Adaptive Expectation For Network Goods

Published Date: 02 Nov 2017

In this chapter, we consider a utility function that is influenced by the value of network externality function at the consumer’s expected market size/share. Using this utility function, a market share adjustment function is introduced and is utilized to develop dynamic models of existing rational and static expectation processes. In addition, the role and scope of dynamic models of market share adjustment process are extended to the well-known adaptive expectation and its extension processes. The properties of equilibrium states of dynamic models are investigated which include location, stability, oscillation and the initial states in systematic and unified way. The most significant byproduct of presented results is that the properties of equilibrium states depend on the type of consumer expectation of a network good and the parameters of dynamic market share adjustment processes.

Introduction

With regard to network goods, the utility function is known to be affected by the market size/share (Shy, 2001). The fundamental concept of network externality is that consumers gain more benefit when more consumers consume the same good. Hence, the market size/share is a factor that affects the consumer’s decision to adopt a network good. In practice, consumers do not have information about the actual market size/share. In view of this, consumers need to speculate their expected market size/share.

It is well-known (Katz and Shapiro, 1985) that in the network goods economy, the idea of dependence of consumer surplus function on the value of network externality function at the consumer’s expected market size/share was introduced. Moreover, under the fulfilled expectation assumption, the rational equilibrium is characterized for compatible products. Katz and Shapiro (1985) ideas and results are extended by Easley and Kleinberg (2010) to consumer demand goods and Amir and Lazzati (2011) to industry performance. Furthermore, under the rational expectation assumption, the effects of the market structure of the equilibrium states are analyzed.

In this work, we extend the idea of Katz and Shapiro (1985) to modify the utility function that depends on a value of network externality function at an expected market size/share at a real time . The consumer’s expected market size/share is speculated, and it is based on either the current or the past market share, or the combination of the current and past market share. Instead of directly using properties of modified utility function, we utilize the modified function to introduce a concept of market share adjustment function. Employing the market share adjustment function, we develop mathematical models corresponding to consumer rational, static, and current and lagged adaptive expectations in a systematic and unified way. Furthermore, we establish the existence and location of equilibrium states and study the qualitative properties of equilibrium states in a systematic way. The qualitative properties (stability and oscillation) are analyzed in the context of the parameters of dynamic systems. In fact, the parameters of the dynamic systems are decomposed into subsets that are characterized by the stability and oscillatory regions. Moreover, the presented dynamic model formation provides a suitable design to develop an agent-based simulation model (Paothong, 2013).

In short, the chapter is organized as follows. By introducing a market share adjustment function, dynamic models for rational and static expectation processes are developed in Section 4.2. In addition, sufficient conditions are given to exhibit the existence of corresponding rational and static equilibrium states. In Section 4.3, by considering a market share adjustment function in the context of adaptive expectations, dynamic models of current and lagged adaptive expectation processes are developed. Furthermore, it is shown the static equilibrium states are invariant under the current and lagged adaptive expectation processes. In Section 4.4, the sufficient conditions are given to establish the qualitative properties (stability and oscillatory) of dynamic systems. We note that the stability conditions are in terms of parameters of dynamic systems. In Section 4.5, by extending the domain of the speed of adjustment parameter, we develop the results parallel to the results of Section 4.4. Moreover, we decompose the domain of dynamic systems and the speed of adjustment parameters into regions according to qualitative properties of dynamic systems. In Section 4.6, by using simulations, we exhibit the influence of the initial states on the solution paths of dynamic models under static, current and lagged adaptive expectation processes.

Dynamic Models of Rational and Static Expectation Processes

We analyze an underlying network good model in discrete time, , and all variables are time varying or time invariant. For each , a utility function is composed of the consumer’s individual preference, the price of the network good and the value of network externality function at a consumer’s expected market size/share. In this study, for simplicity, we treat the market size as the market share,, that is, it has value between zero and one, . Hence,

(4.2.1)

where is the consumer’s individual preference; is the consumer’s expected market share at time ; is the price of the network good and is the generalized network externality function (GNEF) (Paothong and Ladde, 2012). We further make two assumptions:

The individual preference is distributed according to a cumulative distribution function, (cdf) with a probability density function, (pdf).

Each consumer has an identical expectation type and network externality function.

