Contents lists available at ScienceDirect

Journal of Economic Dynamics & Control

Journal of Economic Dynamics & Control 35 (2011) 462–478

0165-18

doi:10.1

� Cor

E-m

journal homepage: www.elsevier.com/locate/jedc

Two state capital accumulation with heterogenous products:Disruptive vs. non-disruptive goods

Jonathan P. Caulkins a,�, Gustav Feichtinger b, Dieter Grass b,Richard F. Hartl c, Peter M. Kort d,e

a Carnegie Mellon University, Heinz College and Qatar Campus, 5000 Forbes Ave., Pittsburgh, PA 15237, USAb Technical University Vienna, Austriac University of Vienna, Austriad Tilburg University, the Netherlandse University of Antwerp, Belgium

a r t i c l e i n f o

Article history:

Received 15 April 2009

Accepted 14 September 2010Available online 22 September 2010

JEL classification:

C61

D24

D92

L10

Keywords:

Investment

Product innovation

Maximum principle

Skiba curve

89/$ - see front matter & 2010 Elsevier B.V. A

016/j.jedc.2010.09.008

responding author.

ail address: caulkins@andrew.cmu.edu (J.P. C

a b s t r a c t

The paper considers the problem of a firm that, while producing a standard product, has

the option to introduce an innovative product. The innovative product competes with

the standard product and will therefore reduce revenues of the standard product. A

distinction is made between innovative products that do or do not become even

more relatively appealing as their market share grows (e.g., because of network

externalities). It is shown that in the former case, which we call a ‘‘disruptive’’ good,

history dependent long run equilibria can occur, which are in line with recent real life

economic examples.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

This paper considers the problem of a firm having the option to produce different products for the same market.One product is an old fashioned ‘‘standard’’ that is already established. The other product is a recent innovation, which isqualitatively better and thus brings strong competition to the older product. At the same time, producing this innovativeproduct is more complicated for the firm, because it requires skilled labor, advanced technology, or other inputs that are inlimited supply. An example is Philips, which had produced the ‘‘standard’’ CRT television sets for decades, when in 1999 itstarted a joint venture with its Korean partner LG to produce LCD (flatscreen) television sets. Another example (TheEconomist, May 20th, 2006, p. 75) is Neumann, a traditional microphone producer that is investing 1.4 m Euro to developdigital microphones. In a similar fashion, Motorola is developing WiMax, an emerging wireless standard that will linkmobile devices to the internet at broadband speeds. In a way this is surprising, since WiMax is widely seen as a challengerto existing mobile phone technologies (The Economist, October 7th, 2006, p. 74).

Microsoft faces an analogous problem (see, e.g., The Economist, April 1st, 2006, pp. 55–56). Currently, Microsoftdominates the PC market, but it knows that its position is under threat from online applications, which are changing how

ll rights reserved.

aulkins).

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478 463

people use these computers. Rather than relying on an operating system, and its associated application software—boughtin a box from Microsoft and then loaded onto a PC—computer users are increasingly able to call up the software theyneed over the internet. Software companies are now selling software as a subscription service that can be accessed via aweb-browser. Salesforce.com, an example of this trend, offers salesforce management tools; other firms offer accountingand other back-office functions.

Since Microsoft’s two main products—Windows and Office—remain fabulously profitable quasi-monopolies, Microsoftis still in a position that most firms would kill for. However, these online applications clearly threaten the way Microsoftmakes money. For that reason Microsoft is reorienting itself by adding an online component into virtually all its products.The first steps in that direction are ‘‘Windows Live’’ and ‘‘Office Live’’. The difficulty will be getting the timing right, so thatnew products do not undermine existing cash cows.

The aim of this paper is to establish the optimal speed at which the firm should start bringing the new product tomarket, given that it is already actively producing a standard good. In other words, we try to answer the question of howquickly to introduce an upgraded product that will cannibalize part of an existing product’s market. The paper is original inthat it looks at the question in continuous time and with state variables for two capital stocks, each of them governing theproduction processes of the two products.

We are interested in understanding not only whether a firm should introduce the new product but also what is the idealrate at which to introduce it and how that affects sales of the standard product. Hence, we need to employ an optimalcontrol model to focus on ‘‘dynamics’’. This inevitably raises complexities relative to a discrete, two period model, so wefocus on the case of a monopolist or someone for whom pricing drives innovation but not competition. This approachyields an extra benefit. As the examples in this introduction reflect, competition is often invoked as the reason forintroducing a new product, but we will see that even a monopolist can benefit from product innovation and, hence, facestructurally similar choices.

In this paper we make a distinction between two types of goods. One type behaves according to a usual linear demandstructure. The second, which we call a disruptive good, has the characteristic that small volumes have negligible influence butthe consumer utility of using the standard product significantly decreases when the disruptive good is dominantly present inthe market. A typical example would involve network externalities. When VHS became common, films stopped being releasedon Betamax so the utility of having a Betamax VCR declined once the suppliers of complementary products focused efforts onserving people using the new technology. A contemporary example is the use of a mobile phone as a way to pay. TheEconomist (February 17th, 2007, pp. 67–70), notes that this ‘‘new system could prove to be a ‘disruptive technology’. Bankscould be ‘disintermediated’ if, say, the payments for the train ticket, newspaper and coffee made every day by a commuterwith his mobile phone appear not on their monthly credit-card statements, but on those of a mobile-phone service provider.’’The basic punchline will be that non-disruptive goods can coexist or not, depending smoothly on the parameter values, butthat with disruptive goods this coexistence can also depend on the size of the firm’s existing investment in the standardproduct. It turns out that the presence of non-disruptive goods leads to saddle point behavior, while disruptive goods cangenerate a history dependent (Skiba) solution.

Our inquiry is motivated by the case of product innovation, where the initial stock of capital dedicated to producing thenew product is zero. However, we will also provide solutions for other initial values of this capital stock. This generality isrelevant if, for example, a firm producing two different products that are imperfect substitutes experiences unanticipatedprocess innovation that alters one or more relevant problem parameters and needs to know how optimally to steer to anew equilibrium.

The analysis rests on two streams of literature. First we have the static heterogeneous product market games as in Dixit(1979), Singh and Vives (1984), Vives (1985) and Qiu (1997). These papers consider two firms competing either in a Cournot orBertrand fashion, and use the same linear demand structure as we adopt in our model for non-disruptive goods. However, ourpaper differs in that we have a dynamic framework where one firm produces two different goods. In other words, whereas thejust described literature focuses on competition in a static context, we study a monopolist in a dynamic framework.

