Venture capital financed investments in intellectual capital

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Journal of Economic Dynamics & Control 30 (2006) 23392361Venture capital nanced investmentspartially nanced by venture capital. Among the questions that the paper addresses are: howARTICLE IN PRESSwww.elsevier.com/locate/jedcCorresponding author. Department of Econometrics & Operations Research and CentER, TilburgUniversity, P.O. Box 90153, 5000 LE Tilburg, The Netherlands. Tel.: +31 13 4662062;0165-1889/$ - see front matter r 2005 Elsevier B.V. All rights reserved.doi:10.1016/j.jedc.2005.07.005fax: +3113 4663280.E-mail address: kort@uvt.nl (P.M. Kort).much venture capital should be acquired to help nance the development of the rm? Howshould a wish to grow, with the aim of making a breakthrough in the IT area, be balancedagainst the stockholders wish to consume? The problem is studied as an optimal controlproblem with a random time horizon and we derive a series of prescriptions for investmentand nancial decisions.r 2005 Elsevier B.V. All rights reserved.JEL classification: D92; G24; O32Keywords: Intellectual capital accumulation; Investment; Dividends; Venture capital; Stochastic optimalcontrolin intellectual capitalSteffen Jrgensena, Peter M. Kortb,c,, Engelbert J. DocknerdaDepartment of Business and Economics, University of Southern Denmark, Odense, DenmarkbDepartment of Econometrics & Operations Research and CentER, Tilburg University,Tilburg, The NetherlandscDepartment of Economics, University of Antwerp, Antwerp, BelgiumdDepartment of Finance and Vienna Graduate School of Finance, University of Vienna, Vienna, AustriaReceived 29 March 2004; accepted 11 July 2005Available online 29 September 2005AbstractThe paper considers a new company in the IT industry, founded by a management team andARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 2339236123401. IntroductionThe information technology (IT) industry has been characterized by a rapidgrowth of the number of new rms entering the industry. New product and serviceinnovations are being developed and marketed in increasing numbers. Growth hasbeen fast in software development, e-business, internet services, and web design, butalso the telecommunications industry has seen a rapid increase in new equipmentand services. Many new rms have experienced a fast growth in terms of the numberof employees, but the growth has necessarily not been nancially sound. Somecompanies have used substantial parts of their equity, and the venture capital theyattracted, to cover losses incurred by attempts to grow rapidly, and others headedfor bankruptcy due to severe operating losses. Quite a few businesses were liquidatedwithout having generated a single dollar of revenue.This paper sets up a model of investment and nancial decision making in a newrm, and derives a series of prescriptions for its growth and nancing. Suppose thatthe company is founded at time t 0, by acquiring another rm in the industry. Therm thus acquires a stock of intangible assets: specialized employee skills,knowledge, expertise, experience, and information. It has been argued that weare entering the knowledge society in which the basic economic resource is no longercapital. . . but is and will be knowledge (Drucker, 1995). Zingales (2000) positedthat during the last decade, knowledge has replaced capital stocks as the main assetof a rm. Supposing knowledge can be capitalized, we summarize the rmsintangible assets in a stock of intellectual capital (IC) (see also McAdam andMcCreedy, 1999).We consider a particular objective of the rm, being the development of a specicinnovation (an IT product or service) which will make a breakthrough in its specicarea. The date of the innovation, T, is unknown at time zero where the rm mustmake a plan for its investments in IC. During the time period until T , makinginvestments in its stock of IC will increase the probability that the innovation ismade by time T. At the initial instant of time, the rm, represented by a team ofentrepreneurs who manage the rm, can attract venture capital. The purpose of thisis to increase the stock of IC that is acquired at the initial instant of time.The venture capitalist in our model plays no active role in the sense that sheneither is involved in the management of the rm nor does she choose the exit timestrategically. The venture capitalist in our model is a stockholder of the rm who hasa predetermined exit strategy. She either exits when the breakthrough is made andthe reward is divided up among stockholders, or she exits at a predetermined latestexit time where she receives a xed return on her invested capital.The paper deals with two main problems, essentially seen from the point of view ofthe entrepreneurs/founders. The rst problem concerns how much, if any, venturecapital should be attracted initially. The second problem is how to design a strategyfor the rms later investments in IC, in order to raise the probability of making thediscovery. Each of the two topics is very interesting on its own. Our analysis hereinterlinks both so that we can get some understanding of what the impact of venturecapital nancing on innovations and research and development is.ARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 23392361 2341Although the rst US-based venture capital rm was established shortly after thesecond world war, academic research started to focus on venture capital nancingonly 1015 years ago. Central to academic research is the simple setup where abusiness founder (entrepreneur) is looking for outside equity holders. Since such anancing-relationship is characterized by contracting under asymmetric informationmost of the theoretical research on venture capital nancing focusses either onadverse selection or moral hazard (see Amit et al. (2000) for a survey of VC-modelswith asymmetric information). The role of asymmetric information in venture capitalnancing is widely recognized and dominates theoretical research. Sahlman (1988,1990) argued that contracting practises in the venture capital industry reectuncertainties about future payoffs and informational asymmetries betweenentrepreneurs and venture capitalists. Moreover, the lack of operational history ofthe rms aggravates the adverse selection problem.Admati and Peiderer (1994) deal with the venture capital cycle and characterizethe contract that allows optimal continuation decisions with staged nancing (seealso Kaplan and Stromberg, 2003). Bergemann and Hege (1997) analyze a situationin which there is a link between moral hazard and gradual learning about the projectquality. Withholding effort (moral hazard) in such a setting not only puts the ventureat risk but causes the entrepreneur and the VC to learn differently about the quality(payoffs) of the project. They propose a dynamic sharing rule that