s}
(h_{F} + h_{R} + t  d_{K}(1  e))))  F^{'}_{b}(t) (219)
11.2.
Capital tied up in the operating system
Depending on the firm's level of activity
capital will be tied up in the operating system. Capital will be tied up in
trade accounts payable, raw materials 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 (88), (95), (98) and (99)
being used:
d_{D}
V_{deb}(t) = ò p(1  q)S^{'}_{u}(t  x) dx (220)
0
.
 53 
In this model it is assumed that equation
(179) applies. From this equation combined with (220) follows:
d_{D}
V_{deb}(t) = p(1  q) S^{'}_{u0} ò (1  a_{s}(t  x))
dx (221)
0
The computation of the integral in
equation (221) allows equation (221) to be reduced to:
V_{deb}(t) = p(1  q) S^{'}_{u0} d_{D} (1 + a_{s}(t  0.5 d_{D})) (222)
11.2.2.
Trade accounts payable
The volume of trade accounts payable is
defined by the following equation, equations (89), (101), (106) and (107)
being used:
d_{K}
V_{kre}(t) = ò q_{1} b_{R}(1  e) V^{'}_{i}(t  x)) dx (223)
0
Assuming that sales satisfy equation
(179) and that equation (208) applies, equation (223) develops the
following expression:
d_{K}
V_{kre}(t) = q_{1} b_{R}(1  e) S^{'}_{u0} ò (1 + a_{s}(h_{F} + h_{R} + t  x)) dx (224)
0
By computing the integral in equation
(224) this equation is reduced to:
V_{kre}(t) = q_{1} b_{R}(1  e) S^{'}_{u0} d_{K} (1 + a_{s}(h_{F} + h_{R} + t  0.5 d_{K})) (225)
l1.2.3
Raw materials inventory
The volume of the raw materials inventory
is given by equation (193). The value of the raw materials inventory R_{L,1}(t) satisfies the equation:
R_{L,1}(t) = q_{1} b_{R} R_{L}(t)
(226)
.
 54 
If equations (193) and (179) are
substituted into equation (226), we have:
R_{L,1}(t) = q_{1} b_{R} h_{R} S^{'}_{u0}(1 + a_{s} t) (227)
11.2.4.
Finished goods inventory
The volume of the finished goods
inventory is given by equation (182). The calculated consumption of
materials and labor hours per unit of finished goods is given by q_{F}, cf. equation (124). The
definition equa tion applies:
q_{F} = b_{R} q_{1} + b_{a} q_{2} (228)
The value of the finished goods inventory
F_{L,1}(t) satisfies the equati on:
F_{L,1}(t) = q_{F} F_{L}(t) (229)
If equations (182), (179) and (228) are
substituted into equation (229), the following expression is obtained:
F_{L,1}(t) = (b_{R} q_{1} + b_{a} q_{2}) h_{F} S^{'}_{u0}(1 + a_{s} t) (230)
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 (222), (225), (227) and (230) given by:
K(t) = V_{deb}(t)  V_{kre}(t) + R_{L,1}(t) + F_{L,1}(t) (231)
or by substituting into the relevant
places
K(t) = p(1  q)S^{'}_{u0} d_{D} (1 + a_{s}(t  0.5 d_{D}))
 q_{1} b_{R} (1 + e)S^{'}_{u0} d_{K} (1 + a_{s}(h_{F} + h_{R} + t  0.5 d_{K}))
+ q_{1} b_{R} h_{R} S^{'}_{u0}(1 + a_{s} t)
+ (b_{R} q_{1} + b_{a} q_{2}) h_{F} S^{'}_{u0}(1 + a_{s} t)
or
.
 55 
K(t) = S^{'}_{u0}(1 + a_{s} t)(h_{F} (b_{R} q_{1} + b_{a} q_{2}) + q_{1} b_{R} h_{R})
+ p(1  q) d_{D} S^{'}_{u0}(1 + a_{s}(t  0.5 d_{D}))
 q_{1}
b_{R} (1  e) d_{K} S^{'}_{u0}(1 + a_{s}(h_{F} + h_{R} + t  0.5 d_{K})) (232)
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 (96), (98), (111), (112) and (113) and is expressed as:
S^{'}_{u,1}(t) = p S^{'}_{u}(t)
(233)
Using equation (233) equation (179)
gives:
S^{'}_{u,1}(t) = p S^{'}_{u0}(1 + a_{s} t) (234)
Raw materials consumed corresponding to
sales S^{'}_{u}(t) are given by the equation:
V^{'}_{for}(t) = q_{1} b_{R} S^{'}_{u}(t)
(235)
or by using equation (179):
V^{'}_{for}(t) = q_{1} b_{R} S^{'}_{u0}(1 + a_{s} t) (236)
The wages paid, time rates, corresponding
to sales S^{'}_{u}(t) are given by the equation:
a^{'}_{for}(t) = q_{2} b_{a} S^{'}_{u}(t)
(237)
or by using equation (179)
a^{'}_{for}(t) = q_{2} b_{a} S^{'}_{u0}(1 + a_{s} t) (238)
.