Second, we have the literature on capital accumulation, e.g., Lucas (1967), Davidson and Harris (1981), Dechert (1983) andBarucci (1998). In these models the firm is the decision maker. This firm owns one capital stock, which can be increased byinvestment. The capital stock, which is the only state variable in the model, is employed to produce a single homogeneous good.Due to a non-concavity in the revenue function, Davidson and Harris (1981) and Dechert (1983) obtain history dependentequilibria, i.e., two different long run steady states are derived, and the initial capital stock level determines which steady stateis optimal in the long run. The two domains of attraction are separated by a Skiba (1978) point. Haunschmied et al. (2003)extend this literature by considering a two state model, where the second state is the amount of investment. Now, since thestate space has two dimensions, the non-concavity in the revenue function adopted in that paper leads to the existence of aSkiba curve separating the domains of attraction of initial levels of investment and capital stock. While all these contributionshave firms producing identical goods, our paper considers a firm producing two different goods. For each good a differentcapital stock is needed. The paper shows that the resulting two state model, with the two capital stocks being the two states,also can lead to the existence of a Skiba curve, as in Haunschmied et al. (2003). However, instead of being caused by non-concave revenue, in the present paper history dependence results from the interaction terms in the two demand functions.

The paper is organized as follows. Section 2 presents our general framework. This framework is directed towards non-disruptive goods in Section 3 and towards disruptive goods in Section 4. Section 5 concludes. Proofs of propositions can befound in Appendix A.

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478464

2. General model

There are S identical consumers who all have the same utility function U. We start out very generally by introducingpositive constants y,g1,g2,Z,a,b in this utility function in the following way:

Uðq1,q2Þ ¼ q1þyq2�ðg1q2

1þg2q22þ2Zqa

1qb2Þ

2þM

in which q1 is the quantity of the standard product, q2 is the quantity of the innovative product, and M=Y�p1q1�p2q2

denotes the expenditure on outside goods. We assume that each consumer spends only a small part of her income on theindustry’s products. Since the representative consumer has utility linear in income, this implies that income effects on theindustry under consideration can be ignored and partial equilibrium analysis can be applied (Vives, 1999, p. 145;Symeonidis, 2003; Dixit, 1979, p. 22).

Via the first order conditions for the maximization of U(q1,q2) with respect to q1 and q2, this utility function gives rise tothe following demand structure, where inverse demand functions are given by

p1 ¼ 1�g1q1�aZqa�11 qb

2 and

p2 ¼ y�g2q2�bZqa1qb�1

2 :

In this way we make sure that consumer utility is maximized at each point of time.Following Dixit (1979), we assume

y41

to capture the idea that the innovative product has an absolute advantage in demand.State variable K1 is the input to the production process of the standard product, while K2 is the input to the innovative

product. We have spoken of product 2 as being more ‘‘complicated’’ than the standard product in the sense that productionrequires a scarce resource such as skilled labor or advanced technology. This implies a decreasing returns to scaleproduction function. For simplicity we assume that this production function is given by

q2ðK2Þ ¼ bK1=22 ,

where b is a positive constant. In contrast, since the production process of the standard product is not complicated,components can easily be obtained in any volume, e.g., by outsourcing, so we have a linear production function (with abeing a positive constant):

q1ðK1Þ ¼ aK1:

Capital stock Ki (i=1,2) can be increased by investing Ii and it decreases because of depreciation diKi, where di (constant andgreater than zero) is the depreciation rate:

_K 1 ¼ I1�d1K1 and

_K 2 ¼ I2�d2K2:

Investments are modeled as irreversible (for economic arguments supporting this assumption, see, e.g., Dixit and Pindyck,1994); so

I1Z0 and

I2Z0:

We further introduce investment costs CiðIÞ ¼12I2

i (i=1,2) and a discount rate r. The firm maximizes profits, which leads tothe following objective function:

maxI1 ,I2

Z 10

e�rt p1aK1þp2bK1=22 �

1

2I21�

1

2I22

� �dt

in which

p1 ¼ 1�g1aK1�aZaa�1bbK ða�1Þ1 K ð1=2Þb

2 ,

p2 ¼ y�g2bK1=22 �bZaabb�1Ka

1K ð1=2Þðb�1Þ2 :

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478 465

3. Standard goods

3.1. The model

Here we set a=b=1. This results in the familiar quadratic consumer utility function:

Uðq1,q2Þ ¼ q1þyq2�ðg1q2

1þg2q22þ2Zq1q2Þ

2þM

and linear inverse demand functions for products 1 and 2:

p1 ¼ 1�g1q1�Zq2,

p2 ¼ y�g2q2�Zq1:

According to Symeonidis (2003), the parameter Z, Z 2 ð0,1Þ, is an (exogenous) inverse measure of the degree of horizontalproduct differentiation, see also Qiu (1997). In the limit as Z-0, the goods become independent. On the other hand, in thelimit as Z-g, they become perfect substitutes, when g¼ g1 ¼ g2 and y¼ 1. Vives (1999, p. 145) adds that ‘‘the goods aresubstitutes, independent, or complements according to whether g is positive, zero, or negative’’; see also Dixit (1979). Inour case, we have substitutes, which is why Z40. Since we do not have not perfect substitutes, we assume that Z issignificantly less than 1.

Concavity of the consumer utility function requires that (see, e.g., Dixit, 1979)

g140, g240, Z2rg1g2: ð1Þ

In general it makes sense that price is more sensitive to the quantity of its own product, so that

0rZrgi for i¼ 1,2:

This leads to the following dynamic model of the firm (with states K1 and K2 and controls I1, I2):

max

Z 10

e�rt ð1�g1q1�Zq2Þq1þðy�g2q2�Zq1Þq2�1

2I21�

1

2I22

� �dt,

q1 ¼ q1ðK1Þ ¼ aK1, q2 ¼ q2ðK2Þ ¼ bK1=22 ,

_K 1 ¼ I1�d1K1,

_K 2 ¼ I2�d2K2,

I1Z0,

I2Z0:

3.2. Analysis

To determine the necessary optimality conditions we first write down the current value Hamiltonian:

H¼ 1�g1aK1�ZbK1=22

� �aK1þ y�g2bK1=2

2 �ZaK1

� �bK1=2

2

�1

2I21�

1

2I22þl1ðI1�d1K1Þþl2ðI2�d2K2Þ

while taking into account the irreversibility constraints we have the Lagrangian:

L¼Hþn1I1þn2I2:

Since HI1I1o0, HI2I2

o0, and HI1I2¼ 0, maximization of the Hamiltonian w.r.t. the investment rates yields a unique solution.