Following the argument used by Katz and Shapiro (1985), we conclude that the consumer joins the network whenever his/her utility is greater than zero, that is, ; otherwise he/she stays out of the network market. Let be the individual preference level at which the consumer is indifferent between joining and staying out of the network market, that is, . This interpretation motivates us to introduce a concept of a market share adjustment function. It is defined by

(4.2.2)

where , , and are defined in (4.2.1), and are defined in assumption 1.

(a)

(b)

Figure 4. Sketches of (a) GNEF and (b) pdf of consumer’s individual preference.

Remark 4.2. We observe that the market share is illustrated by the shaded area under in Figure 4. Sketches of (a) GNEF and (b) pdf of consumer’s individual preference.b. The function in (4.2.2) is continuous. Since and are pdf and network externality functions, respectively, the first derivative of the function is

(4.2.3)

Thus, the function is increasing on . Moreover, from (4.2.2), the mathematical description of an equilibrium state of network market share at a time is determined by . Clearly, the existence of the equilibrium states is determined by the solution of the algebraic equation (4.2.2).

Proposition 4.2. , if and only if .

Proof From (4.2.2), and the increasing property of , for , we have

This implies that

This completes the proof of "if" part. The proof of the "only if" part follows by imitating the argument in the reverse order.

Proposition 4.2. If , then the function defined in (4.2.2) has a fixed point, .

Proof. For and the increasing property of functions and , we have

,

and

.

Since and are continuous functions, is a real-valued continuous function defined on the interval . Define . Then, is also continuous function with and . Therefore, by application of Theorem 1.6.1 of Ladde et al (1985), there exists such that . Hence, .

Remark 4.2. From the economists’ view, a straight forward interpretation of Proposition 4.2. is as follows. If the price satisfies the condition, , then all consumers join the network with their expected market share, and hence, for all , that is, . On the other hand, if the price satisfies the condition, , then all consumers leave the network with their expected market share, and hence, for all , that is, .

We note that the properties of equilibrium states are affected by the nature of the function and the type of consumer expectation. In the following subsections, we outline two well-known expectations for a network good.

4.2.1 Rational Expectation (RE)

The rational expectation is defined as the optimal forecast. All consumers use all available information to form their expectations. We remark that RE was originally introduced by John F. Muth (1961). Also, in a network good, Katz and Shapiro (1985) used RE in their work, and the equilibrium state is called the fulfilled expectation equilibrium. In this scenario, all consumers have perfect foresight about the market share, . Consequently, from (4.2.2), a dynamic model of RE process satisfies the following discrete time dynamic iterative adjustment process,

, for

(4.2.4)

In this case, the equilibrium state(s) is(are) the point(s) of intersection(s) of function and the line with slope of one. It is well-known in the fixed-point theory that the solution(s) of (4.2.4) is(are) called the fixed point(s) of the function . Obviously, the existence of fixed points of depends on its analytic properties (Proposition 4.2.).

When consumers have a RE for a network good, the multiple fixed points are commonly discussed. Furthermore, the discussion of these equilibrium states can be found in Amir and Lazzati (2011), Easley and Kleinberg (2010) and Katz and Shapiro (1985). In general, there are two disadvantages for the RE scenario. First, it is a very restrictive assumption on the consumer expectation. It implies that all consumers have perfect information about market equilibrium. Second, it is an instantaneous adjustment process. Therefore, it is a completely static process. The equilibrium states of RE process depend on the parametric variation in algebraic equation (4.2.4) (Proposition 4.2.).

Illustration 4.2. Let us consider a particular example in which , and is a generalized network externality function in Chapter 2 (Paothong and Ladde, 2012) with and , that is, . We observe that , , and . In this simple case, . From Proposition 4.2., the appropriate price range is . For this concrete example, if , there are three equilibrium states at and . We note that , and . and are lower and upper stable states, respectively, but is unstable. For and , there corresponds one stable equilibrium state at and , respectively. This suggests that the price level affects the location and number of equilibrium states. Moreover, as the price increases, the function moves downward, , . See Figure 4. Sketches of (a) GNEF, ; (b) pdf of the consumer’s individual preference , (c) function , and location(s) of equilibrium states that are determined by price = 2.25, 2.5 and 2.75 for Illustration 4.2.1..

