Firm Value in Emerging Network Industries Lorenz Schneider
∗
3 January 2014
Abstract This paper examines emerging industries that exhibit positive network effects. We put forward a dynamic model in which two technologies compete to be the standard. The model provides a quantitative method for the valuation of firms. We use the model to examine the relationship between network effects, consumer heterogeneity, and prices. We show that the firm value depends strongly on the particular choice of the network strength function. We compare three types of such functions, identify shortcomings of traditionally used ones, and propose a more realistic one. Network Industries · Network Effects · Market Expectations · Firm Valuation · Monte Carlo Simulation JEL: L15 · O33
1
Introduction
In the late 1990s, prior to the burst of the dot-com bubble, and then again in the following decade, a large number of relatively young companies in emerging network industries reached extremely high valuations. This phenomenon seems puzzling considering that, in most cases, the financial statements did not reflect the cash flows or profits necessary to justify such high firm values. Both Facebook and Groupon serve as recent prominent examples: a simple analysis based on nonlinear demographic dynamics reveals a huge gap between fundamentals on the one hand and actual valuations on the other hand, at virtually any time in these firms’ histories [Cauwels and Sornette, 2012]. In both cases, the driving force behind the extensive growth in firm value appears to be investors’ expectations that the companies can build up a dominant position and “win” their respective markets in the long run. In general, a technology often becomes more attractive to a potential user if many other users are already employing it. It is clear that the strength of these network effects will have a significant impact on the valuation of such a company. Where network effects are weak, and the company is likely to share the market with one or more other companies, valuations will certainly be different from the case in which network effects are strong and one of the companies is likely to take over the market and drive out the others (for example, the case of MySpace ceding to Facebook). For an innovator launching a new technology, there are two other important factors that will have an impact on its potential value: the overall size of the available market to be captured, and consumer ∗ Center
for Financial Risks Analysis (CEFRA), EMLYON Business School,
[email protected].
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heterogeneity. With respect to the market size, Cauwels and Sornette [2012] write for example that “the future growth of users will be regarded as the key to the future valuation of the company”. And regarding consumer heterogeneity, there are well-known examples where strongly pronounced preferences can counteract even significant network effects, as in the case of Skype vs other Voice over Internet Protocols (VOIP) such as Apple’s FaceTime, which integrates seamlessly into an Apple user’s already existing platform. We provide a quantitative method for valuing a particular firm. This is done by constructing a dynamic model in which two price-setting firms compete by promoting their own technology for establishing a standard. More specifically, by examining the relationship between network effects, consumer heterogeneity, and prices, we find that: (i). Unique Nash equilibrium prices exist at all times, and these prices are always strictly higher than the unit production cost (Proposition 3.1); (ii). For a given difference of the strengths of the two networks, firms will set prices more closely to each other if consumer heterogeneity is higher (Proposition 3.2); (iii). There is an important lower boundary for the price difference, which holds for arbitrarily high consumer heterogeneity (Proposition 3.3). Then we evaluate a firm by considering the sum of all discounted profits made during the market growth period, plus the sum of all discounted further profits in the period when the market is saturated. This allows us to compare three specifications for the network effects. We find that traditional specifications such as linear network strength functions lead to price explosions in the model and therefore unrealistically high firm values. We introduce and compare two other network strength functions, an exponential and a logarithmic one, and conclude that the latter one leads to a more realistic and robust model.
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A Model of a Network Market with Price Setting
We propose a simple model of an emerging network market in which two companies compete for customers.
2.1
Consumers
Two companies, company 1 and company 2, sell two competing “sponsored” products (technologies) in an emerging network market at prices p1 and p2 , which they are allowed to change over time. 1 We consider positive network effects, which furthermore work in the same way for both companies. We assume these effects are given by a network strength function f that depends both on the number N of previous adopters and a network strength parameter α. 2 At time t0 = 0, both networks have size zero. Customers then arrive in groups of size s(ti ) ≥ 0 at times ti , i = 1, ..., m. Every customer comes with his own stand-alone valuations v1 and v2 for the two products, which are independent of prices and network strengths. The numbers of previous adopters of 1 According to Arthur [1989], “sponsored technologies are proprietary and capable of being priced and strategically manipulated; unsponsored technologies are generic and not open to manipulation or pricing.” 2 This specification encompasses Arthur’s model when the strengths are of the same size α for both networks: his network strength function f then measures the return by f (N, α) = αN . Note that Arthur also treats the case of different strengths α1 6= α2 , which we don’t consider here.
2
products 1 and 2 are given by N1 (ti−1 ) and N2 (ti−1 ), respectively. At time ti , each customer bases his choice of product 1 or 2, independently of the other customers concurrently arriving, on his utility for product j, which is given by uj (ti ) = vj + f (Nj (ti−1 ), α) − pj (ti ), j = 1, 2.
(1)
As the customer will be comparing these two utilities, we introduce the difference du := u1 − u2 in utilities, along with the differences dv := v1 − v2 in valuations, dn := f (N1 , α) − f (N2 , α) in network strengths, and dp := p1 − p2 in prices. 3 It then follows from (1) that the difference du in utilities is given by du (ti ) = dv + dn (ti−1 ) − dp (ti ). (2) so that the customer will choose product 1 if du (ti ) > 0, which occurs if the customer’s difference in valuations dv satisfies dv > dp (ti ) − dn (ti−1 ). (3) This difference is assumed to be normally distributed with mean 0 and variance σ 2 , i.e. dv ∼ Φ(0, σ 2 ) = Φσ ,
(4)
which reflects the heterogeneity of customers’ valuations. From equations (3) and (4) it follows that the probability ρ1 of a given customer choosing product 1 will depend on dp and dn , and be given by ρ1 (dp , dn ) := P dv > dp (ti ) − dn (ti−1 ) = Φ −dp (ti ) + dn (ti−1 ) . (5) Similarly, the probability ρ2 = 1 − ρ1 of the customer choosing product 2 will be given by ρ2 (dp , dn ) := P dv ≤ dp (ti ) − dn (ti−1 ) = Φ dp (ti ) − dn (ti−1 ) .