It follows that Ii is continuous over time (see Feichtinger and Hartl, 1986, Corollary 6.2). From the maximum principle it isobtained that

@L=@Ii ¼�Iiþliþni ¼ 0, ð2Þ

with complementary slackness conditions

niIi ¼ 0, niZ0: ð3Þ

The adjoint equations are

_l1 ¼ ðrþd1Þl1�ð1�2g1aK1�2ZbK1=22 Þa, ð4Þ

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478466

_l2 ¼ ðrþd2Þl2�ð12y�g2bK1=2

2 �ZaK1ÞbK�1=22 : ð5Þ

Note that the necessary conditions are only sufficient for optimality if the Hamiltonian is concave overall in the state(K1,K2). This is only the case for sufficiently low values of the states K1 and K2 since we have

HK1K1¼�2g1a2o0, ð6Þ

HK2K2¼�1

4bK�3=22 ðy�2ZaK1Þ, ð7Þ

HK1K1HK2K2

�H2K1K2¼ 1

2g1a2bK�3=22 ðy�2ZaK1Þ�Z2b2a2K�1

2 , ð8Þ

where the last two expressions have the required signs for K1 and K2 sufficiently low.While for finite horizon optimal control problems there are useful existence theorems, these are more complicated and

harder to verify for infinite horizon problems. So we rely on economic intuition to argue that an optimal solution exists. Inparticular, considering the objective function with quadratic investment costs and demand negatively depending on thecapital stocks, it is obvious that neither the states nor the controls can diverge. Since they have to be non-negative, they arebounded. This implies that the trajectories of the canonical system converge to some limit set. These limit sets are steadystates and since we consider the solutions converging to every occurring steady state we get the optimal solution bycomparing the objective values.

On an interior arc investments are strictly positive. Concerning the occurrence of steady states and their stability thefollowing proposition is proved in Appendix A.

Proposition 1. The interior arc contains at most one steady state, from which K2 results from the root of the following third

order polynomial in K1=22 :

K3=22 þ

b2

ðrþd2Þd2g2�

2Z2a2

ðrþd1Þd1þ2a2g1

� �K1=2

2 þb

ðrþd2Þd2

Za2

ðrþd1Þd1þ2a2g1

�1

2y

� �¼ 0: ð9Þ

K1 can then be determined from

K1 ¼a�2ZabK1=2

2

ðrþd1Þd1þ2a2g1

:

The Hamiltonian being concave in this steady state is a sufficient condition for the steady state being a saddle point.

For the boundary arc I1=0 the following proposition can be established, which is proved in Appendix A.

Proposition 2. The boundary arc I1=0 contains at most one steady state, at which K1=0 and K2 satisfies

K2Z1

2Zb

� �2

, ð10Þ

K3=22 þ

g2b2

ðrþd2Þd2K1=2

2 �yb

2ðrþd2Þd2¼ 0: ð11Þ

This steady state is always a saddle point.

In Appendix A we also show that the interior steady state and the boundary steady state do not exist within the samescenario1:

Proposition 3. The interior steady state and the boundary steady state cannot exist simultaneously.

The following proposition (see Appendix A for its proof) shows that a steady state on the boundary arc I2=0 is notsustainable.

Proposition 4. A steady state on the boundary arc I2=0 will never be a long run equilibrium.

The intuition is that long run optimality of such a steady state would imply that K2=0 on this steady state, but the marginalvalue of production is infinite at K2=0.

3.3. Results

Fig. 1 illustrates the results obtained in the previous section by choosing the following representative values for theparameters: r=0.1, a¼ 1, b¼ 0:6, d1 ¼ d2 ¼ 0:2, g1 ¼ 1, g2 ¼ 1:1. Panel (a) depicts a bifurcation diagram for the remainingparameters y, which is the absolute advantage in demand for the innovative product, and Z, which is a measure for theconnectivity of the two markets. This panel shows that in the long run the firm concentrates on producing the innovative

1 The authors thank Andy Novak for this result.

Fig. 1. Panel (a) depicts the two regions in the y�Z parameter space, where the interior steady state is optimal (I) and the boundary steady state is

optimal (B). Above the dashed line our analysis is not valid because the concavity assumption (16) is not satisfied. The panels (b) and (c) represent the

phase diagrams for two typical scenarios where long run convergence to the interior and the boundary equilibrium occurs, respectively. The parameter

values in these two cases are Z¼ 0:6, r=0.1, a¼ 1, b¼ 0:6, d1 ¼ d2 ¼ 0:2, g1 ¼ 1, g2 ¼ 1:1. Furthermore, in panel (b) we have y¼ 1:5 while in panel (c) we

have y¼ 3:5.

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478 467

good, while abandoning the standard product, when the absolute advantage in demand for the innovative product is large,i.e., y is large, and/or when the markets for both products strongly influence each other, i.e., Z is large. If Z is large, the firmlikes to cease production of the standard good. The reason is that the standard product is less profitable than the innovativegood (y41), while it influences demand of the innovative good negatively via the cross-price effect measured by Z(cf. Dixit, 1979). In case Z is small, the markets are more separated. In such a case profits from the innovative product arenot hurt by producing the standard good, so that both goods will remain in production.

Panel (b), where Z¼ 0:6 and y¼ 1:5, shows optimal trajectories when it is optimal for the firm to end up producing bothgoods. In panel (c), the absolute advantage in demand for the innovative product is larger than in panel (b), i.e., y¼ 3:5,while still Z¼ 0:6. Therefore in this case the firm ultimately produces only the innovative good and optimal trajectories fordifferent initial capital stock levels are shown.

4. Disruptive goods

4.1. The model

In the previous section the consumer utility function was symmetric in a qualitative sense, i.e., it contained a linear anda quadratic term of both q1 and q2, while the interaction term was �Zq1q2. However, in this section we consider thepossibility that the extent to which the utility of the standard product is reduced depends strongly on the disruptive good’smarket penetration. As long as the innovative product is confined to a small niche, the traditional product continues to be

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478468

supported more or less as before. However, as the new product becomes dominant the value of the old product falls offsharply. In other words, the marginal utility of the traditional good decreases more than proportionally in the quantity ofthe innovative good. We can embody this difference by making just one change. In particular, we increase the parameter b

from 1 to 2, which gives a consumer utility function of the form

Uðq1,q2Þ ¼ q1þyq2�ðg1q2

1þg2q22þ2Zq1q2

2Þ

2þM: ð12Þ

From this specification it follows that

@Uðq1,q2Þ

@q1¼ 1�g1q1�Zq2

2,

which implies that

@2 @Uðq1,q2Þ

@q1

� �@q2

2

o0: ð13Þ

Inequality (13) is the condition that makes the innovative good disruptive, i.e., the marginal utility of the traditionalproduct progressively decreases with the quantity of the innovative good.

