venture capital financed investments in intellectual capital

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Journal of Economic Dynamics & Control 30 (2006) 2339–2361

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Venture capital financed investmentsin intellectual capital

Steffen Jørgensena, Peter M. Kortb,c,�, Engelbert J. Docknerd

aDepartment 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, Belgium

dDepartment of Finance and Vienna Graduate School of Finance, University of Vienna, Vienna, Austria

Received 29 March 2004; accepted 11 July 2005

Available online 29 September 2005

Abstract

The paper considers a new company in the IT industry, founded by a management team and

partially financed by venture capital. Among the questions that the paper addresses are: how

much venture capital should be acquired to help finance the development of the firm? How

should a wish to grow, with the aim of making a breakthrough in the IT area, be balanced

against the stockholders’ wish to consume? The problem is studied as an optimal control

problem with a random time horizon and we derive a series of prescriptions for investment

and financial decisions.

r 2005 Elsevier B.V. All rights reserved.

JEL classification: D92; G24; O32

Keywords: Intellectual capital accumulation; Investment; Dividends; Venture capital; Stochastic optimal

control

see front matter r 2005 Elsevier B.V. All rights reserved.

.jedc.2005.07.005

nding author. Department of Econometrics & Operations Research and CentER, Tilburg

.O. Box 90153, 5000 LE Tilburg, The Netherlands. Tel.: +31 13 4662062;

663280.

dress: [email protected] (P.M. Kort).

www.elsevier.com/locate/jedc

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S. Jørgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 2339–23612340

1. Introduction

The information technology (IT) industry has been characterized by a rapidgrowth of the number of new firms 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 firms have experienced a fast growth in terms of the numberof employees, but the growth has necessarily not been financially 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 financial decision making in a newfirm, and derives a series of prescriptions for its growth and financing. Suppose thatthe company is founded at time t ¼ 0, by acquiring another firm in the industry. Thefirm 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 firm. Supposing knowledge can be capitalized, we summarize the firm’sintangible assets in a stock of intellectual capital (IC) (see also McAdam andMcCreedy, 1999).

We consider a particular objective of the firm, being the development of a specificinnovation (an IT product or service) which will make a breakthrough in its specificarea. The date of the innovation, T, is unknown at time zero where the firm 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 firm, represented by a team ofentrepreneurs who manage the firm, 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 firm nor does she choose the exit timestrategically. The venture capitalist in our model is a stockholder of the firm 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 fixed return on her invested capital.

The paper deals with two main problems, essentially seen from the point of view ofthe entrepreneurs/founders. The first problem concerns how much, if any, venturecapital should be attracted initially. The second problem is how to design a strategyfor the firm’s 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 financing on innovations and research and development is.

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S. Jørgensen et al. / Journal of Economic Dynamics & Control 30 (2006) 2339–2361 2341

Although the first US-based venture capital firm was established shortly after thesecond world war, academic research started to focus on venture capital financingonly 10–15 years ago. Central to academic research is the simple setup where abusiness founder (entrepreneur) is looking for outside equity holders. Since such afinancing-relationship is characterized by contracting under asymmetric informationmost of the theoretical research on venture capital financing 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 capitalfinancing is widely recognized and dominates theoretical research. Sahlman (1988,1990) argued that contracting practises in the venture capital industry reflectuncertainties about future payoffs and informational asymmetries betweenentrepreneurs and venture capitalists. Moreover, the lack of operational history ofthe firms aggravates the adverse selection problem.

Admati and Pfleiderer (1994) deal with the venture capital cycle and characterizethe contract that allows optimal continuation decisions with staged financing (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 financing is summarized inGompers (2005). He analyzes in detail the venture capital investment processincluding empirically observed exit strategies and the relationship between venturecapital financing 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 flow and can invest inthe development of the asset. The methodology used to evaluate the project is realoptions, using stochastic differential equations and Hamilton–Jacobi–Bellmanequations. The financial 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 firms are engagedin a patent race. Using numerical simulations she finds that strategically competingfirms 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 flexibility tomove from one firm to the other and hence greater ability to extract rents from theiremployer.

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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 firm’s current and/or cumulative investments.

The paper is organized as follows: Section 2 models the firm’s 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 financial problem of how muchventure capital to attract. Section 5 contains our concluding remarks.

2. Optimal control model

Time t is continuous and at time t ¼ 0 the firm starts its operations. To increasereadability, we present the stochastic optimal control model in a series ofsubsections.

2.1. Accounting

At time zero, an entrepreneur contributes a fixed amount of capital X 0X0 and thefirm attracts venture capital in amount of L0X0. It may happen that the firm isentirely financed by venture capital (cf. Bergemann and Hege, 1997), in which casewe have X 0 ¼ 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 þ L0, are spent to acquire a stock ofintellectual capital, z040. Let zðtÞ be a state variable that represents the dollar valueof the firm’s stock of IC by time tX0. To simplify, suppose that the stock zðtÞ is thefirm’s only asset. The firm 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.

2.2. Financing

We suppose that after having obtained the venture capital L0, the firm can attractno more capital. Thus, the firm must be self-financed for t40. This approach tofinancing differs sharply from one in which a firm 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-financed.