(a)

(b)

(c)

Figure 4. Sketches of (a) GNEF, ; (b) pdf of the consumer’s individual preference , (c) function , and location(s) of equilibrium states that are determined by price = 2.25, 2.5 and 2.75 for Illustration 4.2..

4.2.2 Static Expectation

To study the adjustment process in the neighborhood of the equilibrium state, the simplest common type of consumer expectation, namely static (or naive) expectation, is employed. Consumers have asymmetric information in time, and they expect no change for the present time. In this case, all consumers use the immediate past actual market share as their present expectation, . Consequently, from (4.2.2), a dynamic model of SE process is determined by the first order nonlinear difference equation that describes a dynamic model of the market share adjustment process,

,

(4.2.5)

In this case, the sequence is a solution path of discrete time dynamic adjustment process (4.2.5), where . The equilibrium states (steady states: ) of (4.2.5) are determined by Proposition 4.2.. The assumption of a single unit time lag is realistic, and is suitable to derive an empirically tractable strategy to identify the adjustment process for the equilibrium sates.

Illustration 4.2. We illustrate the stability conditions of an equilibrium state by the graphical method. Suppose is the initial market share. For in (4.2.5), clearly, , and so on. In this method, we track the initial state and move vertically until we reach the curve to , then move horizontally until we reach and so on until we reach the equilibrium state . See Figure 4.3a. Using the Illustration 4.2.1, the graphical method tells us that the adjustment process will converge to two stable equilibrium states for which ; and diverge from an unstable equilibrium state for which . Since, , and , and are the lower and upper stable equilibrium states, respectively, but is unstable equilibrium state. The detailed analytic stability conditions will be discussed in Section 4.4. See Figure 4.3b.

(a)

(b)

Figure 4. (a) Adjustment process of stable equilibrium state: graphical method; and (b) shows three equilibrium states, two stable and one unstable.

Remark 4.2. If all consumers follow the SE process with , then the market share adjustment function has a fixed point at . Hence, the solution of (4.2.4) is the long run solution of equation (4.2.5). Moreover, since is an increasing function (Paothong and Ladde, 2012), from (4.2.3), is also an increasing function. Thus, there is no chaotic situation for this model (Chiarella, 1988).

Dynamic models of Adaptive Expectation Processes

In this section, we utilize the well-known adaptive expectation (AE) to form a consumer expectation of market share. The AE is one of the backward-looking expectations. More precisely, the future (present) expectation of an endogenous variable is directly adjusted by the weighted mean of its current (immediate past) actual value and either the current or the past expected value. Thus, in this study, we classify the AE into two simple categories.

4.3.1 Current Adaptive Expectation (CAE)

In this traditional adaptive expectation (Chow, 2011) all consumers speculate the future (present) market share by the weighted sum of the current (immediate past) expectation and the current (immediate past) actual market share, that is,

,

(4.3.1)

where is a speed of adjustment in which SE process is a special case of CAE process . The "ω" measures the significance of and . If , then the consumer is very conservative. From (4.3.1) and (4.2.2), a dynamic model of CAE process is described by the following first order nonlinear difference equation,

, .

(4.3.2)

Of course, by following the argument used in the SE process, the existences of equilibrium states are determined by (4.3.2).

Proposition 4.3. From the dynamic model (4.3.2) of the CAE process, we have

and have the same equilibrium states;

is scalar multiple of ;

If is parallel to the line with slope one, then is parallel to ;

If the speed of adjustment decreases (increase), is rotated clockwise (anti-clockwise) toward the line around equilibrium states.

Proof of (a): The equilibrium states of dynamic CAE process are determined by . This implies . Hence, . Conversely, implies . Thus, if and only if . This establishes the validity of statement (a).

Proof of (b): From the definition of , we have,

(4.3.3)

This establishes the statement (b), that is, is a scalar multiple of .

Proof of (c): From (4.3.3), and applying the concept of implicit differentiation, we get

(4.3.4)

Hence, .

Proof of (d): From (4.3.4) we have, . Hence, if , then is rotated clockwise around equilibrium state whenever decreases. On the other hand, if , then is rotated counterclockwise around equilibrium state whenever decreases. Thus, squeezes toward 45 degree line around equilibrium states whenever decreases.