mitigates theproblems. The dynamic aspect of their model is in line with our approach here.Most of the empirical literature on venture capital nancing is summarized inGompers (2005). He analyzes in detail the venture capital investment processincluding empirically observed exit strategies and the relationship between venturecapital nancing and innovation.In the area of research and development investments Schwartz and Zozaya-Gorostiza (2003) study a setup in which an organization undertakes an ITinvestment project, with a random completion date. The project involves an optionof spending initially an amount of money to acquire an IT asset. After theacquisition, the organization starts to receive a (random) cash ow and can invest inthe development of the asset. The methodology used to evaluate the project is realoptions, using stochastic differential equations and HamiltonJacobiBellmanequations. The nancial aspects are, however, ignored. Venture capital backedstart-ups are studied in a recent paper by Meng (2004). She uses a strategic realoptions model to analyze equilibrium investment strategies when rms are engagedin a patent race. Using numerical simulations she nds that strategically competingrms are likely to overinvest in equilibrium.The relationship between investment in human capital, innovation and competi-tion in the product market is studied in a paper by Fulghieri and Sevilir (2005). Theyformulate a model in which the incentives to acquire human capital, a necessarycondition to innovate, depends on the level of competition in the product market.Workers have a higher incentive to invest in human capital in a competitive (related)environment, because the relatedness of the industry gives them a higher exibility tomove from one rm to the other and hence greater ability to extract rents from theiremployer.2.2. FinancingARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 233923612342We suppose that after having obtained the venture capital L0, the rm can attractno more capital. Thus, the rm must be self-nanced for t40. This approach tonancing differs sharply from one in which a rm can attract external funds (equityand/or debt) at any instant of time (e.g., Schworm, 1980; Bensoussan and Lesourne,1981). On the other hand, Kamien and Schwartz (1982) argued that R&Dinvestments, which are in many respects comparable to investments in IC, shouldbe entirely self-nanced.It is also important to note that the assumed, one-shot venture capital nancingdiffers from the common practice in which ventures are nanced stagewise (see, e.g.,Finally there is a stream of literature in the area of R&D that studies stochastic,dynamic investment problems. Examples are Reinganum (1981, 1982), Kamien andSchwartz (1982), Fudenberg et al. (1983), Grossman and Shapiro (1986) andDoraszelski (2003). The idea here is to model the hazard rate of successfulinnovation as a function of the rms current and/or cumulative investments.The paper is organized as follows: Section 2 models the rms situation as anoptimal control problem with a random time horizon. Section 3 states our mainresults concerning optimal investment policies. All proofs of the results in Section 3are given in the appendix. Section 4 addresses the nancial problem of how muchventure capital to attract. Section 5 contains our concluding remarks.2. Optimal control modelTime t is continuous and at time t 0 the rm starts its operations. To increasereadability, we present the stochastic optimal control model in a series ofsubsections.2.1. AccountingAt time zero, an entrepreneur contributes a xed amount of capital X 0X0 and therm attracts venture capital in amount of L0X0. It may happen that the rm isentirely nanced by venture capital (cf. Bergemann and Hege, 1997), in which casewe have X 0 0;L040.The venture capital is used as an initial, one-shot, investment in the stock of IC.The initial amount of available funds, X 0 L0, are spent to acquire a stock ofintellectual capital, z040. Let zt be a state variable that represents the dollar valueof the rms stock of IC by time tX0. To simplify, suppose that the stock zt is therms only asset. The rm is likely to have relatively few tangible assets (machinery,equipment, materials), compared to the intangible ones, and we have chosen todisregard the tangible assets (cf. Brander and Lewis, 1986). Moreover, the cashbalance is zero at any instant of time.Sahlman, 1990; Kaplan and Stromberg, 2003). Thus, what we have in mind is aARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 23392361 2343situation in which the venture capitalist provides only startup nancing or one inwhich the capitalist provides all of the funding commitment on signing the contract.(The latter arrangement was used only in a minority (15%) of the cases examined inKaplan and Stromberg (2003).)Thus, having obtained the venture capital, the rm must nance any furtherinvestments in IC by retentions. The assumption here is that the rm allocates aconstant part of its IC, say, z40, to operations that are not directly connected to theefforts of making the breakthrough. These operations generate revenue at theconstant rate p40. Any excess intellectual capital, zt z, is devoted to trying tomake the breakthrough, but this capital generates no operating revenue in the timeinterval 0;T , where T is the date at which the breakthrough is made. This instant oftime is a priori unknown. To save on notation, we dene Zt zt z, and imposethe constraint ZtX0.The agreement between the rm and the venture capitalist is a contract accordingto which the venture capitalist gets security for her investment in the rm in the formof stocks (typically, convertible preferred stock). The venture capitalist thus getsstock in amount of L0.The stock holders (the venture capitalist and the founders) may receive returns ontheir investments, if so decided by the board. The payment of dividends, however,decreases the funds available for investments in IC.With regard to the venture capitalists exit, we suppose that the agreement statesthe capitalist exits when the breakthrough is made, that is, at time T , but she will exitat a predetermined instant of time, t const:40, if the breakthrough has not yetbeen made by time t. Hence, since the innovation date is unknown, the exit time ofthe capitalist is random and given by te minfT ; tg. This arrangement gives thecapitalist the opportunity to leave the rm if the breakthrough has not been madewithin reasonable time. The exit of the capitalist at time t is a typical provision in aventure capital contract, and is quite similar to the required repayment of theprincipal at the maturity of a debt claim (Kaplan and Stromberg, 2003). Section 2.4discusses in detail how the venture capitalist is paid off when exiting the rm.2.3. Accumulation of intellectual capitalInvestments in IC are purposeful efforts of the rm (hiring