It is also important to note that the assumed, one-shot venture capital financingdiffers from the common practice in which ventures are financed stagewise (see, e.g.,Sahlman, 1990; Kaplan and Stromberg, 2003). Thus, what we have in mind is a

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situation in which the venture capitalist provides only startup financing 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 firm must finance any furtherinvestments in IC by retentions. The assumption here is that the firm 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, zðtÞ � z, is devoted to trying tomake the breakthrough, but this capital generates no operating revenue in the timeinterval ½0;T �, where T is the date at which the breakthrough is made. This instant oftime is a priori unknown. To save on notation, we define ZðtÞ ¼ zðtÞ � z, and imposethe constraint ZðtÞX0.

The agreement between the firm and the venture capitalist is a contract accordingto which the venture capitalist gets security for her investment in the firm 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 capitalist’s 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 ¼ 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 ¼ minfT ; tg. This arrangement gives thecapitalist the opportunity to leave the firm 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 firm.

2.3. Accumulation of intellectual capital

Investments in IC are purposeful efforts of the firm (hiring new staff, trainingexisting staff), made in order to increase the stock ZðtÞ. The single purpose ofinvesting is to increase the chances of making the breakthrough. Let aðtÞX0 denotethe investment rate at time t. This is a control variable of the firm and is also thefirm’s only operating cost. The accumulation dynamics over the time interval ½0;1Þare given by

_ZðtÞ ¼ aðtÞ; Z040. (1)

The absence of a depreciation term in (1), and non-negativity of aðtÞX0, imply thatthe stock Z cannot decrease. Thus, our assumption is that the firm cannot decrease Z

by laying off employees ðao0Þ; neither does the stock Z decrease for the reason thatknowledge becomes obsolete. To model organizational forgetting one can subtract a

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term dZðtÞ, d ¼ const:40, on the right-hand side of (1). Then the firm’s 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 reflect 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 defined on a probability space ðO;F;Pað�ÞÞ and takes itsvalues in ½0;1Þ. The probability measure P depends on the control (althoughindirectly) in the following way:

Pað�ÞðT 2 ½t; tþ dtÞ j TXtÞ ¼ gZðtÞdtþ oðdtÞ, (2)

where oðdtÞ=dt! 0 uniformly in ðZ; aÞ for dt! 0. The term gZðtÞ on the right-handside of (2) is the stopping rate and g40 is a constant. The stopping rate is increasingin Z, and (2) implies

PrðT 2 ½0; tÞÞ ¼ 1� exp �gZ t

0

ZðsÞds

� �. (3)

Eq. (3) shows that the more intellectual capital that was accumulated during the timeinterval ½0; tÞ, the higher the probability of making the breakthrough by time t. Thus,the firm’s 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 firm’s current efforts.

Recall from (1) that current investments translate linearly into IC. However, dueto (3), there is a diminishing marginal efficiency of IC in increasing the stopping rate(since the probability in (3) grows in a concave fashion).

The specification in (2) also implies PrðT ¼ 1Þ ¼ exp½�gR10 ZðsÞds�. Hence, a

necessary and sufficient condition for having a finite completion time T is thatR t

0 ZðsÞds!þ1 for t!þ1. This is true, as Z0 is positive and Zð�Þ is non-decreasing.

2.4. Objective function

To construct the firm’s 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 firm?

As to the first question, we summarize the firm’s situation in a simple, butstandard, way. Let P ¼ 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 ZðTÞ, thestock of IC at the innovation date. Hence, the reward is the same whether the

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innovation is made by a small or a large firm (cf. Grossman and Shapiro, 1986). Asconcerns the exit time te of the venture capitalist, we distinguish two cases:

Case 1: te ¼ T()Tot.The reward P earned at time T is divided among the founders and the venture

capitalist such that the latter gets yðL0ÞP and the former gets the rest, ð1� yðL0ÞÞP,where yðL0Þ 2 ½0; 1Þ: The share yðL0Þ depends only on the amount of venture capitalprovided; thus the share is time-invariant and does not depend on the innovationtime T . We assume yð0Þ ¼ 0; y0ðL0Þ40; y00ðL0Þa0.

Case 2: te ¼ 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 ¼ t and receives,according to the redemption provision of the contract, an amount of bL0,b ¼ const:40. If b ¼ 1, the capitalist receives the principal amount of stock in thecompany. In the majority of venture capital contracts, however, b is greater thanone. 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 entrepreneur’s positive and constant rate of time preference by i, hisobjective functional, J, can now be constructed

Jðað�Þ;Z0Þ ¼ E að�ÞTot

Z t

0

e�it½p� aðtÞ�dtþ e�iT ð1� yðL0ÞÞP

� �

þ E að�ÞTXt

Z T

te�it½p� aðtÞ�dtþ e�iT P� e�itbL0

� �. ð4Þ

The objective is the maximization of the expected present value (at time zero) of theprofit stream p� aðtÞ and the reward P, minus the expected payment, yðL0ÞP or bL0,upon the exit of the venture capitalist.