Expression (12) implies that

p1 ¼ 1�g1q1�Zq22, ð14Þ

p2 ¼ y�g2q2�2Zq1q2: ð15Þ

The inverse demand functions (14) and (15) only make sense if the underlying utility function (12) is concave.Straightforward calculations show that concavity is assured if

g1g2þ2g1Zq1�4Z2q2240: ð16Þ

Eq. (14) says that a new product really starts to have an influence on the price of the existing product when q2 is large,implying that the new product is ‘‘visible’’ and influential in the marketplace. The practical examples we have in mind forthis kind of non-linearity are the following. First, think of mobile (cell) phones as the innovative product and of landlinephones as the standard product. Before the advent of cell phones, landline telecommunications was ubiquitous in manydeveloped countries—most notably the US (K1(0) very large), but not in many lesser developed countries (K1(0) positivebut small). Furthermore, the introduction of new services and their pricing was often controlled by a state-run or state-regulated monopolist. (Q-Tel is only just now giving up its monopoly on telecommunication services in Qatar.) Thosemonopolists needed to decide how to price the two competing types of phone services or, equivalently, to figure out whatvolumes of each to produce.

If cell phone calling is very expensive, then—at least where landlines already existed—cell phones were used primarilyfor emergency calls, e.g., when one was in a car accident and needed to call the police and a tow truck or when one wasrunning late and wanted to warn the other party. Such calling did almost nothing to reduce demand for landline phoneservice; they were additional phone calls that would not have been placed at all in the era before cell phones.

However, as the cost of cell phone calling declined, it began to replace phone calls that otherwise would have beenmade from a landline. That in and of itself does not produce a non-linearity, but the network externality effects associatedwith having a particular phone number did. Once one started using a cell phone enough to carry it everywhere, one begangiving that number out as a main contact number, and people started calling it, rather than his/her landline. Likewise,when one used the cell phone and people ‘‘captured’’ the phone number one was calling from or simply hit ‘‘call back’’,they were calling the cell phone, not the landline. So the more one used the cell phone, the more useful it became, and theless useful the landline was, to the point that now many people have only a cell phone and have cancelled their landlineservice.

Second, think of regular cable as the normal product and pay-per-view options, such as movies on demand, as theinnovative product. In countries with high market penetration for conventional cable (priced in terms of ‘‘packages’’ ofchannels for various fixed monthly charges), if pay-per-view rates are high enough, then pay-per-view would mostly beused for high-value entertainment (e.g., watching new movie releases in one’s home theater instead of going out to apublic movie theater). As the price of pay-per-view drops, one might start to use it for things like sports and so drop downfrom the ‘‘premium’’ to the ‘‘basic’’ package of cable TV channels. When the price drops still further, one might cancel cableservice altogether.

Third, think of downloading music to an Ipod as the advanced product and buying albums on CDs as the standardproduct. Standard music technology used to be buying CDs, but now one can download music on-line. Downloads arepriced low, but they could have been priced so high that one would only download a few individual songs while stillbuying lots of CDs. But if the price of downloads falls far enough that people buy most songs that way, then one might ditchthe home stereo system altogether at which point there is a complete cessation of buying CDs because that is not how onelistens to music anymore. Indeed, there are claims that bands are now even recording music differently in anticipation ofthe fact that most fans will listen on individual portable devices, not classic room-based stereo systems.

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478 469

4.2. Analysis

Now using (14) and (15) for inverse demand functions the Hamiltonian and Lagrangian become:

H¼ aK1�g1a2K21þybK1=2

2 �g2b2K2�3Zab2K1K2�

12 I2

1�12I2

2

þl1ðI1�d1K1Þþl2ðI2�d2K2Þ,

L¼Hþn1I1þn2I2:

This gives the following necessary optimality conditions:

LI1¼�I1þl1þn1 ¼ 0,

LI2¼�I2þl2þn2 ¼ 0,

n1I1 ¼ 0, n1Z0,

n2I2 ¼ 0, n2Z0,

_l1 ¼ ðrþd1Þl1�aþ2g1a2K1þ3Zab2K2,

_l2 ¼ ðrþd2Þl2�12ybK�1=2

2 þg2b2þ3Zab2K1:

We observe that the Hamiltonian is concave if K2 is sufficiently low, such that it satisfies

HK1K1HK2K2

�ðHK1K2Þ2¼ 1

2g1a2ybK�3=22 �9Z2a2b440: ð17Þ

The next proposition, which is proved in Appendix A, excludes the occurrence of limit cycles.

Proposition 5. Steady states are the only long run optimal solutions.

This proposition assures that paths converging to a steady state of the canonical system are the only candidates for optimalsolutions. This implies that the problem of calculating the optimal solution is reduced to the problem of finding the steadystates of the canonical system and determining the paths converging to these steady states. For the actual computationswe have to resort to numerical techniques, which are described in Appendix B. A more detailed discussion of thesetechniques is provided by Caulkins et al. (2009) or Grass (2010).

In case there is more than one solution candidate, the one is selected that gives the highest value of the objectivefunction. An initial point in the state space (K1,K2) from which multiple trajectories start out that converge to differentsteady states, while at the same time these trajectories exhibit the same objective value, is a so called Skiba point. Theprincipal idea of how to locate such Skiba points and connect them, which results in a Skiba curve, is also described inAppendix B. Again, a more detailed description of Skiba points and the way to numerically compute them, can be found inGrass et al. (2008) and Grass (2010).

From the co-state equations, we obtain that on an interior steady state, where li ¼ diKi, it must hold that

ðrþd1Þd1K1�aþ2g1a2K1þ3Zab2K2 ¼ 0,

ðrþd2Þd2K2�12ybK�1=2

2 þg2b2þ3Zab2K1 ¼ 0: ð18Þ

The first equation gives an expression for K1:

K1 ¼a�3Zab2K2

ðrþd1Þd1þ2g1a2: ð19Þ

Hence, the candidate interior equilibria satisfy a function K1(K2), which is decreasing. The following proposition can beestablished, the proof of which can be found in Appendix A.

Proposition 6. The interior arc contains at most two steady states, at which K2 satisfies

K3=22 �

3Za2b2þg2b

2ððrþd1Þd1þ2g1a2Þ

9Z2a2b4�ððrþd1Þd1þ2g1a2Þðrþd2Þd2

K1=22

þ

1

2ybððrþd1Þd1þ2g1a2Þ

9Z2a2b4�ððrþd1Þd1þ2g1a2Þðrþd2Þd2

¼ 0, ð20Þ

and K1 is given by (19). Concavity of the Hamiltonian in a steady state guarantees that this steady state is a saddle point.

The next proposition shows that the boundary arc I1=0 may contain a stable steady state.

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478470

Proposition 7. On the boundary arc I1=0 a stable steady state can occur for which it holds that K1=0 and K2 is implicitly

given by

K3=22 þ

g2b2

ðrþd2Þd2K1=2

2 �yb

2ðrþd2Þd2¼ 0:

This steady state can only occur for K2 sufficiently large, i.e.,

K2Z1

3Zb2: ð21Þ

Finally, let us check the boundary arc I2=0 where we have

l2 ¼�n2:

This has the following implication for the co-state equation for l2 on the steady state:

_l2 ¼�12ybK�1=2

2 þg2b2þ3Zab2K1�ðrþd2Þn2 ¼ 0:

At this steady state we further have that K2=0. So, we can conclude that this boundary steady state will not occur, since thiswould require that n2 ¼�1. The latter cannot hold, because n2Z0.

4.3. Results

From the previous section it can be concluded that three possible optimal investment patterns can arise. First, both thestandard and the innovative goods are produced in the long run. Second, it can be optimal for the firm to converge to asituation where it solely produces the innovative good. While both of these cases also occurred in the non-disruptivemodel, here we have an additional scenario where long run optimality depends on the initial capital stock levels.

Fig. 2 depicts the three possible cases, where we employ the parameter values r=0.1, a¼ b¼ 1, d1 ¼ 0:1, d2 ¼ 0:15,g1 ¼ g2 ¼ 1, and Z¼ 0:575.

In panel (a) the absolute advantage for the innovative product, measured by y, is relatively small, i.e., y¼ 1:575.Consequently, all optimal trajectories converge to a situation where the firm produces both the standard and theinnovative goods. The opposite case is depicted in panel (c), where ultimately the firm only produces the innovative good.The latter scenario occurs for y large, i.e., y¼ 1:618.

In terms of qualitative behavior, for low values of y the optimal trajectory converges to an internal equilibrium as inpanel (a), while for high values of y the optimal trajectory converges to a boundary equilibrium where only the high techgood is taken into production. Although the parameter regions where these two types of solution behavior prevail arelarge, we nevertheless selected y�values close to each other. This is to illustrate the high degree of sensitivity in they�region where the qualitative behavior changes. Later on (see Fig. 3) we will demonstrate, that for other parameter values(i.e., high vales of the interaction parameter Z and/or the discount rate r) this sensitivity is not that high.

In an intermediate scenario, e.g., where y¼ 1:605, we have history dependent equilibria. If initially production of thestandard good takes place at a small scale while the firm is sufficiently active in producing the innovative good, then onlythe innovative good is produced in the long run. Otherwise the long run equilibrium involves production of both thestandard and the innovative good. The regions of attraction of these two long run equilibria are separated by a so calledSkiba curve2 (see the upward sloping bold line in panel (b) of Fig. 2).

This history dependent behavior is intuitively plausible. In terms of our landline vs. cell phone example, the Skibaanalysis of this problem might say that, in places with extensive landline penetration (e.g., the US), the telecommunica-tions monopolist might want to keep cell phone service prices high to protect its standard product. However, in placeswithout the initial extensive landline penetration (e.g., Qatar or China), telecommunications service provision profits couldbe maximized by pricing cell phone services lower and more or less abandoning the landline market altogether.

A similar example is paying for public transport by mobile phone (The Economist, February 17th, 2007, pp. 67–70).While in the US adoption of this technology goes slowly, ‘‘the less well-off countries are sometimes the origin of cheaperways to use the technology. Instead of making big investment in having lots of electronic NFC ticket checking devices at theentrances and exits of stations, Croatia has found that tickets can be bought by and delivered directly to a mobile phone.’’

This example can also be examined from the perspective of a bank where credit cards are the standard product andpayment by mobile phone is the innovative product. Such banks would lose revenue if customers paid for goods directlyvia mobile phone rather than with credit cards. However, ‘‘some of the smaller banks, which do not have payment-cardportfolios to protect, might have the most to gain from offering customers a way to use their mobile phones to pay foritems directly from their accounts’’ (The Economist, February 17th, 2007, pp. 67–70).

The Skiba curve is upward sloping in the K1�K2 space. From a profit maximization point of view it is not wise for thefirm to end up there. This ‘‘dead zone’’ in K1�K2 space comes directly from the consumers’ utility functions. If they are

2 The Skiba curve occurred for the first time in the literature in Haunschmied et al. (2003). Some more methodological contributions on Skiba points

include Wagener (2003) and Kiseleva and Wagener (2010).

Fig. 2. Phase diagrams in the disruptive model for three different scenarios with (a) y¼ 1:575, (b) y¼ 1:605, and (c) y¼ 1:618. The remaining parameter

values are r=0.1, a¼ b¼ 1, d1 ¼ 0:1, d2 ¼ 0:15, g1 ¼ 1, g2 ¼ 1. In all three scenarios three steady states exist, where two (the boundary and one interior

steady state) exhibit a two-dimensional stable manifold. In panel (a) the interior steady state is the long run optimal solution, whereas in panel (c) the

boundary steady state is optimal. In panel (b) either the boundary or one of the interior steady states can be optimal in the long run, depending on the

initial capital stock values. A Skiba curve separates the corresponding domains of attraction.

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478 471

buying little of the new product so q2 is small, then q22 is very small, and the first product’s inverse demand function is not

much affected by the presence of the second (newer) product. But if q2 gets larger, then q22 gets very big, and consumers

begin to derive negative value from even the first unit of product 1 that they might buy. Also, the second product’s price isnegatively affected by the term 2Zq1q2 (see (15)).

When the ‘‘damage’’ done to the first product’s profit potential grows non-linearly in the amount of the new productsold, the firm has to make a discrete choice to either sell a lot of the new product, knowing that this will entirely destroythe market for the first product, or the new product must be priced very high so it is purchased only in quite limitedquantities.

Furthermore, traversing this ‘‘dead zone’’ can be quite expensive. So if the firm’s initial capital stock places it on one sideor the other of the dead zone and the firm is relatively short sighted, i.e., r is large, then it might not be willing to invest inways that take it across that dead zone even if it would ideally prefer to be on the other side of that dead zone. This is nicelyillustrated in Fig. 3 which shows that Skiba behavior is more prominent if the discount rate is large. The region where suchbehavior occurs is denoted by S. For each parameter constellation in this region we have history dependence, where a Skibacurve separates the two different types of long run behavior.

In fact, Fig. 3 contains a sensitivity analysis w.r.t. this discount rate and the other parameters y, which is the absoluteadvantage in demand for the innovative product, and Z, which measures the connectivity of the two markets. As in thenon-disruptive case, in the long run the firm only produces the innovative good if both the markets for the standard andthe innovative goods are sufficiently connected and the demand advantage of the innovative product is large. The firmproduces both goods in the long run in the opposite case while the Skiba region is in between. This is reflected by the

Fig. 3. Sensitivity analysis in the parameter space y�Z and r�Z in the disruptive model. Panels (a)–(c) represent the different cases r=0.02, r=0.1, and

r=0.2, respectively. The remaining parameter values are a¼ b¼ 1, d1 ¼ 0:1, d2 ¼ 0:15, g1 ¼ 1, g2 ¼ 1. Below the dashed line the concavity assumption (16)

is satisfied. Above this dashed line, the utility based derivation of the demand system is lost since the utility function is not concave there.

Nevertheless—if we assume that the present demand functions still hold in this area—the remaining analysis remains unchanged. The Skiba region, S, is

limited by the dashed and the solid black lines. The region B is the region where we have long run convergence to the boundary equilibrium while in

region I convergence to the interior equilibrium is optimal. The three +’s in panel (b) from left to right relate to the parameter choices in panels (a)–(c) in

Fig. 2, respectively. In panel (d) the parameter y¼ 1:6 is fixed while we vary the discount rate r. The regions B, S, and I are as before. This illustrates that

the size of the Skiba region increases with r.

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478472

downward sloping boundaries of the Skiba region S in panels a–c while region B (boundary equilibrium) is located at theupper right and region I (interior equilibrium) lies at the lower left.

As we already made clear, the size of the Skiba region increases with r in panels a–c and this is confirmed in panel d,which is a bifurcation diagram w.r.t. the parameters Z and r.

5. Conclusions

This paper studies the problem of a firm that currently produces a standard product and has the option to introduce aninnovative product that is qualitatively better. Examples in modern real life economics are abound. The drawback ofintroducing the innovation to the market is that it has detrimental effects on revenue from the older product. A distinctionis made between standard and what we call disruptive innovation where a disruptive technology is one that progressivelyundermines the utility derived from the old product as the new product gains market share. In both cases the firm eitherends up selling both products, or only the innovative one. This exhibits some kind of structural stability of the model. Onthe other hand, the difference between the two cases is that history dependence can only occur in the disruptive case. Thismakes intuitive sense. If the new product has the potential to severely disrupt revenues from the old product, not justcompete with it linearly, then the wisdom of pushing the new product to center stage can depend on the scale of historicalinvestment in the traditional product.

One possible topic for future research is the following. In this paper we employed decreasing returns to scale for thenew product, motivated by the eventual scarcity of skilled labor and/or advanced technology. Alternatively, also the case ofincreasing returns would be relevant and interesting to investigate.

The analysis is purely focused on choosing the ideal division of production amounts between a standard and aninnovative product such that the firm’s profits are maximized. Another natural extension would be to introducecompetition, which in reality is an important incentive for a firm to innovate.

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478 473

Appendix A. Proofs of propositions

A.1. Proof of Proposition 1

From the state equations (2) and (3) we obtain that on a steady state it should hold that

l1 ¼ d1K1, ðA:1Þ

l2 ¼ d2K2: ðA:2Þ

Eqs. (A.1) and (A.2) can be used to eliminate the co-states from the co-state equations (4) and (5):

ðrþd1Þd1K1�aþ2a2g1K1þ2ZabK1=22 ¼ 0, ðA:3Þ

ðrþd2Þd2K2�ð12y�bg2K1=2

2 �ZaK1ÞbK�1=22 ¼ 0: ðA:4Þ

From the first equation we obtain an expression for K1:

K1 ¼a�2ZabK1=2

2

ðrþd1Þd1þ2a2g1

¼að1�2Zq2Þ

ðrþd1Þd1þ2a2g1

,

which can subsequently be substituted into the second equation. After doing this we can conclude that K2 is the solution ofthe following equation:

K3=22 þ

b2

ðrþd2Þd2g2�

2Z2a2

ðrþd1Þd1þ2a2g1

� �K1=2

2 þb

ðrþd2Þd2

Za2

ðrþd1Þd1þ2a2g1

�1

2y

� �¼ 0,

which is a third order polynomial in K1=22 . This is an equation of the form

x3þbxþc¼ 0, ðA:5Þ

with b40 due to (1). It follows that

D¼ 4b3þ27c240 ðA:6Þ

implying that the polynomial has one real and two complex roots. The real root is given by

K2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

27b3þ

1

4c2

r3

s�

1

2c�

1

3

bffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

27b3þ

1

4c2

r3

s�

1

2c

0BBBB@

1CCCCA

2

:

If the resulting steady state has positive values for K1 and K2 and is stable, we will have saddle point convergence.To check stability we first consider the Jacobian of the dynamic system consisting of the relevant state and co-state

equations:

J¼ d1d2ðrþd1Þðrþd2Þþ14bK�3=2

2 d1ðrþd1Þðy�2ZaK1Þþd2ðrþd2Þg1a2þHK1K1HK2K2

�H2K1K2

:

Furthermore, Dockner’s k (see, e.g., Feichtinger and Hartl, 1986 or Dockner and Feichtinger, 1991) is

k¼�d1ðrþd1Þ�2g1a2�d2ðrþd2Þ�14bK�3=2

2 ðy�2ZaK1Þ:

Clearly, the signs of J and k are, in general, ambiguous, as expected. However, what we do is that concavity of theHamiltonian (cf. (6)–(8)) is sufficient for the steady state being a saddle point.

A.2. Proof of Proposition 2

The maximum principle implies that

@L=@I1 ¼ n1þl1 ¼ 0) l1 ¼�n1:

For the steady state we have

_K 1 ¼ 0, I1 ¼ 0) K1 ¼ 0:

_l1 ¼�ðrþdÞn1�ð1�2ZbK1=22 Þa¼ 0:

Hence, since n1Z0, the steady state on this boundary arc can only occur if

K2Z1

2Zb

� �2

: ðA:7Þ

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478474

The steady state on this boundary arc further satisfies

_K 2 ¼ I2�d2K2 ¼ 0) I2 ¼ l2 ¼ d2K2,

_l2 ¼ ðrþd2Þd2K2�12ybK�1=2

2 þg2b2¼ 0:

The latter equation implies that a steady state should be a root of the following third order polynomial in K1=22 :

K3=22 þ

g2b2

ðrþd2Þd2K1=2

2 �yb

2ðrþd2Þd2¼ 0:

Since the constant of the second term is positive, analogous to the reasoning along (9)–(A.6) we obtain that this polynomialhas one real root.

To establish whether such a steady state can be a long run equilibrium, besides stability this steady state should satisfy(A.7). To check stability, we consider the dynamic system

_K 2 ¼ I2�d2K2,

_I2 ¼ ðrþd2ÞI2�12ybK�1=2

2 þg2b2:

After checking the Jacobian, we straightforwardly conclude that this steady state is always a saddle point.

A.3. Proof of Proposition 3

The interior steady state is a root of the polynomial (9), while the boundary steady state is a root of (11). Bothpolynomials are monotonically increasing and intersect at

K1=22 ¼

1

2Zb:

Now assume the proposition is not true so that we have simultaneous existence of the two steady states. For the interiorsteady state it holds that (cf. (A.3))

ðrþd1Þd1K1�aþ2a2g1K1þ2ZabK1=22 ¼ 0,

which implies that a necessary condition for K1 to be positive is that

K1=22 o

1

2Zb:

This in turn implies that the polynomial (9) has a positive value at K1=22 ¼ 1=2Zb.

On the other hand we know that the boundary steady state has to satisfy (10), meaning that the polynomial (11) is non-positive at

K1=22 o

1

2Zb :

So the polynomials (9) and (11) have unequal sign at K1=22 ¼ 1=2Zb, while at the same time we know that they intersect

there. Hence we have a contradiction and the two steady states do not exist simultaneously.

A.4. Proof of Proposition 4

On a steady state it holds that

_K 2 ¼ 0) I2 ¼ 0) K2 ¼ 0:

The maximum principle implies that

@L=@I2 ¼ l2þn2 ¼ 0) l2 ¼�n2:

This has the following implication for the co-state equation for l2 on a steady state:

_l2 ¼�ðrþd2Þn2�ð12y�g2bK1=2

2 �ZaK1ÞbK�1=22 ¼ 0:

So, we can conclude that this boundary steady state will not occur for

�12yþZaK1o0:

Furthermore, on this boundary steady state n2 should be infinite when

�12yþZaK140:

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478 475

Finally, let us investigate the case where over a longer time interval it holds that

�1

2yþZaK1 ¼ 0) K1 ¼

y2Za : ðA:8Þ

Hence, we have that

_K 1 ¼ 0) _l1 ¼_I1 ¼ d1

_K 1 ¼ 0

so that for this possibility to exist it must hold that

_l1 ¼ ðrþd1Þd1K1�ð1�2g1aK1Þa¼ 0: ðA:9Þ

However, combining (A.8) and (A.9) learns that

_l1 ¼ ðrþd1Þd1y

2Za� 1�yg1

Z

� �a40:

The conclusion is that on a steady state with K2=0 it cannot hold that K1 ¼ y=2Za on a time interval with positive length.

A.5. Proof of Proposition 5

An optimal path is bounded, which immediately follows from the utility function and the state dynamics. Therefore,since strange attractors cannot occur in two-dimensional optimal control problems (Dockner and Feichtinger, 1991), itconverges either to a steady state or a limit cycle.

To disprove the existence of a limit cycle we consider the Jacobian matrix of the canonical system

J¼

�d1 0 1 0

0 �d2 0 1

2g1a2 3Zab2 rþd1 0

3Zab2 1

4ybK3=2

2 0 rþd2

0BBBBB@

1CCCCCA ðA:10Þ

and the term

k¼

@ _K1

@K1

@ _K1

@l1

@ _l1

@K1

@ _l1

@l1

þ

@ _K2

@K2

@ _K2

@l2

@ _l2

@K2

@ _l2

@l2

þ2

@ _K1

@K2

@ _K1

@l2

@ _l1

@K2

@ _l1

@l2

¼�d1 1

2g1a2 rþd1

þ

�d2 11

4ybK3=2

2 rþd2

þ2

0 0

3Zab2 0

o0:

In Dockner and Feichtinger (1991) a complete characterization of the local stability of a steady state is given, using that thecorresponding eigenvalues xi, i=1,2,3,4 are given as

x1,2,3,4 ¼r

27

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2

4�k27

1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2�4detJ

qr:

This formula is then used to show that for ko0 the canonical system cannot undergo a Hopf bifurcation and therefore nolimit cycle can emerge, which ends the proof.

A.6. Proof of Proposition 6

Plugging (19) into (18) ultimately gives

K3=22 �

g2b2ððrþd1Þd1þ2g1a2Þþ3Za2b2

�ððrþd1Þd1þ2g1a2Þðrþd2Þd2þ9Z2a2b4K1=2

2 þ

12 ybððrþd1Þd1þ2g1a2Þ

�ððrþd1Þd1þ2g1a2Þðrþd2Þd2þ9Z2a2b4¼ 0:

This is a third order polynomial in K21/2, which is again of the form

x3þbxþc¼ 0: ðA:11Þ

However, unlike the cases for the non-disruptive model, here it can occur that

bo0

implying that the possibility of three real roots, which happens when

D¼ 4b3þ27c2o0

may arise. However, it is easily seen that bo0 implies that c40 in (A.11). This in turn implies that the smallest root of thethird order polynomial is negative (to see this, realize that the functional value of the third order polynomial equals c40for K2

1/2=0), so that at most two steady states occur within the interior region.

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478476

To check stability of a steady state we depart from the relevant dynamic system:

_K 1 ¼ l1�d1K1,

_K 2 ¼ l2�d2K2,

_l1 ¼ ðrþd1Þl1�aþ2g1a2K1þ3Zab2K2,

_l2 ¼ ðrþd2Þl2�yb2

K�1=22 þg2b

2þ3Zab2K1

from which we derive the Jacobian:

J¼ d1d2ðrþd1Þðrþd2Þþd1ðrþd1Þyb4

K�3=22 þ2g1a2d2ðrþd2Þþ

1

2g1a2ybK�3=2

2 �9Z2a2b4:

A necessary condition for the Jacobian to be positive is that (17) is satisfied, thus when the Hamiltonian is concave. ForDockner’s k we have:

k¼�d1ðrþd1Þ�2g1a2�d2ðrþd2Þ�yb4

K�3=22 o0:

The conclusion is that the steady state is a saddle point whenever the saddle point is situated in the concave domain of theHamiltonian.

A.7. Proof of Proposition 7

It holds that

I1 ¼ 0, l1 ¼�n1:

At the steady state we have that K1=0, which implies for the co-state equation for l1 at the steady state that

_l1 ¼�ðrþd1Þn1�aþ3Zab2K2 ¼ 0:

Since n1Z0, the steady state can occur only for K2 sufficiently large, i.e.,

K2Z1

3Zb2:

Concerning the co-state equation for l2, we get for this boundary arc that

_l2 ¼ ðrþd2Þl2�12ybK�1=2

2 þg2b2:

The steady state on this boundary arc, on which l2 ¼ d2K2 and _l2 ¼ 0, satisfies

K3=22 þ

g2b2

ðrþd2Þd2K1=2

2 �yb

2ðrþd2Þd2¼ 0:

Since

b¼g2b

2

ðrþd2Þd240

we know that

D¼ 4b3þ27c240,

which implies that the polynomial has one real root.To establish whether this steady state can be a saddle point, besides stability this steady state should satisfy (21). To

check stability, we depart from the dynamic system

_K 2 ¼ I2�d2K2,

_I2 ¼ ðrþd2ÞI2�12ybK�1=2

2 þg2b2,

from which it is straightforwardly derived that the Jacobian is negative, so that this steady state is always a saddle point.

Appendix B. Numerical analysis

To numerically compute the solution paths we use a boundary value problem (BVP) approach together with acontinuation technique described in Grass et al. (2008). To numerically handle the infinite time interval we truncate it to afinite time horizon [0, T]. Assuming that the long run optimal solution is a steady state, one can state asymptotic

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478 477

transversality conditions, see, e.g., Beyn et al. (2001) and Pampel (2000), ensuring that the solution ends up at thelinearized stable manifold of the limit set.

If the stable manifold is two-dimensional, the asymptotic transversality conditions together with the initial conditionsyield a well-posed BVP. A trivial solution for this BVP is the steady state solution itself. Starting with this trivial solution, anumerical continuation technique can be applied, see, e.g., Krauskopf et al. (2007) and Allgower and Georg (2003) allowingone to follow the (optimal) solution for arbitrary initial states ðK1ð0Þ,K2ð0ÞÞ 2 R

þ

0 �Rþ .For notational simplicity we subsequently write

x :¼ ðK1,K2,l1,l2Þu

and xi, i=1,y,4 refer to the corresponding coordinates. Furthermore x�ð�Þ is the path in the state–costate spacecorresponding to the solution path ðK1ð�Þ,K2ð�Þ,I1ð�Þ,I2ð�ÞÞ.

To compute and continue a Skiba point (curve), with the two optimal solution paths x1ð�Þ and x2ð�Þ approaching twodistinct steady states xi, i=1,2, the BVP is given by

1.

The asymptotic transversality conditions for both steady states. 2. Coincidence of the initial states at time zero, i.e., x1

1(0) = x12(0), x2

1(0)=x22(0).

3.

The same objective value for both paths represented by the Hamiltonian at the initial point, i.e.,

Hðx1ð0ÞÞ ¼Hðx2ð0ÞÞ:

A phase condition identifying a unique solution in the case of a Skiba curve.

4.

This is a well-posed BVP and the simplest way of realizing the phase condition is to vary one of the coordinates by a fixedstep width.

To find the boundaries of the Skiba region we use the fact that this region is limited by the bifurcation lines of aheteroclinic connection between the two steady states xi, i=1,2,. For the actual calculation of the heteroclinic connectionline hð�Þ we refer to the definition of a heteroclinic connection saying that

limt-1

xðtÞ ¼ x1,

with

xð0Þ 2 hð�Þ:

This can immediately be reformulated as a BVP and as a codim-2 bifurcation be continued by varying one parameter valuewhile freeing a second one.

References

A cash call, The Economist, February 17th, 2007.Allgower, E.L., Georg, K., 2003. Introduction to Numerical Continuation Methods. In: SIAM Classics in Applied Mathematics, SIAM Press, Philadelphia, PA.Barucci, E., 1998. Optimal investment with increasing returns to scale. International Economic Review 39, 789–808.Beyn, W.J., Pampel, T., Semmler, W., 2001. Dynamic optimization and Skiba sets in economic examples. Optimal Control Applications and Methods 22,

251–280.Caulkins, J.P., Feichtinger, G., Grass, D., Tragler, G., 2009. Optimal control of terrorism and global reputation: a case study with novel threshold behavior.

Operations Research Letters 37, 387–391.Davidson, R., Harris, R., 1981. Non-convexities in continuous time investment theory. Review of Economic Studies 48.Dechert, W.D., 1983. Increasing returns to scale and the reverse flexible accelerator. Economics Letters 13.Dixit, A.K., 1979. A model of duopoly suggesting a theory of entry barriers. Bell Journal of Economics 10, 20–32.Dixit, A.K., Pindyck, R.S., 1994. Investment Under Uncertainty. Princeton University Press, Princeton.Dockner, E.J., Feichtinger, G., 1991. On the optimality of limit cycles in dynamic economic systems. Journal of Economics 53, 31–50.Feichtinger, G., Hartl, R.F., 1986. Optimale Kontrolle Okonomischer Prozesse: Anwendungen des Maximumprinzips in den Wirtschaftswissenschaften. de

Gruyter, Berlin.Grass, D., 2010. Numerical computation of the optimal vector field in a fishery model. ORCOS, Institute of Mathematical Methods in Economics, Vienna

University of Technology.Grass, D., Caulkins, J.P., Feichtinger, G., Tragler, G., Behrens, D.A., 2008. Optimal Control of Nonlinear Processes: With Applications in Drugs, Corruption,

and Terror. Springer Verlag, Berlin.Haunschmied, J.L., Kort, P.M., Hartl, R.F., Feichtinger, G., 2003. A DNS-curve in a two-state capital accumulation model: a numerical analysis. Journal of

Economic Dynamics and Control 27 (4), 701–716.Kiseleva, T., Wagener, F.O.O., 2010. Bifurcations of optimal vector fields in the shallow lake model. Journal of Economic Dynamics and Control 34,

825–843.Krauskopf, B., Osinga, H.M., Galan-Vioque, J., 2007. Numerical Continuation Methods for Dynamical Systems: Path Following and Boundary Value

Problems. Springer Verlag, Dordrecht.Lucas, R.E., 1967. Adjustment costs and the theory of supply. Journal of Political Economy 75, 321–334.Pampel, T., 2000. Numerical approximation of generalized connecting orbits. Ph.D. Thesis, University of Bielefeld.Qiu, L.D., 1997. On the dynamic efficiency of Bertrand and Cournot equilibria. Journal of Economic Theory 75, 213–229.Singh, N., Vives, X., 1984. Price and quantity competition in a differentiated duopoly. Rand Journal of Economics 15.Skiba, A.K., 1978. Optimal growth with a convex–concave production function. Econometrica 46, 527–540.Spot the dinosaur, The Economist, April 1st, 2006.Symeonidis, G., 2003. Comparing Cournot and Bertrand equilibria in a differentiated duopoly with product R&D. International Journal of Industrial

Organization 21, 39–55.The problem with solid engineering, The Economist, May 20th, 2006.

J.P. Caulkins et al. / Journal of Economic Dynamics & Control 35 (2011) 462–478478

The cutting edge, The Economist, October 7th, 2006.Vives, X., 1985. On the efficiency of Cournot and Bertrand equilibria with product differentiation. Journal of Economic Theory 36, 166–175.Vives, X., 1999. Oligopoly Pricing: Old Ideas and New Tools. The MIT Press, Cambridge, USA.Wagener, F.O.O., 2003. Skiba points and heteroclinic bifurcations, with applications to the shallow lake system. Journal of Economic Dynamics and Control

27, 1533–1561.