2.2
(6)
Firms
If firm j sells nj (ti ) products at time ti , it will make a profit πj given by πj (ti ) = (pj (ti ) − c) nj (ti ), j = 1, 2,
(7)
where c is the production cost per unit of each product, which we assume to be equal for both companies. If the prevailing network sizes N1 (ti−1 ) and N2 (ti−1 ) are known, the expected profit at time ti is given by Eti−1 [πj (ti )] = E [πj (ti )|N1 (ti−1 ), N2 (ti−1 )] . (8) We assume that each firm j = 1, 2 sets its price pj (ti ) based on the knowledge of prevailing network sizes Nj (ti−1 ), such that it maximises its expected profit for ti as given by (8). Companies are therefore myopic (i.e. not forward-looking). In this competition ` a la Bertrand, the equilibrium prices are those for which both firms simultaneously and non-cooperatively maximise their expected individual profits at ti . In order to obtain firm valuations with our model, we distinguish two periods. Let r > 0 be the interest rate. The discount factor for one time-period of length 4t = T /m is given by δ := e−r4t . First, for firm j, in the competition period from t1 to tm = T , the first profit component is given by the sum of all profits πj (ti ) = (pj (ti ) − c)nj (ti ). The net present value (npv) of the first profit component is Πj (0, T ) =
m X
δ i πj (ti ).
(9)
i=1 3 For the reader’s convenience, the parameters used to describe the market evolution and the network model are given in Tables 4 and 5 of the appendix.
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Second, after the end of the competition period at time T , both networks have established sizes N1 (T ) and N2 (T ), and the market is saturated at a level S(T ) = N1 (T ) + N2 (T ). For this postcompetition period from time T on, we introduce a rent factor θ ≥ 0 that reflects a yield for the company per customer per time period. This can be thought of as arising from periodic rent or subscription or maintenance payments, or alternatively as periodic renewal or replacement costs for the customer. We assume that each company j updates its price one last time at the very end of the competition stage, taking into account the final sizes of the established networks, and that θ is applied to this final price pj (T ). 4 After this, both firms leave their P prices constant, i.e. pj (T + i) = pj (T ) for all ∞ i ≥ 1. This second income stream is given by N (T ) j i=1 θ(pj (T + i) − c), and its npv at time t0 , P∞ i m Πj (T, ∞) = δ Nj (T ) i=1 δ θ(pj (T + i) − c), can be calculated as a geometric series to be Πj (T, ∞) = Nj (T )(pj (T ) − c)
θδ m+1 . 1−δ
(10)
Linking this income stream to the final price pj (T ) ensures that it is also linked to the firm’s final market share, which is naturally reflected in this price. The npv of both profit components added together is then Πj (0, ∞) = Πj (0, T ) + Πj (T, ∞). (11) We refer to the sum in equation (11) as the value of firm j obtained with our model.
2.3
Market Growth
The way in which the market grows over time is an important feature we want to capture. It will make a difference if most of the customers arrive very early in the market, or if most of them arrive very late, or if they arrive evenly spread out over time. Apart from this changing the nature of the competition, it will also have an effect on firm values due to the presence of discount factors. A growth function that gives us this flexibility is the incomplete Beta-function Ia,b : [0, 1] → [0, 1] given by Z x 1 Ia,b (x) = ta−1 (1 − t)b−1 dt, 0 ≤ x ≤ 1, (12) B(a, b) 0 for parameters a, b > 0, where B(a, b) is Euler’s Beta-function Z 1 B(a, b) := ta−1 (1 − t)b−1 dt, 0
which is related to the well-known Gamma-function [Remmert, 1997] by B(a, b) = Γ(a)Γ(b)/Γ(a + b). The function Ia,b gives a two-parameter family of smooth, strictly increasing curves from (0, 0) to (1, 1) in the unit square, whose shape can be like an “S”, an inverted “S”, or a straight line, for example. Note that this family also includes power functions, since for b = 1 we have Ia,1 (x) = xa . And for a function with given parameters a, b, the family also includes the function reflected at the diagonal from (0, 0) to (1, 1), since Ia,b (x) + Ib,a (1 − x) = 1. In terms of interpretation, a can be thought of as the procrastination parameter and b as the anticipation parameter. A large value of a compared to b, say a = 5 and b = 2, will lead to a curve that grows very slowly in the early stages, and in which customers therefore tend to arrive late in the market. In contrast, a small value of a compared to b, say a = 2 and b = 5, will lead to a curve that grows rapidly in the early stages and then flattens out, and in which most customers therefore arrive early. 4 The assumption that both companies update their prices at the end of the competition period is important for including the special case m = 1, where this competition only lasts one period, in a meaningful way.
4
In order to obtain a growth function for our model, the function Ia,b (x) : is re-scaled to fit the time horizon T := tm and the market saturation level S(T ) := Ssat to give the market growth function S := Sa,b S : {t0 , ..., tm } → {0, ..., Ssat }, ti 7→ S(ti ), S(t0 ) = 0, S(tm ) = Ssat , where the values S(ti ) are rounded to the nearest integer. The total number of customers who have arrived up to, and including, a time tk , is given by S(tk ), and we define s(ti ) := S(ti ) − S(ti−1 ), i = 1, ..., m. 5
2.4
Model Implementation
In practical terms, Ia,b is easy and fast to compute, for example in Excel or with matlab. Press et al. [2003] give an algorithm in C++. When we calculate the firm value with our model, we do this by Monte Carlo simulation: we evaluate (11) 25, 000 times (which is a sufficiently large number to ensure that the simulation has converged) and then take the average of these outcomes.
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Network Strength and Consumer Heterogeneity
In this section, we analyse the relationship between network effects and consumer heterogeneity.
Figure 1: Niche Market Share 5 Note that our sequential entry model is valid for m = 1, i.e. where the time horizon T = t , and therefore includes the 1 special case in which the competition is over after just one round.
