FEATURES of FINANCING
of INNOVATION PROJECTS
Ivanov N.V. ([email protected]), Kirsanov A.P.,
Shomova E.N. National Research University Higher School of Economics, Moscow, Russia
INNOVATION PROJECTS
 We shall understand as innovation projects such as
a result of which performance the product turns out, or
the service, not having analogues in the world, or
created with application of bursting, unknown before
methods and technologies, and, hence, including in
their development a significant share of research
works (R&D)[2-5].
 Research stages of the project frequently consist in
checking of applicability of some ideas, hypotheses,
materials, methods for achievement of the purposes of
the project [2,3]. Checks are carried out with use of
experiments, modelling, testing and other methods of
research [4,5].
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Results of researches
Results of researches can be the following:
 applicability of the objectively true hypothesis
is confirmed;
 applicability of the objectively false hypothesis
is denied;
 applicability of the objectively false hypothesis
is confirmed;
 applicability of the objectively true hypothesis is
denied.
3
Designations
Further, for definiteness,like in [1], we shall consider,
that at a research stage some hypothesis is checked. If a
hypothesis objectively true then we shall carry it to a class
1, otherwise to a class 2. We shall designate through P1 a
priori probability of the validity of a hypothesis checked
at a research stage. Then P2=1-P1 is probability of
falseness of the hypothesis. We shall designate through
p11(r), p22(r), p21(r), p12(r) a posteriori probabilities of
reception of the results of a research stage described
above at the volume of financing r, and through
c11,c22,c21,c12 the appropriate additional charges at
later stages of the project.
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Basic equation
 It is natural, that charges are higher at erroneous
conclusions about the validity of the checked
hypothesis, made on a research stage, that is c12>c11,
c21>c22 .
 The mathematical expectation of the charges,
connected to results of a research stage is described by
expression:
C(r) =
  P p (r)c
i
ij
ij + r
(1)
j=1,2 i=1,2
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Dependence on financing r
 Clearly, that with increase of a research stage
financing r the volume of researches and correctness of
received results grows. Differently, at increase r monotonously grow and come nearer to 1 probabilities p11(r)
and p22(r) of reception of correct results at a research
stage and simultaneously probabilities p12(r)=1-p11(r)
and p21(r)=1-p22(r) of reception of erroneous conclusions
decrease.
 At absence of financing correctness of results of
researches is reduced up to levels 1 and 2 which were
achieved prior to the beginning of the researches, i.e.
p11(0) = 1, p22(0) = 2.
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Function C(r)
 Expression (1) can be written down as follows:
C(r)  P1c12  P2c21  r  [P1 p11(r)(c12  c11)  P2 p22 (r)(c21  c22 )]
 From this representation it is visible, that function C(r)
represents a difference of linear growing function
P1c12+P2c21+r and monotonously growing function
P1p11(r)(c12-c11)+P2p22(r)(c22-c21) limited from above.
Two variants of behaviour of function C(r) are possible.
Either C(0) is the minimal value of function C(r) at r ≥0 ,
or the minimum is reached at r0>0 for which C(r0)< C(0)
is the minimal value of function.
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Optimization of financing ropt
For a finding of optimum volume of financing ropt we
shall equate a derivative of expression (1) to zero and
we shall receive the equation:
P1 p11 (r)(c12  c11)  P2 p22 (r)(c21  c22 )  1
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Existence of a minimum of C(r)
Previous equation is true when
 (r)(с12 - с11 ), p22 (r)(с21 - с22 ) > 1
min p11
In figure 1 two possible variants of behaviour of function C(r) are
represented: existence of a minimum at
dC(r)
dr
(0)> 0
(c11=1, c22=1, c21=8, c12=20) and its
absence
dC(r)
dr
(0)< 0
(c11=2, c22=8, c21=2, c12=10).
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Figure 1. Mathematical expectation of the
charges connected to results of a research stage.
11,5
10,5
9,5
8,5
C'(r)>0
C'(r)<0
7,5
6,5
5,5
4,5
3,5
0
0,3 0,6 0,9 1,2 1,5 1,8 2,1 2,4 2,7
3
3,3 3,6 3,9 4,2 4,5 4,8
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Two variants of an innovational stage
It is necessary to note two variants of development of an
innovational stage of the project. The first variant assumes
progressive motion aside achievements of necessary
results. Gradually, in process of achievement of the concrete
scientific purposes, additional financial injections do not
bring necessary equivalent results. It is explained by wellknown practice, that for 20 % of time 80 % of work is made,
and in the rest of 80 % of time - is completed 20 % of work.
For this case typically following behaviour of probabilities pij
(r), i, j=1,2, having property of monotonous decrease of a
derivative:
 ii r
pii (r )  1   i e
pij (r )   i e
 ij r
i=1, 2
i=1,2, j=1,2, i≠j
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Figure 2. Dependence pij on financing r. Var1.
1,1
1
0,9
0,8
0,7
P11
0,6
P22
0,5
P12
0,4
P21
0,3
0,2
0,1
0
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
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Var 2 of an innovational stage
By analogy to the first case work start, having initial probabilities pij,
but process of reception of necessary results passes in another way. At
the initial stages of work, having enclosed in the project of means for
purchase of the necessary equipment (measuring), on development of
an infrastructure, on escalating of potential of the project and so forth,
the developer does not come nearer to the purposes of the project.
And only on passing time and after passage of this "preparatory" stage
"the work itself" begins (the diagram sharply goes upwards; probabilities
P11, P22 tends to 1; Figure 3), giving result. Thus the equations
describing behaviour of probabilities, look as logistic curve:
pii (0)
pii (r ) 
pii (0)  (1  pii (0))e r
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Figure 3. Dependence pij on financing r. Var2.
1,2
1
0,8
P11
0,6
P22
P12
0,4
P21
0,2
0
0
0,5
1
1,5
2
2,5
3
3,5
4
4,5
5
-0,2
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Conclusion
Conditions of existence of optimum volume of
financing ropt , minimizing the general expenses on
performance of the project C(r), are found. The
special cases appropriate to the most natural kinds of
dependences of probabilities pij(r) from volume of
financing r of research stages of the innovational
project are considered.
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References:
1. Ackoff R.L., Sasieni M.W. Fundamentals of operations
research, N.Y., Wiley, 1968, p.455
2. Bergemann D. and Hege U. 2005. “The Financing of
Innovation: Learning and Stopping,”Rand Journal of
Economics, The RAND Corporation, vol. 36(4), pages 719752, Winter.
3. Hall B. H. and J. van Reenen. 2000. “How Effective are Fiscal
Incentives for R&D? A New Review of the Evidence,”
Research Policy 29: 449-469.
4. Kortum S. and Lerner J. 2000. “Assessing the Contribution
of Venture Capital to Innovation,” Rand Journal of
Economics 31(4): 674-92.
5. Scherer F. M. 1998. “The Size Distribution of Profits from
Innovation,” Annales d'Economie et de Statistique 49/50:
495-516.
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