new staff, trainingexisting staff), made in order to increase the stock Zt. The single purpose ofinvesting is to increase the chances of making the breakthrough. Let atX0 denotethe investment rate at time t. This is a control variable of the rm and is also therms only operating cost. The accumulation dynamics over the time interval 0;1are given by_Zt at; Z040. (1)The absence of a depreciation term in (1), and non-negativity of atX0, imply thatthe stock Z cannot decrease. Thus, our assumption is that the rm cannot decrease Zby laying off employees ao0; neither does the stock Z decrease for the reason thatknowledge becomes obsolete. To model organizational forgetting one can subtract aARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 233923612344term dZt, d const:40, on the right-hand side of (1). Then the rms recentexperiences would be more important for making the innovation than the distantexperiences.It can be argued that there should be a concave relationship in (1) between theinvestment rate and the rate of change of the stock, to reect decreasing marginaleffects of investment. We shall capture such a feature in another way (cf. (3)).Consider the time at which a breakthrough occurs. This instant is represented by arandom variable T , which is dened on a probability space O;F;Pa and takes itsvalues in 0;1. The probability measure P depends on the control (althoughindirectly) in the following way:PaT 2 t; t dt j TXt gZtdt odt, (2)where odt=dt! 0 uniformly in Z; a for dt! 0. The term gZt on the right-handside of (2) is the stopping rate and g40 is a constant. The stopping rate is increasingin Z, and (2) impliesPrT 2 0; t 1 exp gZ t0Zsds . (3)Eq. (3) shows that the more intellectual capital that was accumulated during the timeinterval 0; t, the higher the probability of making the breakthrough by time t. Thus,the rms current chances of making the innovation depend on its stock ofknowledge; knowledge has value. This hypothesis was employed by, e.g., Fudenberget al. (1983) and Doraszelski (2003) and seems more plausible than the one used inother models of innovation (e.g., Reinganum, 1981, 1982; Bergemann and Hege,1997; Dockner et al., 2000). In these works, the probability of making abreakthrough depends on current investment only and hence is independent ofaccumulated knowledge. Then knowledge is irrelevant for the rms current efforts.Recall from (1) that current investments translate linearly into IC. However, dueto (3), there is a diminishing marginal efciency of IC in increasing the stopping rate(since the probability in (3) grows in a concave fashion).The specication in (2) also implies PrT 1 expg R10 Zsds. Hence, anecessary and sufcient condition for having a nite completion time T is thatR t0 Zsds!1 for t!1. This is true, as Z0 is positive and Z is non-decreasing.2.4. Objective functionTo construct the rms objective function, we need to discuss the following issues.What happens when the breakthrough is made, and what will the venture capitalistreceive, in return on her investment, upon her exit from the rm?As to the rst question, we summarize the rms situation in a simple, butstandard, way. Let P const:40 denote the expected value of an uncertain rewardto be earned at time T . (One can think of P as being the value of a patent.)The reward being constant means, in particular, that it does not depend on ZT, thestock of IC at the innovation date. Hence, the reward is the same whether theb const:40. If b 1, the capitalist receives the principal amount of stock in thecompany. In the majority of venture capital contracts, however, b is greater thanARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 23392361 2345one. If the contract provides for accruing dividends, our assumption is that thecumulated amount is included in bL0 (which can be accomplished by a suitablechoice of b).Denoting the entrepreneurs positive and constant rate of time preference by i, hisobjective functional, J, can now be constructedJa;Z0 E aTotZ t0eitp atdt eiT 1 yL0P E aTXtZ Tteitp atdt eiT P eitbL0 . 4The objective is the maximization of the expected present value (at time zero) of theprot stream p at and the reward P, minus the expected payment, yL0P or bL0,upon the exit of the venture capitalist.A pair Z; a is feasible if the objective value J is nite, state equation (1) issatised, and the control constraint at 2 0; p is fullled for t 2 0;1.3. Main resultsThis section characterizes an optimal solution of the stochastic optimal controlproblem in Section 2, but without stating any proofs. These can be found in theappendix.Using HamiltonJacobiBellman equations is a standard way to solve a stochasticoptimal control problem. It turns out, however, that in the setup at hand thistechnique is less useful. The reason is that the problem is non-autonomous, leadingto a partial differential equation for the value function. Instead, we employ the factthat problems with a random stopping time can be reformulated, under somecircumstances, as deterministic control problems. In particular, we formulate aninnite horizon deterministic optimal control problem, which is equivalent to theinnovation is made by a small or a large rm (cf. Grossman and Shapiro, 1986). Asconcerns the exit time te of the venture capitalist, we distinguish two cases:Case 1: te T()Tot.The reward P earned at time T is divided among the founders and the venturecapitalist such that the latter gets yL0P and the former gets the rest, 1 yL0P,where yL0 2 0; 1: The share yL0 depends only on the amount of venture capitalprovided; thus the share is time-invariant and does not depend on the innovationtime T . We assume y0 0; y0L040; y00L0a0.Case 2: te t()T4t.The breakthrough has not yet been made by time t, which is the predetermined,latest exit time of the venture capitalist. Hence she exits at t t and receives,according to the redemption provision of the contract, an amount of bL0,stochastic optimal control problem with random stopping time that was stated inARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 233923612346Section 2 (cf. Boukas et al., 1990; Carlson et al., 1991; Sorger, 1991). Thisdeterministic dynamic problem can be solved by applying the appropriate maximumprinciple.The deterministic problem has two state variables, Zt and Y t, where Y t issuch that eY t denotes the probability that no breakthrough in the innovationprocess has occurred up until time t. It turns out to be expedient to solve the problembackwards, and to divide it in two subproblems.Given any feasible pair Zt;Y t Zt;Y t that results from using an optimalinvestment policy on the interval 0; t, we rst solve a subproblem on the timeinterval t;1. Denote this subproblem by P2 and dene the state variableY t Y t Z ttgZsds,in which Y t is a xed but, so far, arbitrary real number. From (3) we now obtain thateY t is the probability that the