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Figure 1 shows, for the network strength function f (N, α) = N α , the average niche market share captured at the end T of the simulation period for different levels of strength of network effects α and consumer heterogeneity σ. It can be seen that as α increases, the outcomes move from a dominant heterogeneity scenario (the market is shared) through a niche survival scenario (the lagging firm obtains a reasonable market share), and finally to a standards battle scenario (the laggard is essentially excluded from the market). The effect of an increasing consumer heterogeneity σ is to mitigate the network effects. Clearly, there is a quite complex interaction of α and σ, that we now characterize formally. We begin our analysis by establishing existence and uniqueness of an equilibrium in prices p1 (t) and p2 (t). Proposition 3.1 There exists a unique Nash equilibrium in prices (p∗1 (ti ), p∗2 (ti )) at all times ti . Furthermore, these prices are strictly greater than the unit production cost c. Proof: See B. The proof of Proposition 3.1 gives some interesting insight into the nature of the equilibrium prices. These prices are computed by considering expected profits at time ti , given firms’ knowledge of N1 (ti−1 ) and N2 (ti−1 ), and they are seen to depend directly on the primitives f and σ of the model. Furthermore, the proof shows how the problem of solving two equations for p∗1 , p∗2 simultaneously can be reduced to finding the equilibrium price difference d∗p by solving one equation, and then the price sum s∗p by solving a second equation, one after the other. Apart from leading to a much stabler and faster model, having just one equation for d∗p will allow us to obtain useful analytical results about the dependence of d∗p on dn and σ. We now assess the effect of the network strength function and of the consumer heterogeneity on prices. There are two mechanisms at play. First, the network strength function f , which depends on the network strength α, maps the network sizes N1 , N2 to the difference in network strengths dn . Second, this difference dn is mitigated by consumer heterogeneity σ as firms determine their optimal prices p∗j , leading to an equilibrium price difference d∗p = p∗1 − p∗2 . In light of Proposition 3.1, we can define the following price difference function Dp : R × R+ → R, (dn , σ) 7→ d∗p , which maps the difference in network strengths dn and the consumer heterogeneity σ > 0 to the price difference d∗p of the equilibrium prices. We have that Dp (0, σ) = 0 for all σ > 0.√Therefore, at any given ti , dn = 0 implies p∗1 (ti ) = p∗2 (ti ). The proof of Proposition 3.1 shows that p∗j = 12 2πσ +c in this case, which means that in a perfectly balanced market, companies will set higher prices when they are faced with a higher consumer heterogeneity. Our next result shows that for any given positive (resp. negative) network strength difference dn , an increase in the consumer heterogeneity σ will lead to a decrease (resp. increase) in the equilibrium price difference d∗p . Proposition 3.2 For any dn > 0 (dn < 0), Dp is a decreasing (increasing) function of σ. Proof: See B. We know that for very small values of σ, dp will be very close to dn , which is in agreement with our ∂ intuition. Now assume dn > 0. Then we know from Proposition 3.2 that ∂σ D(dn , σ) < 0. This, however, does not imply that equilibrium prices converge to the marginal cost as σ grows, nor that d∗p converges to zero. Our main result in this section is that, in fact, as σ increases, d∗p moves smoothly towards 32 dn .
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Proposition 3.3 For any dn > 0 (resp. dn < 0) and for any σ > 0, the price difference function Dp takes values in the open interval ] 23 dn , dn [ (resp. ]dn , 23 dn [). Proof: See B. This means that once the network effects have led to a significant difference in network strengths dn , even an extremely high consumer heterogeneity σ cannot substantially reduce the impact the difference in network sizes has on the price difference d∗p .
Figure 2: Graphs of the Price Difference Function D for three different values of σ Figure 2 shows three graphs of the price difference function D for σ = 10, 25, 100. It can be seen that D is close to a linear function of dn , whose slope decreases as σ increases. In the next section, we will present three types of network strength functions, and examine the effect they have on firm values.
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A Comparison of Network Strength Functions
Our model specification can now be used for the financial valuation of firms. The valuation is done in this section by relying on three different network strength functions. The first one is borrowed from the literature, and we show that it has shortcomings. Then we compare it to two alternatives. In the previous section we have seen that the strength of network effects plays an important role in determining which product an individual customer is likely to choose. We showed that the network effects also have an impact on the prices firms are setting. If we are to build a model that, for realistic
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parameters, gives reasonable values for companies competing in the way described, then we must be careful that these network effects do not lead to “astronomical” prices. The equilibrium prices our model generates can become unrealistic as soon as one company starts to gain a significant advantage over its competitor. In what follows, we consider several examples in which we fix the model parameters, but change the underlying network strength function f . Our base case parameters can be seen in Table 1. In each case, we plot the firm value as a function of strength of network effects α and consumer heterogeneity σ.
4.1
The Linear Case
For a network of size N (t) at time t, Farrell and Saloner [1986] use a linear function u(N (t)) = a + bN (t) to model the network benefits for the user. Similarly, Arthur [1989] considers the return from choosing a technology to be a linear function of the network size: translated into our setting, his network strength function would be of the form f (N, α) = αN . Assuming N1 > N2 , Proposition 3.3 shows that, for any consumer heterogeneity σ, the difference in equilibrium prices d∗p then lies in the interval 2 ] α (N1 − N2 ) , α (N1 − N2 ) [. 3 It is clear that in real-world scenarios, network sizes can be in the millions. For example, Cauwels and Sornette [2012] consider the valuation of Facebook, which currently has more than a billion users. Even for small values of α, this specification of f is therefore unrealistic.
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Figure 3: Firm Value as a Function of Network Strength α and Consumer Heterogeneity σ for f (N, α) = αN . Figure 3 confirms this claim, as the firm value quickly reaches unrealistically high levels.