Remark 4.3. The weighted sum of CAE process (4.3.1) and its dynamic model (4.3.2) can be modified to

and

, ,

respectively. The comments and Proposition 4.3.1 remain valid with regard to this modified CAE process.

4.3.2 Lagged Adaptive Expectation (LAE)

We modify an assumption of the CAE. Here, all consumers use the weighted sum of the current expected market share and the immediate past actual market share to adjust their immediate future market share as

,

(4.3.5)

where is the speed of adjustment defined in (4.3.1). Hence, the dynamic of adjustment process in the context of (4.2.2) and (4.3.5) is described by the second order nonlinear difference equation as

, .

(4.3.6)

Remark 4.3. The weighted sum of LAE process (4.3.5) and its dynamic model (4.3.6) can be modified to

and

, ,

respectively.

Remark 4.3. We observe that the equilibrium states of LAE process are the same as CAE, SE and RE processes, . In other words, the existence of equilibrium states is independent of type of consumer expectations. Of course, the existence of equilibrium states depends on the distribution of consumer preference random variable, price parameter of network good and the network externality function of a network good.

Table 4. Meaning and equilibrium condition of all processes

Process

Meaning

Equilibrium Condition

RE

SE

CAE

or

LAE

or

Stability

In this section, we study the analytic stability conditions of equilibrium states of four types of expectations. For easy reference, we state a few basic definitions of stability concepts.

An equilibrium state, , is called Lyapunov stable or locally uniformly stable (LUS), if for any closed enough initial states to , then stays close to for all time, that is, for each , there exist such that for all .

A Lyapunov stable equilibrium state is called locally uniformly asymptotically stable (LUAS), if for any closed enough initial states to , then converges to , that is, there exist such that .

A Lyapunov stable equilibrium state is called globally uniformly stable (GUS), if for any initial states , then stays close to for all time, that is, for each , there exist such that for all .

A Lyapunov stable equilibrium state is called globally uniformly asymptotically stable (GUAS), if for any initial states , then converges to . That is, for all , .

Now, we are ready to present the stability conditions for each discrete time iterative processes with regard to each types of expectation. We recall that the RE process is absolutely static.

4.4.1 Static Expectation (SE)

The stability of the equilibrium state of SE process depends on the analytic properties of .

Proposition 4.4. We assume that all consumers have static expectation. Further, we assume that the hypothesis of Proposition 4.2. is satisfied. Let and , where is the equilibrium state of (4.2.5).

If , then is GUAS equilibrium state,

If , then is GUS equilibrium state,

If , then is unstable equilibrium state.

Proof of (a): Let be an equilibrium state of (4.2.5). For any and the generalized mean-value theorem (Ladde et al (1985)), we have

.

(4.4.1)

From (4.2.3) and (4.4.1), we have

for .

(4.4.2)

From Proposition 4.2.2, (4.2.5) and (4.4.2), we have

(4.4.3)

By setting , , (4.4.3) is reduced to

, .

(4.4.4)

For some , applying the comparison theorem (Ladde and Sambandham, 1985) to (4.4.4), we get

,

(4.4.5)

where is the solution process of the following comparison iterative process

, .

(4.4.6)

Thus,

,

(4.4.7)

whenever . From (4.4.7) and the assumption that , we conclude that is Lyapunov stable equilibrium state. Moreover, , for all . This exhibits the GUAS property of .

Proof of (b): From (4.1), and , then , for all . Hence, . In view of this, it is obvious that is a GUS equilibrium state.

Proof of (c): Imitating the proof of (a), we have

.

(4.4.8)

Under the condition , and following the similar argument used in (a), we conclude that is an unstable equilibrium state.

Corollary 4.4. We assume that all assumptions of Proposition 4.4. remain valid, except statements (a) and (c) are replaced by:

If , then is a LUAS equilibrium state,

If , then is an unstable equilibrium state,

where is continuously differentiable at .

Proof of (a): Let be an equilibrium state of (4.2.5). We suppose that . Because of the continuity of at , there exists an interval , such that for all . In view of the convexity of an open interval, for all , , we have . Hence, . From this and (4.4.1), we have

, for .

Imitating the proof of Proposition 4.4.(a), we conclude that

.

(4.4.9)

From (4.4.9), for each , one can find , say so that , hence for all . In addition, . Therefore, is a LUAS equilibrium state.

