FEATURES of FINANCING

of INNOVATION PROJECTS

Ivanov N.V. (ivanick@mail.ru), Kirsanov A.P., Shomova E.N. National Research University - Higher School of Economics, Moscow, Russia

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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.

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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:

i ij ijj=1,2 i=1,2

C(r) = Pp (r)c + r (1)

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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 monoto-nously 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:

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.

)])(())(([C(r) 22212221112111212121 ccrpPccrpPrcPcP

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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:

1))(())(( 22212221112111 ccrpPccrpP

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Existence of a minimum of C(r)

Previous equation is true when

In figure 1 two possible variants of behaviour of function C(r) are represented: existence of a minimum at

(c11=1, c22=1, c21=8, c12=20) and its

absence

(c11=2, c22=8, c21=2, c12=10).

11 12 11 22 21 22min p (r)(с - с ), p (r)(с - с ) > 1

dC(r) 0

dr

>

dC(r) 0

dr

<

10

Figure 1. Mathematical expectation of the charges connected to results of a research stage.

3,5

4,5

5,5

6,5

7,5

8,5

9,5

10,5

11,5

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

C'(r)>0

C'(r)<0

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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 well-known 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:

riij

ijerp )( i=1,2, j=1,2, i≠j

i=1, 2 r

iiiiierp 1)(

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Figure 2. Dependence pij on financing r. Var1.

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

1,1

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

P11

P22

P12

P21

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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:

riiii

iiii epp

prp

))0(1()0(

)0()(

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Figure 3. Dependence pij on financing r. Var2.

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

0 0,5 1 1,5 2 2,5 3 3,5 4 4,5 5

P11

P22

P12

P21

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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 719-752, 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.