4.2
The Exponential Case
Next, we consider an exponential network strength function given by f (N, α) = N α . The use of this function can be motivated by the need to capture non-linear network effects. Assuming N1 > N2 , Proposition 3.3 shows that, for any consumer heterogeneity σ, the difference in equilibrium prices d∗p then lies in the interval 2 ] (N1α − N2α ) , N1α − N2α [. 3
9
Figure 4: Firm Value as a Function of Network Strength α and Consumer Heterogeneity σ for f (N, α) = N α. Figure 4 shows how with this choice of f , firm values grow “exponentially” as α increases. As seen previously, the linear case α = 1 leads to extremely high firm valuations. Only a lower value of α can result in a more realistic valuation.
4.3
The Logarithmic Case
Finally, we consider a logarithmic network strength function given by f (N, α) = α ln N, f (0, α) = 0. 6 Assuming N1 > N2 , Proposition 3.3 shows that, for any consumer heterogeneity σ, the difference in equilibrium prices d∗p then lies in the interval 2 N1 N1 ] α ln , α ln [. 3 N2 N2 6 Shurmer and Swann [1995] compare linear to logarithmic network strength functions. They find that either type can lead to a convincing analysis of standards battles. However, they do not consider price setting and firm value.
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Figure 5: Firm Value as a Function of Network Strength α and Consumer Heterogeneity σ for f (N, α) = α ln N . In contrast to to the linear and exponential cases, Figure 5 shows how, with this choice of f , firm values increase much less sharply as α increases.
4.4
Sensitivity Analysis
We have seen that the exponential case (with α < 1) and the logarithmic case both result in realistic firm values. Our model allows a sensitivity analysis that can help identify the most appropriate functional form. In Table 1, we provide a sensitivity analysis, for f (N, α) = N α , in which we shift some of the main model parameters by 20% from the base case scenario and report the changes in firm value. It can clearly be seen that the greatest positive change comes from increasing α. An increase from 0.75 to 0.9 leads to a firm value that is 4 12 times higher! Next in magnitude comes the change caused by increasing the total market size S(T ). We see that a 20% increase in market size leads to a 50% increase in firm value. Anticipating the market growth by increasing the second curve parameter b leads to a small increase of about 2% in firm value. In the other direction, changes in σ and r lead to noticeable decreases in firm value. Procrastinating the market growth by increasing the first curve parameter a also leads to a decrease of 4.5% in firm value.
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Table 1: Sensitivity Analysis of Firm Value to Changed Parameters: Exponential N.S.F.
Parameter T, m S(T ) a b α σ c r θ
Base Parameter Value
Perturbed Parameter Value
Base Firm Value
Perturbed Firm Value
Change in Firm Value
25 1000 5 5 0.75 5 1 0.05 0.01
30 1200 6 6 0.9 6 1.2 0.06 0.012
11074 11074 11074 11074 11074 11074 11074 11074 11074
9865 16686 10576 11314 49612 9325 11016 8992 11559
-10.92% 50.68% -4.50% 2.17% 347.99% -15.80% -0.53% -18.80% 4.37%
Table 2: Sensitivity Analysis of Firm Value to Changed Parameters: Logarithmic N.S.F.
Parameter T, m S(T ) a b α σ c r θ
Base Parameter Value
Perturbed Parameter Value
Base Firm Value
Perturbed Firm Value
Change in Firm Value
25 1000 5 5 5 1 1 0.05 0.01
30 1200 6 6 6 1.2 1.2 0.06 0.012
3363 3363 3363 3363 3363 3363 3363 3363 3363
3251 4016 2851 3774 4708 2813 3305 2873 3437
-3.34% 19.39% -15.24% 12.20% 39.99% -16.38% -1.74% -14.59% 2.18%
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In Table 2, we provide a sensitivity analysis for f (N, α) = α ln N . Again, the greatest positive change comes from increasing α. However, the change of 40% is much less dramatic in this case than the change of almost 350% in the exponential case. Next in magnitude comes, as before, the change caused by increasing the total market size S(T ). We see that a 20% increase in market size now leads to only a 20% increase in firm value. Anticipating the market growth by increasing the second curve parameter b leads to an increase in firm value of over 12%. In the other direction, changes in σ and r lead to noticeable decreases in firm value. Procrastinating the market growth by increasing the first curve parameter a also leads to a decrease of over 15% in firm value. Interestingly, the effect of the curve parameters a and b is considerably stronger in the logarithmic case than in the exponential case. Consideration of these scenarios leads us to the following conclusions. The linear specification leads to explosions of firm value, even for small values of α. Although the specification f (N, α) = N α leads to reasonable firm values when α < 1 is sufficiently far away from the linear case, it still remains extremely sensitive to the choice of α. As we have seen, small changes in α lead to very large changes in firm value. The logarithmic specification f (N, α) = α ln N seems to be much more robust. We therefore conclude that it is the most appropriate of the three specifications we have analysed. Table 3 summarizes our findings: A conventional linear network strength function leads to unrealistically high firm values in the model. An exponential network strength function can lead to reasonable firm values, but the model becomes overly sensitive to the network strength parameter α. A logarithmic network strength function also leads to reasonable firm values, and to a far more robust model overall. Table 3: Network Strength Function and Firm Value Firm Value Realistic Stable
5
Network Strength Function f (N, α) Linear: αN Exponential: N α Logarithmic: α ln N no yes
for α 1 no
yes yes
Contributions to the Literature
In this section, we provide a brief review of two strands of theoretical literature: (1) firm valuation in emerging network industries; and (2) the evolution of market structures. We then establish a link between those streams and highlight our contributions to the existing literature. Valuation in Network Industries. Within the finance and accounting literature, the dot-com boom in the late 1990s and the bust in the early 2000s fuelled many discussions on the most appropriate method for valuing technology companies. Briginshaw [2002], for instance, provides a comprehensive overview of valuation techniques particularly applicable to internet companies in their early stage. For newly founded start-up firms, traditional valuation methods such as DCF (discounted cash flow), CFROI (cash flow return on investment) or EVA (economic value added) are difficult to apply since future earnings are highly uncertain and hard to predict. Schwartz and Moon [2000], McGrath and MacMillan [2000], as well as Howey [2004] therefore advocate the use of real option theory derived from Black and Scholes [1973] to value internet firms since it captures better the risk and highly volatile growth rates in network industries: potential future earnings can be tremendous, whereas potential losses if the firm does not succeed are limited. Total firm value comprises the value of its assets as reflected in the financial statements and the value of its real options [de Andr´es Alonso et al., 2006]. However, the criticism of Corr [2006-2007] is that 13
the real option approach typically does not take into account the factor of industry competition. Cauwels and Sornette [2012] therefore present a more conservative methodology based on nonlinear demographic dynamics. With respect to underlying effects that drive firm value in technology-intensive industries, Rajgopal et al. [2003] examine the relevance of network advantages on value, arguing that network externalities can create significant competitive advantage. Such advantages in turn constitute an important intangible asset which is not recognised in financial statements. Investigating two-year forecast earnings of their sample firms, Rajgopal et al. [2003] find that network advantages are positively correlated with future earnings and could therefore serve as an explanation for comparably high valuations of start-up firms in network industries. Pastor and Veronesi [2006] develop a stock valuation model and calibrate it to several Nasdaq share prices. They allow for competition to arise in a firm’s product market. This competition enters at a random date in the future and abruptly eliminates the firm’s “abnormal” earnings from that date onwards, making the firm a “normal” one whose market value of equity equals its book value. Therefore, this model can be viewed as a two-stage model, having a first stage without competition and a second stage with competition. However, in contrast to the model we present here, the first time-horizon is random, and the element of competition is reduced to a simple “yes-or-no” form. Market Structure in Network Industries. 7 The literature on network economies originates with Katz and Shapiro [1985]. Focusing on a relatively simple static oligopoly model, they study the effects of consumption externalities on competition. With utility depending on future adoptions, consumers form expectations about future network size, and expected network effects will vary with consumer expectations. With no horizontal product differentiation, the firm expected to be dominant will be able to charge higher prices and emerge as market leader. Consumer heterogeneity however can help niche products survive despite a dominant competitor. Farrell and Saloner [1986] show that if the preference for variety in the consumer population is sufficiently large compared to network effects, standardisation may not occur. In addition, small initial advantages may also influence the likelihood of highly asymmetric market outcomes. Arthur [1989] examines market dynamics and the role of random historical events, modelling two technologies competing for adoption. In the presence of network externalities, a small insignificant head start can suffice for a firm to corner the market through earlier adoption and subsequent improvement. Moreover, the timing of market entry as well as a firm’s learning orientation can help a particular technology rise to dominance [Schilling, 2002]. Church and Gandal [1992] examine whether there is de facto standardisation in hardware markets or whether multiple platforms can be supported, and describe different kinds of equilibria that may exist. Contrary to the static approach in Farrell and Saloner [1986], Mitchell and Skrzypacz [2006] incorporate explicit dynamics into a structural duopoly model with strategic firms. Assuming consumers’ utility to be an increasing function of past and present market shares, they analyse the evolution of standards. Particular emphasis is put on discount factors: the lower the discount, the less strongly market shares tend to diverge. In contrast, for sufficiently strong network effects, a standard will always evolve. Indirect network effects are considered in both Markovich [2008] as well as Markovich and Moenius [2009]. They develop structural dynamic oligopoly models and identify market structure as a major determinant of competitive dynamics. Focusing on the short-run, Markovich and Moenius [2009] make an explicit distinction between ex ante and ex post effects of market share changes, which influence competitive efforts and the probability of standardisation. In quality competition, the leading firm will cut investments and thus reduce competitive pressure. Consequently, unlike in models focusing on variety, the market does not “tip” but continuously develops towards standardisation. Cabral [2011] models consumers not as myopic but as forward looking, i.e. purchase decisions are 7 For
a more extensive survey on market structure, see Koski and Kretschmer [2004].
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based not on current, but on future expected network size. Firms’ and consumers’ value functions are increasing functions of current network size. With generations overlapping, consumers sequentially decide on adoption, which in turn can be influenced by network entry prices. Cabral [2011] finds that for sufficiently strong externalities, leading networks tend to increase in size until they capture the whole market but then tend to decrease in size again. Swann [2002] examines the functional form of network effects in telephone networks. He remarks that “there is little empirical or theoretical evidence about the actual functional form”. He finds that it can be linear, but only under strong conditions. In other conditions, he finds that the value to a typical subscriber will be an s-shaped function of network size. In an earlier paper, Shurmer and Swann [1995] compare linear to logarithmic network strength functions. They find that either type can lead to a convincing analysis of standards battles. In general, under network effects, price competition may lead to multiple equilibria. Grilo et al. [2001] and Griva and Vettas [2011] focus on the nature of such equilibria. However, due to the timing of moves and the myopia of firms, in the model presented here we always find a unique equilibrium point. Contributions of the Paper to the Existing Literature. Our model is closest in spirit to Cabral [2011]. However, our approach differs in the way we model consumer adoption. In Cabral’s model, the overall market size, i.e. the number of consumers, stays constant over time, and only the way in which it is divided between the two firms changes as existing customers die and new customers join one of the networks. In our model, the market grows over time from its initial size of zero to its final size, and existing customers cannot die or leave the network. In other words, we model an emerging network market, whereas Cabral models an already established network market. We also incorporate results and approaches from seminal papers on network industries such as Arthur [1989], who considers the dynamics of two unsponsored technologies, and Katz and Shapiro [1985], who compare the abilities of sponsored and unsponsored technologies to force a standard. However, we agree with Shurmer and Swann [1995] that “the models prove difficult to apply to the evidence in particular real world cases in any systematic empirical way.” So, on the one hand, a contribution of our approach to the market structure literature is that we explicitly link our fundamental parameters to firm value. On the other hand, we contribute to the firm valuation literature by explicitly modelling the competition stage. Swann [2002] examines the functional form of network effects. We pursue this line of investigation and, by extending our analysis to include firm value, make a step in the direction of “real world” empirical analysis.