Proof of (b): Let be an equilibrium state of (4.2.5). We suppose that . Again by repeating the reasoning used in the proof of (a), there exists an interval , such that for all . Moreover, we have

, for .

Hence,

.

Because , then is an unstable equilibrium state.

Remark 4.4. From Corollary 4.4., the LUAS condition of SE process is

(4.4.10)

Moreover, is an increasing function, . Thus, the solution is oscillation-free. See Figure 4. Stability and oscillating regions of CAE process with regard to (a) Remark 4.5.1 and (b) a.

4.4.2 Current Adaptive Expectation (CAE)

The stability of the equilibrium state of CAE process is parallel to the stability of the equilibrium state of the SE process. For the sake of completeness, we formulate the results. The detailed proofs can be reconstructed by repeating the arguments used in Section 4.4.1.

Proposition 4.4. We assume that all hypotheses of Proposition 4.4.1 remain true. Let and , where be an equilibrium state of (4.3.2).

If , then is a GUAS equilibrium state,

If , then is a GUS equilibrium state,

If , then is an unstable equilibrium state.

Proof of (a): Let be an equilibrium state of (4.3.2). For any and the generalized mean-value theorem, we have

.

Hence,

.

(4.4.11)

By repeating the argument used in the proof of Proposition 4.4.1, inequalities (4.4.2), (4.4.3) and (4.4.7) are reduced to

,

(4.4.12)

,

(4.4.13)

and

,

(4.4.14)

respectively. Moreover, under the assumption, , is a GUAS equilibrium state.

Proof of (b): The proof of (b) can be reconstructed from the proof of Proposition 4.4.1(b).

Proof of (c): Imitating the arguments used in the proofs of (a) and Proposition 4.4.1(c), we have

.

(4.4.15)

Thus, from the assumption, , one concludes that is unstable equilibrium state.

In the following, we formulate a result parallel to Corollary 4.4..

Corollary 4.4. We assume that all assumptions of Proposition 4.4. remain valid, except statements (a) and (c) are replaced by:

If, then is a LUAS equilibrium state,

If , then is an unstable equilibrium state,

where is continuously differentiable at .

Proof: The proofs of (a) and (b) can be constructed based on the proofs of Corollary 4.4.1 and Proposition 4.4.. The details are omitted.

Remark 4.4. We observe that the conditions , and , in Proposition 4.4. are equivalent to , and , respectively. Moreover, the conditions and in Corollary 4.4. are also equivalent to and .

Remark 4.4. The stability and instability results with respect to discrete time dynamic process described in Remark 4.3. can be formulated. In this case, the stability conditions (a), (b) and (c) in Proposition 4.4. reduce to , and , respectively, where is defined in Proposition 4.4.. Moreover, the stability and instability conditions of Corollary 4.4. become and , respectively. A remark similar to Remark 4.4. is valid with regard to the process described in Remark 4.3..

4.4.3 Lagged Adaptive Expectation (LAE)

First, we develop elementary notations and framework to rewrite the second order nonlinear difference equation (4.3.6).

For and . Let be the fixed state of function in (4.3.6), and let us define

.

From this, we have

,

Hence,

.

(4.4.16)

Let us define a transformation,

.

(4.4.17)

From (4.4.17) and (4.4.18), equation (4.3.6) is rewritten as

,

(4.4.18)

where ,

We note that is a linear operator. Therefore, , where is the eigenvalue of for which .

where and .

We note that for ,

(4.4.19)

We present the following Proposition:

Proposition 4.4. Let be an equilibrium state of (4.3.6).

From (4.4.19) and if , then is a LUAS equilibrium state,

From (4.4.19) and if , then is an unstable equilibrium state,

If and , then the solution process is a damped oscillatory,

If , then the solution process is an undamped oscillatory.

Proof: Using the analytic argument, the proof of these statements can be reconstructed. The technical details are omitted.

Remark 4.4. From and , we observe that the condition in Proposition 4.4. is equivalent to .

Remark 4.4. The stability and instability conditions with respect to discrete time dynamic process described in Remark 4.3. can be formulated. In this case, the eigenvalues are modified to

and .

A remark similar to Remark 4.4. is valid with regard to the process defined in Remark 4.3..