innovation breakthrough did not take place beforetime t. This can be employed to reformulate the objective functional (4) in such away that it ts in an innite horizon deterministic optimal control problem. Theobjective functional of problem P2, denoted by J2, represents the expected presentvalue at time t t and is given byJ2a;Zt;Y t EaZ Tteittp atdt eiTtP bL0 bL0 eitY tZ 1teitY tp at PgZtdt, 5and the state dynamics are_Zt at; Zt Zt._Y t gZt; Y t Y t. 6When a solution of P2 has been obtained, one can determine the optimal value of J2,which will be denoted by J2Zt;Y t.Then we solve the full problem on the time interval 0;1: This problem has thefollowing objective functional, when cast as a deterministic optimal control problem:Ja;Z0 Z t0eitY tp at 1 yL0PgZtdt eitJ2Zt;Y t. 7For t 2 0; t, the dynamics of the problem are given by the state equations_Zt at; Z0 Z0._Y t gZt; Y 0 0: 8The calculations will proceed as follows. Section 3.1 provides a solution of P2 andSection 3.2 solves the full problem.ARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 23392361 23473.1. Solution of P2Dene costate variables l1t; l2t associated with state variables Z and Y,respectively. In (5), the term eitY t is constant and can be disregarded in thecharacterization of an optimal solution. The Hamiltonian isH eitY p a PgZ l1a l2gZ,which is linear in the control a, and hence an optimal investment policy is bangbangor singular. There are three candidate policies. In terms of the costate l1t they canbe stated asDividend : at 0)l1teitY to1.Investment : at p)l1teitY t41.Singular : at 2 0; p)l1teitY t 1. 9The time function l1teitY t has an interpretation as a shadow price of the stock Z,which provides an intuition for the policies in (9). Notice that since stockholders arerisk neutral, the value of a marginal dollar of net prot equals one. A dividend policypays maximally to the stockholders and since there are no investments, the stock ofIC remains constant. This happens if a unit of IC has a shadow price less than oneand then the marginal dollar should be paid to the owners. Since the stock Z cannotbe decreased (to get additional funds to pay as dividends), it is kept constant. Usingan investment policy, the stock of intellectual capital grows at its maximal rate; thereis no payout to stockholders. The reason is that the shadow price exceeds the valueto the stockholders of having the marginal dollar paid out as dividends. Then thedollar is spent entirely on investments. Finally, if the shadow price equals one,stockholders are indifferent between investing the marginal dollar and having itpaid out.A particular interest relates to the stock level Z associated with a singular policy.Suppose that one chooses the singular control as a 0. Then the singular stock level,ZS, say, is constant and equalsZS 1g i giP pq . (10)To have a positive ZS, we introduce the assumptionP4pi ig. (11)Note that (11) implies P4p=i. This inequality means that the expected rewardexceeds the present value of a perpetuity of p. Otherwise the rm would have noincentive to try to make the innovation.The instant of time at which the stock Zt reaches the singular level ZS, whenstarting out in state Zt and using the investment policy, is given bytS ZS Zt. (12)pSin is constant, equal to, say ~Z. Theelem dt for the stopping time T then iso-investment policy, as of time t, isARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 233923612348eitY tZ 1tfeitY tp Pg ~Zdt bL0 p Pg ~Zi g ~Z bL0. (13)Now suppose, hypothetically, that a marginal investment were made. This wouldincrease marginally the stock of IC. Differentiation in (13) with respect to ~Z providesthe expected marginal revenue of such investmentgiP p2.Thegiveexpected present value of employing a zern byce there is no investment, the stock Ztentary probability of the time interval t; tg ~ZeR ttg ~Zds g ~Zeg ~Ztt.Proposition 1 characterizes an optimal solution of P2. The reader should be awarethat the optimality conditions applied in the proposition are necessary only andrequire the existence of an optimal control. Due to the fact that the discount factorinvolves the state variable Y , it is not possible to verify standard sufciencyconditions based upon concavity of Hamiltonians.Proposition 1. For any ZtXZ0, an optimal solution of P2 is(i) If ZtXZS then at 0 for t 2 t;1.(ii) If ZtoZS, thenat p0 for t 2t; tStS;1( ).The intuition for the zero-investment policy in Case (i) is that the stock level Zt ishigh (ZtXZS). Then no further investments are needed. (If Zt4ZS, the dividendpolicy is used. If Zt ZS, the singular policy is used). Case (ii) occurs if the stocklevel Zt is low (ZtoZS). Investment then is worthwhile, and is continued until thestock Zt reaches the singular level ZS.The policy stated in Proposition 1(ii) works as follows. Investing at the rate a pmakes the state Z increase at the maximal rate. The purpose is to get as quickly aspossible from the initial level Zt to the singular level ZS. Once the latter is reached, itis optimal to stay there forever. In Case (i) one would also like to approach thesingular level as fast as possible. In our setup, where disinvestment is impossibleand there is no exogenous decay of the capital stock, ZS clearly cannot bereached if Zt4ZS. The best option then is to stay forever at the level Zt. Clearly,if Zt ZS the rm is already at the singular stock level. This is, however, a hair-line case.For a further interpretation of the results of the proposition, consider Case (i).i gZARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 23392361 2349Denote this subproblem by P1 and note (also here) that one cannot verify sufciencyconditions since the concavity requirements are not fullled.The characterization of an optimal investment policy in problem P1 is similar tothat in P2 and it sufces to note the following. Dening a costate m1t associatedwith state variable Zt, one can identify three candidate policies:Investment : at p)m1teitY t41.Dividend : at 0)m1teitY to1.Singular : at 2 0;p)m1teitY t 1.An optimal solution of P1 is characterized in Proposition 2, in which the singularstock level is given byZSA 1 i gi1 yL0P pq . (17)Equating this expected marginal revenue to the marginal cost of investment, which isone, yieldsgiP p i gZ2 0. (14)Using (10), (14), and Proposition 1 we conclude the following. For ~Z4ZS, aninvestment would make marginal cost exceed expected marginal revenue. This is whyno investment should take place. If Z ZS, expected marginal revenue equalsmarginal cost. This is a singular arc, along which a 0. Finally, if ZoZS,expected marginal revenue of investment exceeds marginal cost, at least duringsome initial interval of time, say, t; t D. In that case, a dividend policy issuboptimal and investment is worthwhile. Thus, at p on the time interval t; tS.Investment is stopped at time tS because at this instant, the stock reaches the singularlevel ZS.3.2. Complete solution of the investment problemThe objective functional of this problem, when cast as a deterministic optimalcontrol problem, isJa;Z0 Z t0eitY tp at 1 yL0PgZtdt eitJ2Zt;Y t.