6
Conclusion
Our simulation model rests on two main features of emerging network markets that render firms in network industries different from conventional ones. First, we expect that emerging network industries exhibit strong consumption externalities. This implies that early market leaders can expect a dominant market position in the long run. Second, many young technology firms are expected to become profitable much later, making their firm value depend strongly on the final market share they manage to acquire. In our simulation, we incorporate these market features and study the behaviour of firm value as a function of several fundamental parameters, most notably the strength of network effects, the degree of consumer heterogeneity, and the eventual market size. With this model, we examine the relationship between network effects, consumer heterogeneity, and
15
prices. We show that a unique equilibrium in prices always exists, and that prices have a dampening effect on the tendency of markets to generate highly asymmetric outcomes. We also show that a large difference in network strengths will inevitably lead to a large difference in the prices the two competing firms set, regardless of the degree of consumer heterogeneity. The model gives a method for determining the value of a firm in an emerging network market, based on discounted expected future profits. The first part of these profits are made during the growth phase of the markets, and the second in the saturated phase. The relative sizes of these two profit terms depend on the way the market evolves and also on the sizes of the discount factors applied. The overall value of the firm is shown to depend strongly on the function used to model the influence of the network effects on consumers and companies. We compare three different types of network strength functions and conclude that while the traditional linear function leads to unrealistically high firm values in the model, a logarithmic function can lead to a robust model. A more realistic model would include more than two competing companies. It would also be interesting to look at strategic price setting in such a market. Finally, we believe that for any real-world valuation of companies, the task of analysing and comparing different network strength functions should be carried out in more detail in future research.
A
Model Summary
Table 4 gives an overview of the market evolution parameters and Table 5 of the network model parameters. Table 6 shows the timeline of the model. Note that the events are supposed to take place sequentially from top to bottom. Table 4: Market Evolution Parameters t0 t1 , ..., tm T = tm
today market entry times ti , i = 1, ..., m time horizon for competition period
s(ti ) S(tk ) S(T ) = Ssat
number of draws at time ti Pk sum of all draws up to time tk , i.e. S(tk ) = i=1 s(ti ) market saturation level
nj (ti ) Nj (tk )
draws at time ti for product j = 1, 2 sum of all draws up to time tk for product j, Pk i.e. Nj (tk ) = i=1 nj (ti )
a, b
growth curve parameters (procrastination, anticipation)
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Table 5: Network Model Parameters α σ φσ , Φ σ
network strength parameter consumer heterogeneity Gaussian probability density, distribution with variance σ 2
uj vj f pj p∗j
consumer’s utility function for product j consumer’s stand-alone valuation for product j network strength function f : (N, α) 7→ f (N, α) price of product j equilibrium price of product j
du dv dn dp d∗p
difference difference difference difference difference
c ρj πj (ti ) Πj (T1 , T2 )
unit production cost of each product (j = 1, 2) probability of a sale of product j profit for firm j at time ti , i.e. πj (ti ) = (pj (ti ) − c)nj (ti ) profit for firm j between times T1 and T2
r δ θ
interest rate discount factor for one period 4t = T /m, i.e. δ = e−r4t rent/renewal factor
between between between between between
utilities u1 − u2 consumer’s valuations v1 − v2 network strengths f (N1 , α) − f (N2 , α) prices p1 − p2 equilibrium prices p∗1 − p∗2
Table 6: Timeline of the Model At time t0 we have N1 (t0 ) = N2 (t0 ) = 0. For the following times ti , i = 1, ..., m: H H H H H
Calculate dn (ti−1 ) = f (N1 (ti−1 ), α) − f (N2 (ti−1 ), α) Both firms set myopic prices p1 (ti ), p2 (ti ) using (14), (15) Calculate dp (ti ) = p1 (ti ) − p2 (ti ) s(ti ) new customers arrive For each new customer k = 1, ..., s(ti ): O O O O
H
Draw dv from Φ(0, σ 2 )-distribution (independently) Calculate du (ti ) using (2) If du (ti ) > 0: customer k chooses product 1 Else: customer k chooses product 2
Update network sizes N1 (ti ) and N2 (ti )
17
B
Proofs
If the prevailing network sizes N1 (ti−1 ) and N2 (ti−1 ) are known, the expected profit at time ti is given by Eti−1 [πj (ti )] = E [πj (ti )|N1 (ti−1 ), N2 (ti−1 )] = E [(pj (ti ) − c) nj (ti )|N1 (ti−1 ), N2 (ti−1 )] = (pj (ti ) − c) E [nj (ti )|N1 (ti−1 ), N2 (ti−1 )] = (pj (ti ) − c) s(ti ) ρj (dp (ti ), dn (ti−1 )).
(13)
We have assumed here that the price pj (ti ) is set by firm j based on the knowledge of N1 (ti−1 ) and N2 (ti−1 ), and that it can therefore be taken out of the conditional expectation. We have also assumed that the s(ti ) sales occur independently of each other, so that we have a repeated Bernoulli experiment of s(ti ) independent draws with success probability ρj . The equilibrium prices are found by solving the equations ∂Eti−1 [π1 (ti )|N1 (ti−1 ), N2 (ti−1 )] = 0, ∂p1 (ti ) ∂Eti−1 [π2 (ti )|N1 (ti−1 ), N2 (ti−1 )] = 0, ∂p2 (ti )
(14) (15)
for p1 and p2 . We show in the proof of Proposition 3.1 that this system can be separated into two equations in one variable each, and that the equilibrium prices can be calculated numerically in a very fast and robust manner, which will be important for numerical simulations. Proof of Proposition 3.1. Calculating the partial derivatives using equations (5), (6) and (13) and setting them to zero leads to the following system of equations: Φσ (−dp + dn ) = (p1 − c)φσ (−dp + dn ),
(16)
Φσ (dp − dn ) = (p2 − c)φσ (dp − dn ).
(17)
Since Φσ and φσ are both strictly positive functions, it immediately follows that we must have p1 > c and p2 > c. Now, taking the difference (17) - (16) and the sum (16) + (17) gives: 2Φσ (dp − dn ) + dp φσ (dp − dn ) − 1 = 0,
(18)
(p1 + p2 − 2c)φσ (dp − dn ) − 1 = 0.
(19)
To solve (18) for the equilibrium price difference d∗p , introduce the function Hdn ,σ given by Hdn ,σ (dp ) := 2Φσ (dp − dn ) + dp φσ (dp − dn ) − 1,
(20)
If dn = 0, then d∗p = 0 is the unique solution to (18). Next, if dn > 0, it is clear that Hdn ,σ (dp ) < 0 for dp ≤ 0 and Hdn ,σ (dp ) > 0 for dp ≥ dn . The derivative of H (note that φ0σ (dp ) = −(dp /σ 2 )φσ (dp )), given by hdn ,σ (dp ) = (−d2p /σ 2 + dn dp /σ 2 + 3)φσ (dp − dn ), (21) is strictly positive in the interval [0, dn ], and it follows that (18) has a unique solution d∗p ∈ [0, dn ]. Finally, if dn < 0, then Hdn ,σ (dp ) < 0 for dp ≤ dn and Hdn ,σ (dp ) > 0 for dp ≥ 0. Also, hdn ,σ is strictly positive in the interval [dn , 0], and it follows that (18) has a unique solution d∗p ∈ [dn , 0].
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Solving for the equilibrium price sum s∗p = p∗1 + p∗2 , we have from (19) p∗1 + p∗2 = 1/φσ (d∗p − dn ) + 2c, and finally p∗1 = (s∗p + d∗p )/2 and p∗2 = (s∗p − d∗p )/2.
Figure 6: Graphs of H for three different values of σ Figure 6 shows three graphs of the function Hdn ,σ for dn = 100 and σ = 10, 25, 100. It can be seen that in each case the solution Hdn ,σ (dp ) = 0 lies between 2/3·100 and 100, and that the solution becomes smaller as σ increases. For σ = 100, the solution is roughly 67.45, which is already very close to the lower boundary of 66.67. Proof of Proposition 3.2. Let dn > 0. The case dn < 0 follows in the same way. We must show ∂Dp ∗ ∗ ∂σ (dn , σ) < 0. By (20), the equilibrium price difference dp is given by H(dn , σ, dp ) = 0. We know from ∂H (21) that ∂dp (dn , σ, dp ) > 0 for all dp ∈ [0, dn ]. It follows from the implicit function theorem that there is a continuously differentiable function Dp such that d∗p = Dp (dn , σ). This is the price difference function ∂D introduced in section 3. Furthermore, the partial derivative ∂σp is given by . ∂H ∂Dp ∂H (dn , σ) = − (dn , σ, d∗p ) (dn , σ, d∗p ). ∂σ ∂σ ∂dp
(22)
From (21), we know that the denominator is positive. Therefore, to prove the proposition, it remains to ∗ show ∂H ∂σ (dn , σ, dp ) > 0. Calculating the numerator gives Z d∗p −dn 2 (d∗p − dn )2 − σ 2 − (d∗p −dn )2 ∂H t − σ 2 − t22 2σ 2 √ √ (dn , σ, d∗p ) = 2 e 2σ dt + d∗p e . ∂σ 2πσ 4 2πσ 4 −∞ 19
We see immediately that if σ ≤ dn − d∗p , then the first summand is positive and the second summand is non-negative, and we are done in this case. To see that our claim also holds when σ > dn − d∗p , rewrite the partial derivative as Z d∗p −dn t2 ∂H 2 t2 ∗ √ e− 2σ2 dt − Φσ (d∗p − dn ) (dn , σ, dp ) = 2 4 ∂σ σ 2πσ −∞ (d∗p − dn )2 − (d∗p −dn )2 d∗p 2σ 2 + d∗p √ e − φσ (d∗p − dn ), σ 2πσ 4 and then use the relation 2Φσ (d∗p − dn ) + d∗p φσ (d∗p − dn ) = 1 from (20) to obtain 1 ∂H (dn , σ, d∗p ) = 3 ∂σ σ
Z
d∗ p −dn
2 −∞
(d∗p − dn )2 − (dp −dn ) t2 t2 2σ 2 √ e− 2σ2 dt + d∗p √ e − σ2 2πσ 2πσ ∗
Since by definition we have for the variance σ 2 =
R∞ −∞
2
! .
2
t 2 √t e− 2σ2 2πσ
dt, we can write
! Z dn −d∗p 2 ∗ 2 2 (d∗ p −dn ) (d − d ) t2 ∂H t 1 n p − − 2σ 2 √ − e e 2σ2 dt (dn , σ, d∗p ) = 3 d∗p √ ∂σ σ 2πσ 2πσ d∗ p −dn ! Z dn −d∗p Z dn −d∗p ∗ d∗p (dp − dn )2 − (d∗p −dn )2 t2 1 t2 − 2 2 2σ √ √ = 3 dt − e e 2σ dt . σ 2(dn − d∗p ) d∗p −dn 2πσ 2πσ d∗ p −dn d∗
p We will see in Proposition 3.3 that we always have d∗p ∈] 23 dn , dn [, which implies 2(dn −d ∗ ) > 1, so that we p obtain Z dn −d∗p 2 (d∗ p −dn ) t2 1 ∂H (dn , σ, d∗p ) > √ (d∗p − dn )2 e− 2σ2 − t2 e− 2σ2 dt. 4 ∂σ 2πσ d∗p −dn √ t2 The function f : t 7→ t2 e− 2σ2 has a minimum at t = 0 and maxima at t = ± 2σ. Therefore, for √ dn − d∗p < 2σ, and in particular for dn − d∗p < σ, we have that, for all t with d∗p − dn < t < dn − d∗p ,
(d∗p − dn )2 e− for the integrands. It follows that proves the proposition.
∂H ∗ ∂σ (dn , σ, dp )
2 (d∗ p −dn ) 2σ 2
t2
> t2 e− 2σ2
> 0, and hence from (22) that
∂Dp ∂σ (dn , σ)
< 0, which
Proof of Proposition 3.3. Assume dn > 0. We can evaluate the function H from (20) directly at dp = 32 dn and obtain 2 1 2 1 Hdn ,σ ( dn ) = 2Φσ (− dn ) + dn φσ (− dn ) − 1 3 3 3 3 Z 31 dn 2 1 = d n φσ ( d n ) − φσ (t)dt 3 3 − 13 dn Z 31 dn 1 = φσ ( dn ) − φσ (t) dt 3 − 13 dn < 0, since φσ ( 31 dn ) < φσ (t) for all t ∈] − 13 dn , 13 dn [. From the proof of Proposition 3.1 we already know H(dn ) > 0 and that the derivative of H is positive on [0, dn ], which shows that the solution dp must lie in the interval ] 23 dn , dn [. The case dn < 0 is shown in the same way. 20
C
Comparison of the Two Profit Components
Figure 7: Relative Size of the First Profit Component Πj (0, T ) as Fraction of Total Firm Value Πj (0, ∞) Figure 7 shows the relative size of the first profit component Πj (0, T ) as a fraction of the total firm value Πj (0, ∞) for different values of r and θ. It is clear that as the interest rate r increases, the second profit component Πj (T, ∞) is more heavily discounted, leading to an increase in the relative importance of Πj (0, T ).
Acknowledgements I would like to thank Tobias Kretschmer for his invaluable input and our many stimulating discussions. I owe my interest in this subject to him. Special thanks also go to my colleague Bruno Versaevel, whose very helpful comments and suggestions greatly improved the paper. I am also extremely grateful to two anonymous referees, whose constructive and detailed comments added significant value to the content of the paper and enhanced the clarity of the exposition. Finally, I thank Philipp N¨agelein, Cassio Neri, Christian Peukert, Richard Ruble, Leon Zucchini, and seminar participants at LMU Munich for their helpful comments and suggestions, and the Institute for Strategy, Technology and Organization at LMU Munich and the Interdisciplinary Institute of Management at the LSE for their hospitality.
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References W. Brian Arthur. Competing technologies, increasing returns, and lock-in by historical events. The Economic Journal, 99(394):116–131, March 1989. Fisher Black and Myron Scholes. The pricing of options and corporate liabilities. Journal of Political Economy, 81:637–654, May-June 1973. John Briginshaw. Internet Valuation: The Way Ahead. Finance And Capital Markets. Palgrave Macmillan, 2002. Luis Cabral. Dynamic price competition with network effects. The Review of Economic Studies, 78(1): 83–111, 2011. Peter Cauwels and Didier Sornette. Quis pendit ipsa pretia: Facebook valuation and diagnostic of a bubble based on nonlinear demographic dynamics. Journal of Portfolio Management, 38(2):56–66, Winter 2012. Jeffrey Church and Neil Gandal. Network effects, software provision, and standardization. Journal of Industrial Economics, 40(1):85–103, March 1992. Thomas A. Corr. Boom, bust, boom: Internet company valuations - from Netscape to Google. The Financier, 13/14:66–84, 2006-2007. Pablo de Andr´es Alonso, Valentin Azofra-Palenzuela, and Gabriel De la Fuente-Herrero. The real options component of firm market value: The case of the technological corporation. Journal of Business Finance and Accounting, 33(1-2):203–219, January/March 2006. Joseph Farrell and Garth Saloner. Installed base and compatibility: Innovation, product preannouncements, and predation. The American Economic Review, 76(5):940–955, December 1986. Isabel Grilo, Oz Shy, and Jacques-Fran¸cois Thisse. Price competition when consumer behavior is characterized by conformity or vanity. Journal of Public Economics, 80:385–408, 2001. Krina Griva and Nikolaos Vettas. Price competition in a differentiated products duopoly under network effects. Information Economics and Policy, 23(1):85–97, 2011. Richard A. Howey. Valuing technology. Knowledge, Technology & Policy, 17(3/4):44–64, 2004. Michael L. Katz and Carl Shapiro. Network externalities, competition, and compatibility. The American Economic Review, 75(3):424–440, June 1985. Heli A. Koski and Tobias Kretschmer. Survey on competing in network industries: Firm strategies, market outcomes, and policy implications. Journal of Industry, Competition and Trade, 4(1):5–31, 2004. Sarit Markovich. Snowball: A dynamic oligopoly model with indirect network effects. Journal of Economic Dynamics and Control, 32(3):909–938, 2008. Sarit Markovich and Johannes Moenius. Winning while losing: Competition dynamics in the presence of indirect network effects. International Journal of Industrial Organization, 27(3):346–357, 2009. Rita Gunther McGrath and Ian C. MacMillan. Assessing technology projects using real options reasoning. Research Technology Management, 43(4):35–49, July/August 2000.
22
Matthew F. Mitchell and Andrzej Skrzypacz. Network externalities and long-run market shares. Economic Theory, 29(3):621–648, November 2006. Lubos Pastor and Pietro Veronesi. Was there a Nasdaq bubble in the late 1990s? Journal of Financial Economics, 81(1):61–100, 2006. William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical Recipes in C++. Cambridge University Press, second edition, 2003. Shivaram Rajgopal, Mohan Venkatachalam, and Suresh Kotha. The value relevance of network advantages: The case of e-commerce firms. Journal of Accounting Research, 41(1):135–162, March 2003. Reinhold Remmert. Classical Topics in Complex Function Theory. Graduate Texts in Mathematics. Springer, 1997. Melissa A. Schilling. Technology success and failure in winner-take-all markets: The impact of learning orientation, timing, and network externalities. Academy of Management Journal, 45(2):387–398, 2002. Eduardo S. Schwartz and Mark Moon. Rational pricing of internet companies. Financial Analyst Journal, 56(3):62–75, May/June 2000. Mark Shurmer and Peter Swann. An analysis of the process generating de facto standards in the PC spreadsheet software market. Journal of Evolutionary Economics, 5(2):119–132, June 1995. G. M. Peter Swann. The functional form of network effects. Information Economics and Policy, 14(3): 417–429, September 2002.
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