Speed of Adjustment

The speed of adjustment, , plays an important role in CAE and LAE processes. In (4.3.1) and (4.3.5), it is assumed to be . In this section, however, we relax this condition by allowing to be bigger than zero .

4.5.1 Current Adaptive Expectation (CAE)

Under the new range of , the stability and instability conditions of CAE process in Proposition 4.4. are modified in the following results. We simply state the results without the proofs. The proofs can be constructed by imitating the proof of Proposition 4.4..

Proposition 4.5. Let be an equilibrium state of (4.3.2) and and are defined in Proposition 4.4.2

If , then is a GUAS equilibrium state,

If , then is a GUS equilibrium state,

If , then is an unstable equilibrium state.

Likewise, the stability and instability conditions of Corollary 4.4. are modified and presented in the following corollary:

Corollary 4.5. Let be an equilibrium state of (4.3.2).

If, then is a LUAS equilibrium state,

If , then is an unstable equilibrium state,

where is continuously differentiable at .

In the following, we illustrate the significance of this corollary by providing the relationship between the speeds of adjustment and the speed of the market share adjustment process in the context of stability and oscillatory properties of the market share adjustment process.

Remark 4.5. The stability condition of for CAE process in Corollary 4.5.(a) is equivalent to the following condition:

.

(4.5.1)

Moreover, if , then the solution process oscillates with respect to the equilibrium state . This condition is equivalent to

.

(4.5.2)

In particular,

If , then is non-oscillatory LUAS. See region (A) in Figure 4.4a.

If , then is a damped oscillatory LUAS. See region (B) in Figure 4.4a.

If , if and only if then is an unbound non-oscillatory solution. See region (C) in Figure 4.4a.

If , if and only if then is an undamped oscillatory solution. See region (D) in Figure 4.4a.

Remark 4.5. The stability and instability conditions with respect to discrete time dynamic process described in Remark 4.3. in the context of arbitrary speed of adjustment process and Proposition 4.5. (a), (b) and (c) are described by , and , respectively. Moreover, the stability and instability conditions of Corollary 4.5. reduce to and , respectively. Moreover, in the light of Remark 4.5., the stability and oscillation conditions with regard to this case are as follows:

We assume that . Under this assumption, the stability conditions are reduced to the conditions in Remark 4.4..

We assume that , we draw the following conclusions:

if and only if . Under this assumption, is a non-oscillatory LUAS. See region (A) in Figure 4.4b.

if and only if . Under this condition, is a damped oscillatory LUAS. See region (B) in Figure 4.4b.

if and only if . Under this assumption, is unstable. See region (C) in Figure 4.4b.

if and only if . In this case, is an undamped oscillatory. See region (D) in Figure 4. Stability and oscillating regions of CAE process with regard to (a) Remark 4.5.1 and (b) b.

(a)

(b)

Figure 4. Stability and oscillating regions of CAE process with regard to (a) Remark 4.5. and (b) Remark 4.5.

4.5.2 Lagged Adaptive Expectation (LAE)

From and . Let be the dominant root, and hence,

Because of and , we have and

If , is a LUAS equilibrium state. This stability condition reduces to (Kocic and Ladas, 1993).

Remark 4.5. The stability condition of for LAE process is equivalent to the following condition:

and .

(4.5.3)

In addition, if , then the solution process oscillates with respect to the equilibrium state . This condition is equivalent to

.

(4.5.4)

In particular,

If and , then is a non-oscillatory LUAS. See region (A) in Figure 4.5c.

If and , then is an unbound and non-oscillatory solution. See region (C) in Figure 4.5c.

If and , then is a damped oscillatory LUAS. See region (B) in Figure 4.5c.

If and , then is an undamped oscillatory solution. See region (D) in Figure 4.5c.

(a)

(b)

(c)

Figure 4. Stability and oscillating regions of (a) SE (b) CAE and (c) LAE process.

Figure 4.5 exhibits the decomposition of the first quadrant into the subsets consisting of the ordered pair of parameters that create the various characteristics of equilibrium states. The parametric subset (A) is a non-oscillatory LUAS region; the parametric subset (B) consists of a damped oscillatory LUAS region; the subset (C) characterizes the unstable region and subset (D) consists of the undamped oscillatory region.

From (4.4.10), (4.5.1)-(4.5.4), the stability and oscillating conditions of SE, CAE and LAE processes are the same when , and are different when . See Figure 4.6 and Table 4.2.

Figure 4. Stability and oscillating regions of all processes

Table 4. Stability of equilibrium state and oscillatory of all processes

Region

SE

CAE

LAE

Stability

Oscillation

Stability

Oscillation

Stability

Oscillation

E

Yes

No

Yes

No

Yes

No

F

Yes

No

Yes

No

Yes

Yes

G

Yes

No

Yes

Yes

Yes

Yes

H

Yes

No

Yes

No

No

Yes

I

Yes

No

Yes

Yes

No

Yes

J

Yes

No

No

Yes

No

Yes

K

No

No

No

No

No

No

L

No

No

No

No

No

Yes

In the following remark, we compare the expectation type in the context of stability and oscillatory properties.

Remark 4.5.

In the case of SE, the parametric subset is composed two non-oscillatory regions, stable and unstable. Moreover, the stable region is the union of subsets and , while the unstable region is the union of subsets and , .

In the case of CAE, the stable region is the union of subsets and , while the unstable region is the union of subsets and , . The effects of the CAE process reduce the stability region and increase the instability region. In addition, the stability region of CAE process decomposes into two types of regions, namely non-oscillatory and damped oscillatory. The CAE process has destroyed the stability region of SE process, and it is replaced by the undamped oscillatory region.

In the case of LAE, the stable region is the union of subsets and , while the unstable region is the union of subsets and , . The effects of LAE process have further diminished the stability region and increased the instability region of LAE process.

In summary, the stability regions (SR), the instability regions (IR) and the oscillatory regions (OR) of SE, CAE and LAE processes are ordered as , and , respectively.

Initial State

In the previous sections, we discuss the stability of equilibrium states and found that the actual and expected network market share converge to the stable equilibrium state. However, in network goods, the multiple equilibriums are commonly assumed, so then the location of equilibrium is determined by the initial expected network market share of a consumer, . Using the Illustration 4.2., there are three equilibrium states. Figure 4. Solution path for various initial points of SE (a) and CAE (b) when , , price = 2.5 and is the solution paths of SE and CAE processes for various initial states.

(a)

(b)

Figure 4. Solution path for various initial points of SE (a) and CAE (b) when , , price = 2.5 and

For the SE process along with the CAE process, if the initial state,, then the solution paths will converge at ; if the initial state, , then the solution paths will be a fixed point ; if the initial state, , then the solution paths will converge at . Moreover, the adjustment process of the SE process is faster than the CAE process. In the case of the LAE process, we need a pair of initial states at a first and second period, . Figure 4.8 is a phase diagram of LAE process for various initial states. The solution paths will converge at either the lower or upper stable equilibrium state, if any pairs of initial states start in area (1) and area (2), respectively.

Figure 4. Phase diagram for various pairs of initial points of LAE process when , , price = 2.5 and

Conclusions

In the study of network goods, the market share has an influence on the utility. In this work, employing the idea of Katz and Shapiro (1985), we modify the utility function that depends on the value of network externality function at an expected market size/share at a real time, . Therefore, consumers must speculate the market share through their expectations. In this chapter, we briefly review two well-known expectations, namely, rational and static. We introduced a well-known expectation, namely, adaptive expectation with current information about the market share. We further extend this adaptive expectation by employing lagged information. By utilizing the modified utility function, the concept of market share adjustment function is introduced, and further dynamic models corresponding to consumer rational, static, and current and lagged adaptive expectation processes are developed in a systematic and unified way. By using mathematical tools, we establish the existence of equilibrium state and its independence of adaptive expectations. Furthermore, the qualitative properties of equilibrium states (stability, oscillatory) depend on the dynamic models and speed of adjustment parameter. The introduction of adaptive processes generates both damped and undamped oscillations. Moreover, for , the stability results are independent of the types of expectation. However, when , the stability and the oscillation properties depend on both the speed of adjustment and the speed of the market share adjustment process at the equilibrium state . In fact, the positive quadrant described by can be decomposed into mutually disjointed subsets. Moreover, this leads to the decomposition of the positive quadrant that is based on the properties of the equilibrium states of the market share adjustment process. These results provide tools for policy and decision making processes for network goods.