 56 
By using equations (234), (236) and (238)
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) (239)
or
O^{'}(t) = S^{'}_{u0}(1 + a_{s} t)(p  (q_{1} b_{R} + q_{2} b_{a}))  F^{'}_{b} (240)
12.2.
Operating profit (computed on the basis of Fig. 1.1)
In this section the operation profit will
as an alternative be compu ted directly on the basis of Fig. 2.1.
The costs U^{'}_{d}(t) in connection with sales S^{'}_{u}(t) are given by equation
(118). If equations (89), (102), (104), (106), (109), (110), (114), (115),
(116), (117), (208) and (208a) are substituted into equation (118), the
following equation is developed:
U^{'}_{d}(t) = q_{1} b_{R} S^{'}_{u0}(1 + (h_{F} + h_{R} + t) a_{s})
+ q_{2}
b_{a} S^{'}_{u0}(1 + (h_{F} + t)a_{s}) + F^{'}_{b} (241)
Computed with a plus or minus sign
(positive for inventory) the fol lowing value is added to the raw materials
inventory, cf. equation (35):
Q^{'}_{i,1}(t) = q_{1} b_{R} Q^{'}_{i}(t)
(242)
or equation (203) may be used:
Q^{'}_{i,1}(t) = q_{1} b_{R} h_{R} S^{'}_{u0} a_{s} (243)
Here the definition equation for cost
prices of raw materials per unit of finished goods has been used:
.
 57 
q_{R} = q_{1} b_{R} (244)
The following value is added to the
finished goods inventory, cf. equa tion (124):
Z^{'}_{i,1}(t) = q_{F} Z^{'}_{i}(t) (245)
or equation (189) may be used:
Z^{'}_{i,1}(t) = (q_{1} b_{R} + q_{2} b_{a}) h_{F} S^{'}_{u0} a_{s} (246)
The total value flow to inventories now amounts
to, cf. equations (131) and (132):
U^{'}_{tl}(t) = q_{1} b_{R} h_{R} S^{'}_{u0} a_{s} + (q_{1} b_{R} + q_{2} b_{a}) h_{F} S^{'}_{u0} a_{s} (247)
or by reduction
U^{'}_{tl}(t) = S^{'}_{u0} a_{s}(q_{1} b_{R} h_{R} + (q_{1} b_{R} + q_{2} b_{a}) h_{F}) (248)
The total operating profit is obtained by
using equations (234), (241) and (248) and is expressed as:
O^{'}(t) = S^{'}_{u}(t)  (U^{'}_{d}(t)  U^{'}_{tl})) (249)
or by substituting into the right hand
side:
O^{'}(t) = p S^{'}_{u0}(1 + a_{s} t)
 (q_{1} b_{R} S^{'}_{u0}(1 + (h_{F} + h_{R} + t) a_{s})
+ q_{2}
b_{a} S^{'}_{u0}(1 + (h_{F} + t) a_{s}) + F^{'}_{b}
 S^{'}_{u0} a_{s} (q_{1} b_{R} h_{R} + (q_{1} b_{R} + q_{2} b_{a}) h_{F})) (250)
or by reduction:
.
 58 
O^{'}(t) = S^{'}_{u0}(1 + a_{s} t)(p (q_{1} b_{R} + q_{2} b_{a}))  F^{'}_{b} (251)
It will be seen that equations (240) and
(251) are identical, i.e. a systematic use of Fig. 2.1. gives here the same
result as the use of a simple "logical" accounting method.
12.2.1.
Operating profit incl. inventory depreciation
If equations (248) and (145) are
substituted into equation (147), U^{'}_{tl,1}(t) being computed with a plus
or minus sign, the following equation is obtained:
B^{'}_{ln}(t) = a_{n} S^{'}_{u0} a_{s }(q_{1} b_{R} h_{R} + (q_{1} b_{R} + q_{2} b_{a}) h_{F}) (252)
The operating profit incl. inventory
depreciation is given by equation (149). If equations (251) and (252) are
substituted into this equation, the following expression is derived:
O^{'}_{DS}(t) = S^{'}_{u0}(1 + a_{s} t)(p (q_{1} b_{R} + q_{2} b_{a}))  F^{'}_{b}
 a_{n} S^{'}_{u0} a_{s }(q_{1} b_{R} h_{R} + (q_{1} b_{R} + q_{2} b_{a}) h_{F}) (253)
or by reduction:
O^{'}_{DS}(t) = S^{'}_{u0}((1 + a_{s} t)(p (q_{1} b_{R} + q_{2} b_{a}))
 a_{n} a_{s }(q_{1} b_{R} h_{R} + (q_{1} b_{R} + q_{2} b_{a}) h_{F})
 F^{'}_{b} (253a)
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 59 
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) = K_{0} + K_{inc}(t)
(254)
where K_{0} is the net working capital
tied up at time t = 0, and K_{inc}(t) is the change in the working capital
tied up at time t. It is assumed that equation (255) applies:
d K_{inc}(t)
¾¾¾¾¾
= i_{B}(t)
(255)
d t
This means that the increase in the
capital tied up in the operating system is financed by the bank overdraft.