A pair ðZ; aÞ is feasible if the objective value J is finite, state equation (1) issatisfied, and the control constraint aðtÞ 2 ½0; p� is fulfilled for t 2 ½0;1Þ.

3. Main results

This 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 Hamilton–Jacobi–Bellman 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 aninfinite horizon deterministic optimal control problem, which is equivalent to thestochastic optimal control problem with random stopping time that was stated in

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Section 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, ZðtÞ and Y ðtÞ, where Y ðtÞ issuch that e�Y ðtÞ 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 ðZt;Y tÞ ¼ ðZðtÞ;Y ðtÞÞ that results from using an optimalinvestment policy on the interval ½0; tÞ, we first solve a subproblem on the timeinterval ½t;1Þ. Denote this subproblem by P2 and define the state variable

Y ðtÞ ¼ Y t þ

Z t

tgZðsÞds,

in which Y t is a fixed but, so far, arbitrary real number. From (3) we now obtain thate�Y ðtÞ 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 fits in an infinite horizon deterministic optimal control problem. Theobjective functional of problem P2, denoted by J2, represents the expected presentvalue at time t ¼ t and is given by

J2ðað�Þ;Zt;Y tÞ ¼ Eað�Þ

Z T

te�iðt�tÞ½p� aðtÞ�dtþ e�iðT�tÞP� bL0

� �

¼ � bL0 þ eitþY t

Z 1t

e�½itþY ðtÞ�½p� aðtÞ þ PgZðtÞ�dt, ð5Þ

and the state dynamics are

_ZðtÞ ¼ aðtÞ; ZðtÞ ¼ Zt.

_Y ðtÞ ¼ gZðtÞ; Y ðtÞ ¼ Y t. ð6Þ

When a solution of P2 has been obtained, one can determine the optimal value of J2,which will be denoted by J�2ðZt;Y tÞ.

Then we solve the full problem on the time interval ½0;1Þ: This problem has thefollowing objective functional, when cast as a deterministic optimal control problem:

Jðað�Þ;Z0Þ ¼

Z t

0

e�½itþY ðtÞ�½p� aðtÞ þ ð1� yðL0ÞÞPgZðtÞ�dt

þ e�itJ�2ðZt;Y tÞ. ð7Þ

For t 2 ½0; tÞ, the dynamics of the problem are given by the state equations

_ZðtÞ ¼ aðtÞ; Zð0Þ ¼ Z0.

_Y ðtÞ ¼ gZðtÞ; Y ð0Þ ¼ 0: ð8Þ

The calculations will proceed as follows. Section 3.1 provides a solution of P2 andSection 3.2 solves the full problem.

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3.1. Solution of P2

Define costate variables l1ðtÞ; l2ðtÞ associated with state variables Z and Y,respectively. In (5), the term eitþY t is constant and can be disregarded in thecharacterization of an optimal solution. The Hamiltonian is

H ¼ e�ðitþY Þ½p� aþ PgZ� þ l1aþ l2gZ,

which is linear in the control a, and hence an optimal investment policy is bang–bangor singular. There are three candidate policies. In terms of the costate l1ðtÞ they canbe stated as

Dividend : aðtÞ ¼ 0¼)l1ðtÞeitþY ðtÞo1.

Investment : aðtÞ ¼ p¼)l1ðtÞeitþY ðtÞ41.

Singular : aðtÞ 2 ½0; p�¼)l1ðtÞeitþY ðtÞ ¼ 1. ð9Þ

The time function l1ðtÞeitþY ðtÞ 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 profit equals one. A dividend policy

pays 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 ¼ 0. Then the singular stock level,ZS, say, is constant and equals

ZS ¼1

g�i þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigðiP� pÞ

q� �. (10)

To have a positive ZS, we introduce the assumption

P4piþ

i

g. (11)

Note that (11) implies P4p=i. This inequality means that the expected rewardexceeds the present value of a perpetuity of p. Otherwise the firm would have noincentive to try to make the innovation.

The instant of time at which the stock ZðtÞ reaches the singular level ZS, whenstarting out in state Zt and using the investment policy, is given by

tS ¼ZS � Zt

p. (12)

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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 sufficiencyconditions based upon concavity of Hamiltonians.

Proposition 1. For any ZtXZ0, an optimal solution of P2 is

(i)
If ZtXZS then aðtÞ ¼ 0 for t 2 ½t;1Þ. (ii) If ZtoZS, then
aðtÞ ¼p

0

� �for t 2

½t; tSÞ

½tS;1Þ

( ).

The intuition for the zero-investment policy in Case (i) is that the stock level Zt is‘high’ (ZtXZS). Then no further investments are needed. (If Zt4ZS, the dividendpolicy is used. If Zt ¼ 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 ZðtÞ reaches the singular level ZS.

The policy stated in Proposition 1(ii) works as follows. Investing at the rate a ¼ 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 ¼ ZS the firm 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).Since there is no investment, the stock ZðtÞ is constant, equal to, say ~Z. Theelementary probability of the time interval ½t; tþ dtÞ for the stopping time T then is

g ~Ze�R t

tg ~Zds¼ g ~Ze�g

~Zðt�tÞ.

The expected present value of employing a zero-investment policy, as of time t, isgiven by

eitþY t

Z 1tfe�½itþY ðtÞ�½pþ Pg ~Z�dt� bL0 ¼

pþ Pg ~Z

i þ 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 investment

gðiP� pÞ

ði þ gZÞ2.

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Equating this expected marginal revenue to the marginal cost of investment, which isone, yields

gðiP� pÞ � ði þ gZÞ2 ¼ 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 ¼ ZS, expected marginal revenue equalsmarginal cost. This is a singular arc, along which a ¼ 0. Finally, if ZoZS,expected marginal revenue of investment exceeds marginal cost, at least duringsome initial interval of time, say, ½t; tþ D�. In that case, a dividend policy issuboptimal and investment is worthwhile. Thus, aðtÞ ¼ p on the time interval ðt; tS�.Investment is stopped at time tS because at this instant, the stock reaches the singularlevel ZS.

3.2. Complete solution of the investment problem

The objective functional of this problem, when cast as a deterministic optimalcontrol problem, is

Jðað�Þ;Z0Þ ¼

Z t

0

e�½itþY ðtÞ�½p� aðtÞ þ ð1� yðL0ÞÞPgZðtÞ�dtþ e�itJ�2ðZt;Y tÞ.

(15)

For t 2 ½0; tÞ, the objective to be maximized is the integral on the right-hand side of(15). The dynamics are

_ZðtÞ ¼ aðtÞ; Zð0Þ ¼ Z0.

_Y ðtÞ ¼ gZðtÞ; Y ð0Þ ¼ 0. ð16Þ

Denote this subproblem by P1 and note (also here) that one cannot verify sufficiencyconditions since the concavity requirements are not fulfilled.

The characterization of an optimal investment policy in problem P1 is similar tothat in P2 and it suffices to note the following. Defining a costate m1ðtÞ associatedwith state variable ZðtÞ, one can identify three candidate policies:

Investment : aðtÞ ¼ p¼)m1ðtÞeitþY ðtÞ41.

Dividend : aðtÞ ¼ 0¼)m1ðtÞeitþY ðtÞo1.

Singular : aðtÞ 2 ½0;p�¼)m1ðtÞeitþY ðtÞ ¼ 1.

An optimal solution of P1 is characterized in Proposition 2, in which the singularstock level is given by

ZSA ¼1

g�i þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigðið1� yðL0ÞÞP� pÞ

q� �. (17)

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To have a positive singular level ZSA, we strengthen the assumption in (11) byintroducing the hypothesis

P4i

gþ

piþ yðL0ÞP.

Note

ZSAoZS,

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 ZðtÞ reaches ZSA, when starting out fromZ0 and using the investment policy aðtÞ � p, is given by

tSA ¼ZSA � Z0

p. (18)

Proposition 2. Assume that ZSA � pt40. For any Z040, an optimal solution is

(i)
If Z0XZSA, then
aðtÞ ¼ 0 for t 2 ½0; tÞ.

(ii)
If ZSA � ptpZ0oZSA, then
aðtÞ ¼p

0

� �for t 2

½0; tSAÞ

½tSA; tÞ

( ).

(iii)
If Z0oZSA � tp, then
aðtÞ ¼ p for t 2 ½0; tÞ.

The intuition of Case (i) is that it occurs if the initial stock Z0 is already sufficientlyhigh; 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 insufficient and investments are used toincrease it (maximally). In Case (iii), investments continue throughout the timeinterval ½0; tÞ. 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 that

tSA

4

¼

o

8><>:

9>=>;t () Z0

o¼

4

8><>:

9>=>;ZSA � pt. (19)

In Case (iii) we have tSA4t()Z0oZSA � pt. This means that even wheninvestment is done at the maximal rate p during the entire time interval ½0; tÞ, it

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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 interval½0; tÞ would overshoot the singular level. Therefore, investment goes on only duringthe time interval ½0; tSAÞ. 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, then
aðtÞ ¼ 0 for t 2 ½0;1Þ.

(ii)
If ZSApZ0oZS, then
aðtÞ ¼

0

p

0

8><>:

9>=>; for t 2

½0; tÞ

½t; tSÞ

½tS;1Þ

8><>:

9>=>;.

(iii)
If ZSA � ptoZ0oZSA, then
aðtÞ ¼

p

0

p

0

8>>><>>>:

9>>>=>>>;

for t 2

½0; tSAÞ

½tSA; tÞ

½t; tSÞ

½tS;1Þ

8>>>><>>>>:

9>>>>=>>>>;.

(iv)
If Z0pZSA � pt, then
aðtÞ ¼p

0

� �for t 2

½0; tSÞ

½tS;1Þ

( ).

Proposition 3 is derived from Propositions 1 and 2. The four cases in Proposition 3are ordered in accordance with increasing investment efforts, or, equivalently,decreasing initial stocks of IC.

In Case (i), the initial stock level Z0 is sufficiently 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 Z0

exceeds 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. Theninvestment is resumed and is continued until the long-term singular stock level is

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reached. 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 reflectsthe fact that the firm’s past efforts contribute to its chances of making thediscovery. Due to that effect, the firm 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 policy

The derivations of Section 3 proceeded under the assumption that the amount ofventure capital, L0, was fixed. It remains to discuss the choice of L0. This amount canbe seen as a parameter, determined at time zero, that influences 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 ¼

X 0 þ L0 shows that the venture capital decision will influence 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 capitalist’sshare, yðL0Þ, 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 firm does not invest at all.The intuition is that the amount of venture capital attracted, and used in theinitial acquisition of IC, is sufficiently large. In Cases (ii)–(iv) there is an initialphase 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 JðL0Þ the firm’s payoff over the time interval ½0;1Þ. Using (7) and (15)yields

JðL0Þ ¼

Z t

0

e�½itþY ðtÞ�½p� aðtÞ þ ð1� yðL0ÞÞPgZðtÞ�dt

þ e�it eitþY t

Z 1t

e�½itþY ðtÞ�½p� aðtÞ þ PgZðtÞ�dt� bL0

� �. ð20Þ

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To 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 by

y0ðL0Þ ¼1

K½l1ð0Þ � be�it�, (21)

in which

K ¼

Z t

0

e�½itþY ðtÞ�PgZðtÞdt40.

The condition in (21) compares the costate l1ð0Þ (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 capitalfinancing should be attracted in accordance with (21). On the other hand, ifbe�it

Xl1ð0Þ, no capital should be attracted.The optimality condition in (21) can be rewritten to yield

y0ðL0ÞK þ be�it ¼ l1ð0Þ.

Hence, the optimal amount of venture capital financing requires that the marginalexpected value of the firm 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 ¼ 0 is more likely for t small(since the condition in (21) then can be satisfied by a larger range of b’s). 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 ¼ 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 yðL0Þ is convex. To be specific, let yðL0Þ ¼12L20.

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 capitalist’s investment.

Using (21) yields L0 ¼ ðl1ð0Þ � be�itÞ=K . In this expression, a direct and anindirect effect can be identified. The direct effect is that the amount of venture capitaldecreases if the reward P increases. The reason is that a marginal increase in L0 leads

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to a larger monetary loss, y0ðL0ÞP, 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 reflected 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 l1ð0Þ is sufficiently large to influence positively the amount of venturecapital L0 (cf. (21)).

5. Concluding remarks

The paper has considered a dynamic optimization problem of a firm 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 financed 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 bang–bang 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 firm’s 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 firm 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 finding in the IT area is subject to allkinds of developments in the economic environment that cannot be foreseen.This can be modeled by introducing a random diffusion on the market valueof 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.,Reinganum, 1981, 1982; Haurie, 1994; Dockner et al., 2000).

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Acknowledgement

The authors like to thank two anonymous referees and editor Carl Chiarella fortheir helpful comments.

Appendix

Proof of Proposition 1. In (5), the term eitþY t is constant and can be disregarded inthe characterization of an optimal solution. Let l1ðtÞ; l2ðtÞ be costate variablesassociated with state variables Z and Y ; respectively, and Z1ðtÞ; Z2ðtÞ be Lagrangianmultipliers associated with the control constraint aðtÞ 2 ½0; p�. The Hamiltonian is

H ¼ e�ðitþY Þ½p� aþ PgZ� þ l1aþ l2gZ

and the Lagrangian then becomes

L ¼ H þ Z1aþ Z2ðp� aÞ.

Suppose that there exists an optimal solution of problem P2. Necessary optimalityconditions then are

l1ðtÞ ¼ Z2ðtÞ � Z1ðtÞ þ e�ðitþY ðtÞÞ, (A.1)

_l1ðtÞ ¼ �e�ðitþY ðtÞÞgP� gl2ðtÞ, (A.2)

_l2ðtÞ ¼ e�ðitþY ðtÞÞ½p� aðtÞ þ PgZðtÞ�, (A.3)

Z1ðtÞaðtÞ ¼ 0; aðtÞX0; Z1ðtÞX0,

Z2ðtÞðp� aðtÞÞ ¼ 0; aðtÞpp; Z2ðtÞX0,

limt!1

l2ðtÞ ¼ 0. (A.4)

Due to the linearity of the Hamiltonian in the control, an optimal investmentpolicy is bang–bang or singular. Clearly, both Z1 and Z2 cannot be positive and hencethree candidate policies remain. Characterized by the signs of the Lagrangianmultipliers they are

Investment (a ¼ p)
Dividend (a ¼ 0) Singular (a ¼ 0)
Z1
0 þ 0 Z2 þ 0 0
The singular investment policy is indeterminate in the interval ½0; p� and we choosea ¼ 0. The stock level ZS associated with this singular policy is constant and given by(10). The instant of time at which the stock ZðtÞ reaches the singular level ZS, when

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starting out in state Zt, is given by

tS ¼ZS � Zt

p.

In what follows we characterize in detail the three candidate policies.

Investment policy (Path 1):Z1ðtÞ ¼ 0; Z2ðtÞ40; _ZðtÞ ¼ aðtÞ ¼ p:From (A.1) and (A.3) it follows that

l1ðtÞ ¼ e�ðitþY ðtÞÞ þ Z2ðtÞ, (A.5)

_l2ðtÞ ¼ e�ðitþY ðtÞÞPgZðtÞ. (A.6)

Differentiating l1ðtÞ in (A.5) twice with respect to time yields

_l1ðtÞ ¼ �e�ðitþY ðtÞÞ½i þ gZðtÞ� þ _Z2ðtÞ, (A.7)

€l1ðtÞ ¼ e�ðitþY ðtÞÞ½ði þ gZðtÞÞ2 � gp� þ €Z2ðtÞ. (A.8)

Differentiating in (A.2) with respect to time, and using (A.6) and (A.8), provides

€l1ðtÞ ¼ e�ðitþY ðtÞÞiPg,

€Z2ðtÞ ¼ e�ðitþY ðtÞÞfgðiPþ pÞ � ½i þ gZðtÞ�2g. ðA:9Þ

Dividend policy (Path 2):Z2ðtÞ ¼ 0; Z1ðtÞ40; _ZðtÞ ¼ aðtÞ ¼ 0; ZðtÞ ¼ ZD ¼ const:40.From (A.1) and (A.3) it follows that

l1ðtÞ ¼ e�ðitþY ðtÞÞ � Z1ðtÞ, (A.10)

_l2ðtÞ ¼ e�ðitþY ðtÞÞ½pþ PgZD�. (A.11)

Differentiating l1ðtÞ in (A.10) twice with respect to time yields

_l1ðtÞ ¼ �e�ðitþY ðtÞÞ½i þ gZD� � _Z1ðtÞ, (A.12)

€l1ðtÞ ¼ e�ðitþY ðtÞÞ½i þ gZD�2 � €Z1ðtÞ. (A.13)

Differentiating in (A.2) with respect to time and using (A.11) yields

€l1ðtÞ ¼ e�ðitþY ðtÞÞgðiP� pÞ

and using (A.13) then provides

e�ðitþY ðtÞÞ½i þ gZD�2 � €Z1ðtÞ ¼ e�ðitþY ðtÞÞgðiP� pÞ¼) €Z1ðtÞ

¼ e�ðitþY ðtÞÞ½ði þ gZDÞ2� gðiP� pÞ�. ðA:14Þ

Singular policy (Path 3):Z1ðtÞ ¼ Z2ðtÞ ¼ 0; _ZðtÞ ¼ aðtÞ ¼ 0; ZðtÞ ¼ ZS ¼ const:40.

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The characterizations of costates and Lagrangian multipliers follow from those ofPath 2, by putting Z1ðtÞ equal to zero. Thus

l1ðtÞ ¼ e�ðitþY ðtÞÞ, (A.15)

_l2ðtÞ ¼ e�ðitþY ðtÞÞ½pþ PgZS�, (A.16)

_l1ðtÞ ¼ �e�ðitþY Þði þ gZSÞ, (A.17)

gðiP� pÞ � ði þ gZSÞ2¼ 0 () ZS ¼

1

g�i þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigðiP� pÞ

q� �. (A.18)

We construct a solution that satisfies the necessary optimality conditions, by usinga path-coupling procedure which starts by considering Path 2 or Path 3 as a candidatefor being a final path. A final path is one that is employed as of some instant of timetill infinity. Path 1 will not be considered a candidate for being a final 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 final 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 l1ðtÞ is always non-negative. Note that (A.10) implies

limt!1

l1ðtÞ ¼ 0, (A.19)

limt!1

Z1ðtÞ ¼ 0 (A.20)

along Path 2.Now suppose that Path k (k ¼ 1 or 3) were to precede Path 2. Then, at the

coupling point t between Path k and Path 2, it must be true that

Z1ðtÞ ¼ 0,

which follows from the required continuity of l1ðtÞ; and using (A.5), (A.15). By definition,Z1ðtÞ is positive on Path 2 for t 2 ðt;1Þ. Then, using Z1ðtÞ ¼ 0 and (A.20), it follows thatZ1ðtÞ must have at least one maximum along Path 2. In such a point we have €Z1ðtÞo0.From (A.14) and the fact that ZðtÞ ¼ const: ¼ ZD along Path 2, one therefore obtains

gðiP� pÞ � ði þ gZDÞ240

everywhere along Path 2. Hence, by (A.14) it holds that €Z1ðtÞo0 everywhere on Path2. This means that _Z1ðtÞ 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

_Z1ðtÞ ¼ e�ðitþY ðtÞÞ½Pg� ði þ gZDÞ� þ gl2ðtÞ ¼) limt!1

_Z1ðtÞ ¼ 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. &

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Lemma 5. When Path 2 is used for t 2 ðt;1Þ it holds that

gðiP� pÞ � ði þ gZDÞ2o0.

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 €Z1ðtÞ40 everywherealong Path 2, which via (A.14) leads to the result. &

Next, suppose that Path 3 is the final path, in which (A.18) is satisfied 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! Path

3, where on Path 1 it holds that

gðiP� pÞ � ði þ gZðtÞÞ240.

Proof. From (A.9) and (A.18) one obtains €Z2ðt�13Þ40, where t13 is the moment of

time at which Path 1 passes into Path 3. Since Z1ðtÞ increases on Path 1, it followsthat

€Z2ðtÞ40

everywhere along Path 1.From (A.5), (A.15), continuity of l1ðtÞ; and the fact that Z2ðtÞ40 along Path 1, it

follows that

Z2ðt�13Þ ¼ 0, (A.22)

_Z2ðt�13Þp0. (A.23)

Now, €Z2ðtÞ40 on Path 1, (A.22) and (A.23) imply that Z2ðtÞ 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 Z2ð~tÞ ¼ 0 sinceotherwise continuity l1ðtÞ would be violated. However, this is incompatible with(A.22) and the fact that Z2ðtÞ 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 ZðtÞ 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 sinceZðtÞ is constant on both paths. This implies, using (A.14), that €Z1ðtÞ ¼ 0 along Path 2and hence _Z1ðtÞ is constant.

Now, continuity of l1ðtÞ requires that just before a coupling point t23 betweenPaths 2 and 3 it must hold that Z1ðt

�23Þ ¼ 0, cf. (A.10) and (A.15). Hence _Z1ðt

�23Þ must

be negative, and since _Z1 is constant, it is negative everywhere along Path 2.From (A.2) follows that _l1ðtÞ 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. &

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We summarize the results of the above lemmas as follows. Consider problem P2.Sequences of paths that satisfy the necessary optimality conditions are the following:

gðiP� pÞ � ði þ gZtÞ2o0 () Zt4ZS: Path 2.

gðiP� pÞ � ði þ gZtÞ2¼ 0 () Zt ¼ ZS: Path 3.

gðiP� pÞ � ði þ gZtÞ240 () ZtoZS : Path 1! Path 3.

The two first sequences generate the control path aðtÞ ¼ 0 for t 2 ½t;1Þ. The last onegenerates the control path aðtÞ ¼ p for t 2 ½t; tSÞ, aðtÞ ¼ 0 for t 2 ½tS;1Þ. Thiscompletes the proof of Proposition 1. &

Proof of Proposition 2. Essentially, a proof of this proposition is superfluous; inmost respects it would be similar to that of Proposition 1.

Proposition 1deals with an infinite horizon problem with a fixed 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 firm invests maximally to reach thesingular stock level at time tS; due to the infinite 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 afinite horizon problem with a fixed 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 firm investsmaximally to try to reach the singular stock level at time tSA. However, due to thefinite 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 ½tSA; tÞ. 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 functions

f ðZÞ ¼ gðiP� pÞ � ði þ gZðtÞÞ2,

f AðZÞ ¼ gðið1� yÞP� pÞ � ði þ gZðtÞÞ2,

which both are decreasing and strictly concave. Note that

f ðZÞ

4

¼

o

8>><>>:

9>>=>>;0 () ZðtÞ

o

¼

4

8>><>>:

9>>=>>;ZS

f AðZÞ

4

¼

o

8>><>>:

9>>=>>;0 () ZðtÞ

o

¼

4

8>><>>:

9>>=>>;ZSA,

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and recall that ZSAoZS. It is convenient to use this notation to summarize theresults of Propositions 1 and 2:

Proposition 1.

(i)
If f ðZtÞp0, then aðtÞ ¼ 0 for t 2 ½t;1Þ. (ii) If f ðZtÞ40, then aðtÞ ¼ p for t 2 ½t; tSÞ, aðtÞ ¼ 0 for t 2 ½tS;1Þ.
Proposition 2. Let Zpt9Z0 þ pt.

(i)
If f AðZ0Þp0, then aðtÞ ¼ 0 for t 2 ½0; tÞ. (ii) If f AðZ0Þ40 and Zpt4ZSA, then aðtÞ ¼ p for t 2 ½0; tSAÞ, aðtÞ ¼ 0 for t 2 ½tSA; tÞ. (iii) If f AðZ0Þ40 and ZptpZSA, then aðtÞ ¼ p for t 2 ½0; tÞ.
Now we can prove Proposition 3.Case (i): The assumption is f ðZ0Þp0, which implies f AðZ0Þo0. Hence, by

Proposition 2(i), we have aðtÞ ¼ 0 for t 2 ½0; tÞ, which implies Zt ¼ Z0. Then, by theassumption of Case (i), one obtains f ðZtÞ ¼ f ðZ0Þp0. Using Proposition 1(i) thenyields aðtÞ ¼ 0 for t 2 ½t;1Þ. The proof is complete.

Case (ii): The assumption is f AðZ0Þp0 and f ðZ0Þ40, The first inequality implies,by Proposition 2(i), that aðtÞ ¼ 0 for t 2 ½0; tÞ. Then one has Zt ¼ Z0, which impliesf ðZtÞ ¼ f ðZ0Þ40. Using Proposition 1(ii) then yields aðtÞ ¼ p for t 2 ½t; tSÞ andaðtÞ ¼ 0 for t 2 ½tS;1Þ. The proof is complete.

Case (iii): The assumption is f AðZ0Þ40 and Z04ZSA � pt. The first inequalityimplies, by Proposition 2(ii), that aðtÞ ¼ p for t 2 ½0; tSAÞ and aðtÞ ¼ 0 for t 2 ½tSA; tÞ.Then Zt ¼ ZSA in Proposition 1 and hence f ðZtÞ ¼ f ðZSAÞ. However, f ðZSAÞ40,and using Proposition 1(ii) one obtains aðtÞ ¼ p for t 2 ½t; tSÞ and aðtÞ ¼ 0 fort 2 ½tS;1Þ. The proof is complete.

Case (iv): The assumption is f AðZ0Þ40. By Proposition 2(iii) we get aðtÞ ¼ p fort 2 ½0; tÞ. This implies ZtoZSAoZS, and hence f ðZtÞ4f ðZSÞ ¼ 0. Then, byProposition 1(ii), we get aðtÞ ¼ p for t 2 ½t; tSÞ and aðtÞ ¼ 0 for t 2 ½tS:1Þ. Theproof is complete. &

References

Admati, A.R., Pfleiderer, P., 1994. Robust financial contracting and the role of venture capitalists. Journal

of Finance 49 (2), 371–402.

Amit, R., Brander, J., Zott, C., 2000. Venture capital financing of entrepreneurship: theory empirical

evidence and a research agenda. In: Sexton, D., Landstrom, H. (Eds.), The Blackwell Handbook of

Entrepreneurship. Blackwell Business, pp. 259–282.

Anderson, W.R., Sundaresan, S., 1996. Design and valuation of debt contracts. Review of Financial

Studies 9 (1), 37–68.

Bensoussan, A., Lesourne, J., 1981. Optimal growth of a firm facing a risk of bankruptcy. INFOR 19 (4),

292–310.

Bergemann, D., Hege, U., 1997. Venture capital financing, moral hazard, and learning. Journal of

Banking and Finance 22, 703–735.

Boukas, E.K., Haurie, A., Michel, P., 1990. An optimal control problem with a random stopping time.

Journal of Optimization Theory and Applications 64 (3), 471–480.

ARTICLE IN PRESS


Brander, J.A., Lewis, T.R., 1986. Oligopoly and financial structure: the limited liability effect. American

Economic Review 75, 956–970.

Carlson, D.A., Haurie, A.B., Leizarowitz, A., 1991. Infinite Horizon Optimal Control: Deterministic and

Stochastic Systems. Springer, Berlin.

Dockner, E.J., Jørgensen, S., Van Long, N., Sorger, G., 2000. Differential Games in Economics and

Management Science. Cambridge University Press, Cambridge, UK.

Doraszelski, J., 2003. An R&D race with knowledge accumulation. Rand Journal of Economics 34, 20–42.

Drucker, P., 1995. The information executives truly need. Harvard Business Review, January–February,

54–62.

Fudenberg, D., Gilbert, G., Stiglitz, J., Tirole, J., 1983. Preemption, leapfrogging and competition in

patent races. European Economic Review 22, 3–31.

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: Empirical

Corporate Finance. North-Holland, Amsterdam, forthcoming.

Grossman, G.M., Shapiro, C., 1986. Optimal dynamic R&D programs. Rand Journal of Economics 17

(4), 581–593.

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. 90–108.

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 empirical

analysis of venture capital contracts. Review of Economic Studies 70, 281–315.

McAdam, R., McCreedy, S., 1999. A critical review of knowledge management models. The Learning

Organization 6 (3), 91–100.

Meng, R., 2004. A patent race in a real options setting: investment strategy, valuation, CAPM beta and

return volatility. Working Paper, The University of Hong Kong.

Reinganum, J., 1981. Dynamic games of innovation. Journal of Economic Theory 25, 21–41.

Reinganum, J., 1982. A dynamic game of R&D: patent protection and competitive behavior.

Econometrica 50, 671–688.

Sahlman, W.A., 1988. Aspects of financial contracting in venture capital. Journal of Applied Corporate

Finance 1, 23–36.

Sahlman, W.A., 1990. The structure and governance of venture-capital organizations. Journal of Financial

Economics 27, 473–521.

Schwartz, E.S., Zozaya-Gorostiza, C., 2003. Investment under uncertainty in information technology:

acquisition and development projects. Management Science 49, 57–70.

Schworm, W.E., 1980. Financial constraints and capital accumulation. International Economic Review

21, 643–660.

Sorger, G., 1991. Maximum principle for control problems with uncertain horizon and variable discount

rate. Journal of Optimization Theory and Applications 70 (3), 607–618.

Zingales, L., 2000. In search for new foundations. Journal of Finance 55 (4), 1623–1653.

Further reading

Bensoussan, A., Lesourne, J., 1980. Optimal growth of a self-financing firm in an uncertain environment.

In: Bensoussan, A., Kleindorfer, P., Tapiero, C.S. (Eds.), Applied Stochastic Control in Econometrics

and Management Science. North-Holland, Amsterdam, pp. 235–269.

Reisinger, H., Dockner, E.J., Baldauf, A., 2000. Examining the interaction of marketing and financing

decisions in a dynamic environment. OR Spektrum 22, 159–171.

venture capital financed investments in intellectual capital

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