(15)For t 2 0; t, the objective to be maximized is the integral on the right-hand side of(15). The dynamics are_Zt at; Z0 Z0._Y t gZt; Y 0 0. 16gpny Z040, an optimal solution is(i)ARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 233923612350(iii) If Z0oZSA tp, thenat p for t 2 0; t.The intuition of Case (i) is that it occurs if the initial stock Z0 is already sufcientlyhigh; in particular, it exceeds the singular level ZSA. Then it does not pay to invest inorder to increase the stock of IC.In Cases (ii) and (iii), the initial stock Z0 is insufcient and investments are used toincrease it (maximally). In Case (iii), investments continue throughout the timeinterval 0; t. In Case (ii), investment is stopped at time tSA, i.e., at the instant oftime where the singular stock level is reached.To see the difference between (ii) and (iii) note thattSA4o8>:9>=>;t () Z0o48>:9>=>;ZSA pt. (19)In Case (iii) we have tSA4t()Z0oZSA pt. This means that even wheninveat p0 for t 20; tSAtSA; t( ).(ii)If Z0XZSA, thenat 0 for t 2 0; t.If ZSA ptpZ0oZSA, thenProposition 2. Assume that ZSA pt40. For aTo have a positive singular level ZSA, we strengthen the assumption in (11) byintroducing the hypothesisP4ig pi yL0P.NoteZSAoZS,that is, the singular stock level in problem P1 is less than in P2. The reason clearly isthat in problem P1, the reward P must be shared with the venture capitalist.If Z0oZSA the instant of time at which Zt reaches ZSA, when starting out fromZ0 and using the investment policy at p, is given bytSA ZSA Z0. (18)stment is done at the maximal rate p during the entire time interval 0; t, itat 0 for t 2 0;1.(ii)(iii)(iv)nd 2. The four cases in Proposition 3vestment efforts, or, equivalently,ARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 23392361 2351decreasing initial stocks of IC.In Case (i), the initial stock level Z0 is sufciently high and there is no need forinvestments at all. In Case (ii), the stock level Z0 is less than the long term singularlevel ZS and it pays to invest to build up intellectual capital. Investments are,however, postponed until the exit of the venture capitalist. The reason is that Z0exceeds the short term singular stock level ZSA. In Case (iii) we see a sort of pulsingstrategy where investment switches twice between maximal effort and zero.Although the long-term singular level has not yet been reached at time tSA,investment is temporarily stopped until the venture capitalist exits. ThenPrareinveat 0for t 2 tS;1 .oposition 3 is derived from Propositions 1 aordered in accordance with increasing inp0>>>: >>>; t; tStS;1>>>>: >>>>;If Z0pZSA pt, thenp 0; tS( )at p0>: >; for t 2 t; tStS;1>: >;.If ZSA ptoZ0oZSA, thenat p08>>>>>=for t 20; tSAtSA; t8>>>>>>>=.If ZSApZ0oZS, then08>< 9>= 0; t8>< 9>=cannot increase the initial stock Z0 to the singular level ZSA. In Case (ii) we havetSApt()Z0XZSA pt, which means that investing throughout the time interval0; t would overshoot the singular level. Therefore, investment goes on only duringthe time interval 0; tSA. Clearly, the initial stock Z0 is larger in Case (ii) than in Case(iii); hence there is less need for investments.We are now ready to state a main result of the paper, the characterization of anoptimal solution of the full problem. This result is stated as Proposition 3.Proposition 3. For any Z040, an optimal solution is(i) If Z0XZS, thenstment is resumed and is continued until the long-term singular stock level isinitial acquisition of IC, is sufciently large. In Cases (ii)(iv) there is an initialARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 233923612352phase of investment, which is warranted by the fact that in these cases the amountof venture capital is smaller. Investments go on for the longest interval of timein Case (iv), which is intuitive since here the amount of venture capital is thesmallest.Denote by JL0 the rms payoff over the time interval 0;1. Using (7) and (15)yieldsJL0 Z t0eitY tp at 1 yL0PgZtdt eit eitY tZ 1eitY tp at PgZtdt bL0 . 20reached. In Case (iv), the initial stock is the lowest in the four cases, and it pays tokeep on investing until reaching the long-term singular level ZS.In Cases (ii)(iv) we see the implications of the knowledge effect, which reectsthe fact that the rms past efforts contribute to its chances of making thediscovery. Due to that effect, the rm scales down its investments in IC as thestock of knowledge (IC) increases (Doraszelski, 2003). In such cases,investment expenditures can be reduced (although rather abruptly) as the stock ofIC increases. (In a non-linear model, one would see a smoother decrease in theinvestment rate).4. Financial policyThe derivations of Section 3 proceeded under the assumption that the amount ofventure capital, L0, was xed. It remains to discuss the choice of L0. This amount canbe seen as a parameter, determined at time zero, that inuences the dynamicoptimization problem.We assumed that venture capital can only be used for an initial, one-shot,investment in the stock of IC. For the stockholders, the intertemporal trade-off isbetween giving up current consumption in order to invest (to increase the probabilityof winning the reward P), and refraining from investment in order to consume now.We have seen that this decision depends on the stock level Z0. Recalling that Z0 X 0 L0 shows that the venture capital decision will inuence the investmentdecision. Thus, increasing the amount of venture capital will increase Z0, the initialstock of IC, and hence reduce the need for later investments (see, for example,Proposition 3, Case (i)). On the other hand, increasing the amount of venture capitalwill increase bL0, the exit payment to the venture capitalist as well as the capitalistsshare, yL0, of the reward.The occurrence of the four cases in Proposition 3 depend on the amount ofventure capital, L0, such that Case (i) occurs if L0XZS X 0; Case (ii) ifZSA X 0pL0pZS X 0, Case (iii) if ZSA pt X 0oL0oZSA X 0, and Case(iv) if L0pZSA pt X 0. Recall that in Case (i) the rm does not invest at all.The intuition is that the amount of venture capital attracted, and used in thetARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 23392361 2353To determine the amount L0, we apply a necessary optimality condition inLeonard and Long (1992, Theorem 7.11.1). Whenever L0 is strictly positive it isgiven byy0L0 1Kl10 beit, (21)in whichK Z t0eitY tPgZtdt40.The condition in (21) compares the costate l10 (the shadow price of Z0) with thepresent value of the constant b. Recall that if the venture capitalist exits at thepredetermined time t, she is paid the amount bL0. If the shadow price exceeds the(present value of the) marginal payment to be made at time t, venture capitalnancing should be attracted in accordance with (21). On the other hand, ifbeitXl10, no capital should be attracted.The optimality condition in (21) can be rewritten to yieldy0L0K beit l10.Hence, the optimal amount of venture capital nancing requires that the marginalexpected value of the rm arising from additional funds is equal to the expectedpresent value of a marginal payment to the venture capitalist.The choice in (21) depends on the exit time t and the rate of time preference, i. Tosee the implications of varying these parameters, we proceed as follows.Case A: Time preference rate fixed: The choice of L0 0 is more likely for t small(since the condition in (21) then can be satised by a larger range of bs). Thus, if thelatest exit time of the venture capitalist is in the near future, the founders do not wishto attract venture capital. The policy could also occur for a large t, but then a largevalue of b is necessary. The intuition here is that although the exit of the capitalist isquite far in the future, the founders do not attract venture capital due to the largeamount, bL0, that must be paid upon exit of the capitalist.Case B: Exit time fixed: The choice L0 0 is more likely for i small. Thus, if thefounders are far-sighted, they are reluctant to attract venture capital. Thepolicy could also occur for a large value of i, but then a large value of b isnecessary. The intuition here is that although the founders are myopic, they do notattract venture capital due to the large amount that must be paid upon exit of thecapitalist.Suppose now that the function yL0 is convex. To be specic, let yL0 12L20.Thus, the share that the venture capitalist receives from the reward increasesprogressively with the amount of venture capital supplied. This could be seen as ameans of hedging the capitalists investment.Using (21) yields L0 l10 beit=K . In this expression, a direct and anindirect effect can be identied. The direct effect is that the amount of venture capitaldecreases if the reward P increases. The reason is that a marginal increase in L0 leadskinds of developments in the economic environment that cannot be foreseen.This can be modeled by introducing a random diffusion on the market valueARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 233923612354of the reward obtained by the breakthrough. The problem could then haveanother dimension related to debt contract valuation (see, e.g., Anderson andSundaresan, 1996).Third, in reality competition plays an important role in the determination ofinnovation strategies. Introducing competition in our setup would require theanalysis of a differential game. There is considerable literature in applied differentialgames dealing with innovations and R&D in competitive environments (see, e.g.,to a larger monetary loss, y0L0P, when P is large. The indirect effect concerns thefact that a higher reward increases the incentive to invest in intellectual capital andthus affects L0 positively. In the model, this is reected in the fact that the thresholdstock ZSA increases with the reward (cf. (17)). According to Proposition 3 this willmake sequences (iii) and (iv) more likely as optimal ones. In these two policies, theshadow price l10 is sufciently large to inuence positively the amount of venturecapital L0 (cf. (21)).5. Concluding remarksThe paper has considered a dynamic optimization problem of a rm that tries tomake a breakthrough with a new IT product. The probability of achieving abreakthrough can be increased by investing in intellectual capital (IC). The initialinvestment in IC can be nanced by venture capital. The paper has determinedoptimal investment policies and an optimal amount of venture capital. The singularstock of IC is a main determinant of the type of the investment policy.Having assumed linearity of every cost and payoff, and having assumed non-evaporating intellectual capital, gives rise to a bangbang control scheme.Alternatively, we could have imposed, e.g., (i) a declining value in time of thebreakthrough, (ii) a non-linear investment cost, or (iii) the aging of intellectualcapital. As such changes without doubt would have their own quantitative effects, itis our belief that the rms optimal behavior will keep its qualitative structure with acentral role for the singular stock of intellectual capital.We conclude by stating three more rigorous extensions that should bepromising areas for future research. First, in reality venture capital is usuallyprovided in stages, and not as a lump sum at the start of the venture. This isperhaps the most limiting assumption of the paper. Capital being providedstagewise means that the rm receives capital as seed money, start up capital,capital for development of prototypes, etc. (see Sahlman (1990, Table 2) for details).A priori, the completion times of the various stages are random, and the state inwhich a new stage starts out is random, too. The methodology of piecewisedeterministic optimal control problems applies here (see, e.g., Carlson et al., 1991;Dockner et al., 2000).Second, in reality the value of a new nding in the IT area is subject to allReinganum, 1981, 1982; Haurie, 1994; Dockner et al., 2000).ir helpful comments.AppendixitY tZ1 0 0ARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 23392361 2355Z2 0 0The singular investment policy is indeterminate in the interval 0; p and we choosea 0. The stock level ZS associated with this singular policy is constant and given byInvestment (a p) Dividend (a 0) Singular (a 0)Proof of Proposition 1. In (5), the term e is constant and can be disregarded inthe characterization of an optimal solution. Let l1t; l2t be costate variablesassociated with state variables Z and Y ; respectively, and Z1t; Z2t be Lagrangianmultipliers associated with the control constraint at 2 0; p. The Hamiltonian isH eitY p a PgZ l1a l2gZand the Lagrangian then becomesL H Z1a Z2p a.Suppose that there exists an optimal solution of problem P2. Necessary optimalityconditions then arel1t Z2t Z1t eitY t, (A.1)_l1t eitY tgP gl2t, (A.2)_l2t eitY tp at PgZt, (A.3)Z1tat 0; atX0; Z1tX0,Z2tp at 0; atpp; Z2tX0,limt!1l2t 0. (A.4)Due to the linearity of the Hamiltonian in the control, an optimal investmentpolicy is bangbang or singular. Clearly, both Z1 and Z2 cannot be positive and hencethree candidate policies remain. Characterized by the signs of the Lagrangianmultipliers they are(10). The instant of time at which the stock Zt reaches the singulatheAcknowledgementThe authors like to thank two anonymous referees and editor Carl Chiarella forr level ZS, whenARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 233923612356starting out in state Zt, is given bytS ZS Ztp.In what follows we characterize in detail the three candidate policies.Investment policy (Path 1):Z1t 0; Z2t40; _Zt at p:From (A.1) and (A.3) it follows thatl1t eitY t Z2t, (A.5)_l2t eitY tPgZt. (A.6)Differentiating l1t in (A.5) twice with respect to time yields_l1t eitY ti gZt _Z2t, (A.7)l1t eitY ti gZt2 gp Z2t. (A.8)Differentiating in (A.2) with respect to time, and using (A.6) and (A.8), providesl1t eitY tiPg,Z2t eitY tfgiP p i gZt2g. A:9Dividend policy (Path 2):Z2t 0; Z1t40; _Zt at 0; Zt ZD const:40.From (A.1) and (A.3) it follows thatl1t eitY t Z1t, (A.10)_l2t eitY tp PgZD. (A.11)Differentiating l1t in (A.10) twice with respect to time yields_l1t eitY ti gZD _Z1t, (A.12)l1t eitY ti gZD2 Z1t. (A.13)Differentiating in (A.2) with respect to time and using (A.11) yieldsl1t eitY tgiP pand using (A.13) then provideseitY ti gZD2 Z1t eitY tgiP p) Z1t eitY ti gZD2 giP p. A:14Singular policy (Path 3):Z1t Z2t 0; _Zt at 0; Zt ZS const:40.ARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 23392361 2357The characterizations of costates and Lagrangian multipliers follow from those ofPath 2, by putting Z1t equal to zero. Thusl1t eitY t, (A.15)_l2t eitY tp PgZS, (A.16)_l1t eitY i gZS, (A.17)giP p i gZS2 0 () ZS 1gi giP pq . (A.18)We construct a solution that satises the necessary optimality conditions, by usinga path-coupling procedure which starts by considering Path 2 or Path 3 as a candidatefor being a nal path. A nal path is one that is employed as of some instant of timetill innity. Path 1 will not be considered a candidate for being a nal path from thefollowing reason: since the probability of making the innovation is concavelyincreasing in the stock of IC, it is clearly suboptimal to keep on investing forever.We start by considering the case where Path 2 is the nal path.Lemma 4. If Path 2 is the final path, it cannot be preceded by any other path.Proof. By contradiction. Since Z is a good stock, we can safely assume that thecostate l1t is always non-negative. Note that (A.10) implieslimt!1l1t 0, (A.19)limt!1Z1t 0 (A.20)along Path 2.Now suppose that Path k (k 1 or 3) were to precede Path 2. Then, at thecoupling point t between Path k and Path 2, it must be true thatZ1t 0,which follows from the required continuity of l1t; and using (A.5), (A.15). By denition,Z1t is positive on Path 2 for t 2 t;1. Then, using Z1t 0 and (A.20), it follows thatZ1t must have at least one maximum along Path 2. In such a point we have Z1to0.From (A.14) and the fact that Zt const: ZD along Path 2, one therefore obtainsgiP p i gZD240everywhere along Path 2. Hence, by (A.14) it holds that Z1to0 everywhere on Path2. This means that _Z1t decreases along Path 2 (or, equivalently, that Z1 is a strictlyconcave function).On the other hand, using the necessary (A.2), (A.4), and (A.12) yields_Z1t eitY tPg i gZD gl2t ) limt!1_Z1t 0. (A.21)However, (A.21) cannot hold if _Z1 is decreasing everywhere on Path 2. Thus,supposing that another path can precede Path 2 leads to a violation of the necessaryoptimality conditions. This observation completes the proof. &ARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 233923612358Lemma 5. When Path 2 is used for t 2 t;1 it holds thatgiP p i gZD2o0.Proof. Since Z is constant along Path 2, (A.14) shows that Z1 does not change itssign. From this fact, and using (A.20) and (A.21) it follows that Z1t40 everywherealong Path 2, which via (A.14) leads to the result. &Next, suppose that Path 3 is the nal path, in which (A.18) is satised everywherealong Path 3. For this situation we have the following result.Lemma 6. If Path 3 is the final path, a solution is given by the sequence Path 1! Path3, where on Path 1 it holds thatgiP p i gZt240.Proof. From (A.9) and (A.18) one obtains Z2t1340, where t13 is the moment oftime at which Path 1 passes into Path 3. Since Z1t increases on Path 1, it followsthatZ2t40everywhere along Path 1.From (A.5), (A.15), continuity of l1t; and the fact that Z2t40 along Path 1, itfollows thatZ2t13 0, (A.22)_Z2t13p0. (A.23)Now, Z2t40 on Path 1, (A.22) and (A.23) imply that Z2t decreases everywherealong Path 1.Suppose that another path were to precede Path 1. This would require that at thecoupling point, ~t, say, between these two paths it must hold that Z2~t 0 sinceotherwise continuity l1t would be violated. However, this is incompatible with(A.22) and the fact that Z2t is decreasing everywhere along Path 1. Thisdemonstrates that no path can be coupled before the sequence Path 1! Path 3.The inequality stated in the lemma follows directly follows from (A.18) and the factthat Zt increases on Path 1. &Lemma 7. Path 2 cannot precede Path 3 if the latter is a final path.Proof. By contradiction. All along the sequence Path 2! Path 3, (A.18) holds sinceZt is constant on both paths. This implies, using (A.14), that Z1t 0 along Path 2and hence _Z1t is constant.Now, continuity of l1t requires that just before a coupling point t23 betweenPaths 2 and 3 it must hold that Z1t23 0, cf. (A.10) and (A.15). Hence _Z1t23 mustbe negative, and since _Z1 is constant, it is negative everywhere along Path 2.From (A.2) follows that _l1t is continuous on every path. Using (A.12) and (A.17)shows that this is violated for a strictly negative _Z1. This completes the proof. &ARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 23392361 2359We summarize the results of the above lemmas as follows. Consider problem P2.Sequences of paths that satisfy the necessary optimality conditions are the following:giP p i gZt2o0 () Zt4ZS: Path 2.giP p i gZt2 0 () Zt ZS: Path 3.giP p i gZt240 () ZtoZS : Path 1! Path 3.The two rst sequences generate the control path at 0 for t 2 t;1. The last onegenerates the control path at p for t 2 t; tS, at 0 for t 2 tS;1. Thiscompletes the proof of Proposition 1. &Proof of Proposition 2. Essentially, a proof of this proposition is superuous; inmost respects it would be similar to that of Proposition 1.Proposition 1deals with an innite horizon problem with a xed initial state Zt. Inthis problem there is a singular stock level ZS and the optimal policy is to reach thislevel as quickly as possible. If ZtoZS, the rm invests maximally to reach thesingular stock level at time tS; due to the innite horizon, ZS will always be reached.On the other hand, if the initial stock Zt exceeds the singular level ZS, no investmentis needed.The situation in Proposition 2 is basically the same. Proposition 2 deals with anite horizon problem with a xed initial state Z0. In this problem there is also asingular stock level, ZSA, and the optimal policy is to reach this level as quickly aspossible, by using maximal investment efforts. If Z0oZSA, the rm investsmaximally to try to reach the singular stock level at time tSA. However, due to thenite horizon, ZSA may not be reached before the end of the horizon, t. Thus, oneneeds to distinguish Cases (ii) and (iii) in Proposition 2. In Case (ii) there is timeenough to reach the singular level, and investment is discontinued during theremaining time interval tSA; t. In Case (iii), investment goes on until the end of thehorizon, because tSA4t. Finally, as in Proposition 1, if the initial stock Z0 exceedsthe singular level ZSA, no investment is needed.Proof of Proposition 3. To establish Proposition 3, we introduce the two functionsf Z giP p i gZt2,f AZ gi1 yP p i gZt2,which both are decreasing and strictly concave. Note thatf Z4o8>>>:9>>=>>;0 () Zto48>>>:9>>=>>;ZSf AZ4o8>>>:9>>=>>;0 () Zto48>>>:9>>=>>;ZSA,of Finance 49 (2), 371402.eview of Financialptcy. INFOR 19 (4),ARTICLE IN PRESSS. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 233923612360292310.Bergemann, D., Hege, U., 1997. Venture capital nancing, moral hazard, and learning. Journal ofBanking and Finance 22, 703735.Boukas, E.K., Haurie, A., Michel, P., 1990. An optimal control problem with a random stopping time.SBensJtudies 9 (1), 3768.oussan, A., Lesourne, J., 1981. Optimal growth of a rm facing a risk of bankruEAndntrepreneurship. Blackwell Business, pp. 259282.erson, W.R., Sundaresan, S., 1996. Design and valuation of debt contracts. RAmit, R., Brander, J., Zott, C., 2000. Venture capital nancing of entrepreneurship: theory empiricalevidence and a research agenda. In: Sexton, D., Landstrom, H. (Eds.), The Blackwell Handbook ofAdmaoti, A.R., Peiderer, P., 1994. Robust nancial contracting and the role of venture capitalists. JournalReferencesNow we can prove Proposition 3.Case (i): The assumption is f Z0p0, which implies f AZ0o0. Hence, byProposition 2(i), we have at 0 for t 2 0; t, which implies Zt Z0. Then, by theassumption of Case (i), one obtains f Zt f Z0p0. Using Proposition 1(i) thenyields at 0 for t 2 t;1. The proof is complete.Case (ii): The assumption is f AZ0p0 and f Z040, The rst inequality implies,by Proposition 2(i), that at 0 for t 2 0; t. Then one has Zt Z0, which impliesf Zt f Z040. Using Proposition 1(ii) then yields at p for t 2 t; tS andat 0 for t 2 tS;1. The proof is complete.Case (iii): The assumption is f AZ040 and Z04ZSA pt. The rst inequalityimplies, by Proposition 2(ii), that at p for t 2 0; tSA and at 0 for t 2 tSA; t.Then Zt ZSA in Proposition 1 and hence f Zt f ZSA. However, f ZSA40,and using Proposition 1(ii) one obtains at p for t 2 t; tS and at 0 fort 2 tS;1. The proof is complete.Case (iv): The assumption is f AZ040. By Proposition 2(iii) we get at p fort 2 0; t. This implies ZtoZSAoZS, and hence f Zt4f ZS 0. Then, byProposition 1(ii), we get at p for t 2 t; tS and at 0 for t 2 tS:1. Theproof is complete. &and recall that ZSAoZS. It is convenient to use this notation to summarize theresults of Propositions 1 and 2:Proposition 1.(i) If f Ztp0, then at 0 for t 2 t;1.(ii) If f Zt40, then at p for t 2 t; tS, at 0 for t 2 tS;1.Proposition 2. Let Zpt9Z0 pt.(i) If f AZ0p0, then at 0 for t 2 0; t.(ii) If f AZ040 and Zpt4ZSA, then at p for t 2 0; tSA, at 0 for t 2 tSA; t.(iii) If f AZ040 and ZptpZSA, then at p for t 2 0; t.urnal of Optimization Theory and Applications 64 (3), 471480.Drucker, P., 1995. The information executives truly need. Harvard Business Review, JanuaryFebruary,ARTICLE IN PRESS5462.Fudenberg, D., Gilbert, G., Stiglitz, J., Tirole, J., 1983. Preemption, leapfrogging and competition inpatent races. European Economic Review 22, 331.Fulghieri, P., Sevilir, M., 2005. Competition, Human Capital, and Innovation Incentives. Working Paper,University of North Carolina.Gompers, P., 2005. Venture capital. in: Eckbo, B.E. (Ed.), Handbook of Corporate Finance: EmpiricalCorporate Finance. North-Holland, Amsterdam, forthcoming.Grossman, G.M., Shapiro, C., 1986. Optimal dynamic R&D programs. Rand Journal of Economics 17(4), 581593.Haurie, A., 1994. Stochastic differential games in economic modelling. In: Henry, J., Yvon, J.-P. (Eds.),System Modelling and Optimization, Lecture Notes in Control and Information Sciences, vol. 197,Springer, Berlin, pp. 90108.Leonard, D., Van Long, N., 1992. Optimal Control Theory and Static Optimization in Economics.Cambridge University Press, Cambridge, UK.Kamien, M.I., Schwartz, N.L., 1982. Market Structure and Innovation. Cambridge University Press,Cambridge, UK.Kaplan, S.N., Stromberg, P., 2003. Financial contracting theory meets the real world: an empiricalanalysis of venture capital contracts. Review of Economic Studies 70, 281315.McAdam, R., McCreedy, S., 1999. A critical review of knowledge management models. The LearningOrganization 6 (3), 91100.Meng, R., 2004. A patent race in a real options setting: investment strategy, valuation, CAPM beta andreturn volatility. Working Paper, The University of Hong Kong.Reinganum, J., 1981. Dynamic games of innovation. Journal of Economic Theory 25, 2141.Reinganum, J., 1982. A dynamic game of R&D: patent protection and competitive behavior.Econometrica 50, 671688.Sahlman, W.A., 1988. Aspects of nancial contracting in venture capital. Journal of Applied CorporateFinance 1, 2336.Sahlman, W.A., 1990. The structure and governance of venture-capital organizations. Journal of FinancialEconomics 27, 473521.Schwartz, E.S., Zozaya-Gorostiza, C., 2003. Investment under uncertainty in information technology:acquisition and development projects. Management Science 49, 5770.Schworm, W.E., 1980. Financial constraints and capital accumulation. International Economic Review21, 643660.Sorger, G., 1991. Maximum principle for control problems with uncertain horizon and variable discountrate. Journal of Optimization Theory and Applications 70 (3), 607618.Zingales, L., 2000. In search for new foundations. Journal of Finance 55 (4), 16231653.Further readingBensoussan, A., Lesourne, J., 1980. Optimal growth of a self-nancing rm in an uncertain environment.In: Bensoussan, A., Kleindorfer, P., Tapiero, C.S. (Eds.), Applied Stochastic Control in Econometricsand Management Science. North-Holland, Amsterdam, pp. 235269.Reisinger, H., Dockner, E.J., Baldauf, A., 2000. Examining the interaction of marketing and nancingdecisions in a dynamic environment. OR Spektrum 22, 159171.Brander, J.A., Lewis, T.R., 1986. Oligopoly and nancial structure: the limited liability effect. AmericanEconomic Review 75, 956970.Carlson, D.A., Haurie, A.B., Leizarowitz, A., 1991. Innite Horizon Optimal Control: Deterministic andStochastic Systems. Springer, Berlin.Dockner, E.J., Jrgensen, S., Van Long, N., Sorger, G., 2000. Differential Games in Economics andManagement Science. Cambridge University Press, Cambridge, UK.Doraszelski, J., 2003. An R&D race with knowledge accumulation. Rand Journal of Economics 34, 2042.S. Jrgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 23392361 2361Venture capital financed investments in intellectual capitalIntroductionOptimal control modelAccountingFinancingAccumulation of intellectual capitalObjective functionMain resultsSolution of P2Complete solution of the investment problemFinancial policyConcluding remarksAcknowledgementAppendixReferencesbm_fur