If equation (255) is used together with
equation (254), the following equation will also apply:
d K(t)
¾¾¾¾¾ = i_{B}(t) (255a)
d
t
It is assumed that:
B(0) = 0 (256)
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 (255) i^{'}_{B}(t) may be both positive and
negative as it is also assumed here that, besides equations such as (152),
(153), (154) and (155), the following equation applies:
y^{'}_{B}(t) = r^{'}_{B}(t) (257)
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 60 
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
(258)
for all t > 0, apart from certain
selected times tq where, in practice, changes take place
in financing conditions, and new investments a re made. Subject to these
assumptions equation (159) may be reduced to
i^{'}_{K}(t) = i^{'}_{B}(t) (259)
with the condition i^{'}_{L,D}(t) = 0
In close connection with the operational financial
possibilities of equations (258) and (259) this model also assumes that
equation (260) applies:
y^{'}_{L}(t) = r^{'}_{L}(t) (260)
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 (162)
is changed into:
i^{'}(t) = i^{'}_{L,i}(t) (261)
where i^{'}_{e}(t) becomes the quantity, cash
flow released, for the following purposes:
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 61 
New investments
Instalments on loans
Etc.
This change of equation (162) is
desirable seen in relation to the possibilities of implementing this
mathematical model on a computer.
12.4.1.
Interest payments
From equations (165), (166), (167), (257)
and (260) the total interest payment is derived:
y^{'}_{B}(t) + y^{'}_{L}(t) = r_{B} B(t) + r_{L} L(t) (268)
where r_{B} is interest rate bank and r_{L} is interest rate lender.
12.4.2. Depreciation
Depreciation to tax computation is
obtained from equation (164) and is expressed as:
d^{'}_{2}(t) = a_{D} D(t) (263)
where a_{D} is the depreciation rate per
time period.
12.4.3.
Tax payments
From equations (168), (262) and (263) the
following equation is deri ved:
f^{'}_{u}(t) = a_{D} D(t) + r_{B} B(t) + r_{L} L(t) (264)
By using equations (169) and (264) total
tax payments are expressed as:
H^{'}_{S,1}(t) = s(O^{'}_{DS}(t)  a_{D} D(t)  (r_{B} B(t) + r_{L} L(t))) (265)
.
 62 
12.4.4. Cashflow released
With the special definition of i^{'}_{e}(t) given in 12.3.3. cash flow
relea sed is defined by:
i^{'}_{e}(t) = O^{'}(t)  H^{'}_{S,1}(t)  (y^{'}_{B}(t) + y^{'}_{L}(t))
which together with equation (175) gives:
i^{'}_{e}(t) = l^{'}(t) + dK(t)/dt  H^{'}_{S,1}(t)  (y^{'}_{B}(t) + y^{'}_{L}(t))
If equation (255a) including the related
assumption is used here, the following equation is obtained:
i^{'}_{e}(t) = l^{'}(t)  H^{'}_{S,1}(t)  (y^{'}_{B}(t) + y^{'}_{L}(t)) + i^{'}_{B}(t)
(266)
or if equations (262) and (265) are used:
i^{'}_{e}(t) = l^{'}(t) + i^{'}_{B}(t)  s O^{'}_{DS}(t) + s a_{D} D(t)
 (1  s)(r_{B} B(t) + r_{L} L(t)) (267)
By using equations (149) and (174), the
following equation is derived from equation (267):
i^{'}_{e}(t) = O^{'}(t)  s O^{'}_{DS}(t) + s a_{D} D(t)
 (1  s)(r_{B} B(t) + r_{L} L(t)) (268)
If the function O^{'}_{L}(t) is defined by the
equation:
O^{'}_{L}(t) = O^{'}(t)  s O^{'}_{DS}(t) (269)
O^{'}_{L}(t)^{ }may be
designated as the profit after tax from the operating system.
Equation (268) is now transformed into:
i^{'}_{e}(t) = O^{'}_{L}(t)(1 + s a_{D} D(t)/O^{'}_{L}(t)
 (1  s)(r_{B} B(t) + r_{L} L(t))/O^{'}_{L}(t) (270)
It appears from equation (270) that it
may be appropriate to define the following managerial ratios:
