Models for market shares by imitation or innovation

We compare deterministic and master-equation treatments of imitation and innovation processes by two types of firms. We expect the deterministic so­lutions to be identical or analogous to those of the mean dynamic equations provided the variances about the mean asymptotically vanish, although we should be aware of the possibility of disappointment, as Feller's example in Chapter 5 shows.

We provide additional illustration of these generating-function techniques in the context of market shares with two types of firms, one technically advanced and the other technically less advanced. The exact natures of the two classes of firms are not important so long as firms of one class can become firms of the second class.

To be concrete, we consider a collection of n firms. This assumption allows us to use one state variable rather than two, by taking the number of firms of the advanced type to be the state variable, k = 0, 1, 2, ∙ ∙ ∙, n, where n is fixed. The disadvantage of this approach is that we must provide separate boundary conditions for k = 0 and k = n, which look different from the master equations valid for 0 < k < n. Since we are merely illustrating the use of generating functions, we do not bother with these boundary conditions here. In Chapter 5 we show how to deal with models with two state variables without assuming that the total number of firms is exogenously fixed. Elsewhere, we have dealt with a model with two types of agents. There, the transition rates w(k, k + 1) = λn(1 - k∕n)η₁(k∕n) and w(k, k - 1) = μn(k∕n)η₂(k∕n) have been used. In other words, the birth and death rates are not constant but state-dependent. Then, the functional form of η₁ is specified as e^eh(k/n)/{e^eh(k/n) + e^-eh(k/n)}, where β is a parameter to incorporate uncertainty or imprecise information about alternative choices, and h(∙) is a function equal to the difference of means of the alternative discounted present values associated with the alternative choices.

7.3.1 Deterministic innovation process

Suppose we divide the firms into two groups: group A of k firms with superior technologies to the firms in group B, consisting of the remainder, i.e., n — k firms. The identities of the firms belonging to the groups are not important. What matters is the number - or the value of the fraction, x = k/n - of firms belonging to group A.

Suppose, for the sake of simplicity, that an innovation, when it occurs to firms of group B, turns them into members of group A. The fraction is then often modeled by

The solution is

This shows that eventually all firms belong to group A, that is, the fraction converges to 1. The reason, of course, is that no firm leaves the market; hence eventually all firms belong to group A. Our purpose is not to implement more realistic assumptions, but rather to justify the ordinary differential equation above, which is often used without much justification.

We now reformulate this process as the birth process with transition rate w(k, k + 1) = λ(n - k). This specifies that firms of the less advanced class can have individual probability rates λ of advancing to the superior class. The master equation is

except for the boundary conditions for) and, with which we do not bother here.

Next, convert this equation into the partial differential equation for the prob­ability generating function,

We can solve this equation by the characteristic-curve method as shown in Section 7.2. When we do this, G(z, t) approaches zⁿ as t goes to infinity for all initial numbers of firms in group B. This shows that all firms become technically advanced as time progresses.

Our purpose is to point out that the posited ordinary differential equation can be derived from the cumulant generating function, to show the relations between it and the partial differential equations, and to show that it can be used by itself, since no coupling exists between the mean and variance dynamics for this simple process.

7.3.2 Deterministic imitation process

We next introduce interactions between firms of different classes. We follow Iwai (1984a,b, 1996) and assume that firms in group B individually imitate firms in group A and succeed in becoming members of group B at the rate μ. Using the deterministic approach, this is expressed by²

Writing this as dx∕x(1 - x) = μ dt and integrating it, we obtain its solution as

Again, we see that all firms succeed in becoming members of the advanced class.

In terms of the master equation, we respecify the transition rate to be w(k, k + 1) = μk(n — k), in

Now, the partial differential equation for the probability generating function is slightly more complicated:

The ordinary differential equations for the mean and variance are now cou­pled:

and

However, as κ₁ approaches n, the effect of the variance on the mean vanishes.

The reader may have noticed the absence of the phenomenon of firms going out of business - the death process in the birth-and-death process models. When bankruptcy is modeled, we must drop the assumption that n is fixed exogenously. See Chapter 10 for discussions of processes with both birth and death effects on firm market shares.

7.3.3 A joint deterministic process

We now combine these two effects into a single equation:

We can solve this directly by noting that

The solution is

where γ = λ + μ, and where C = (λ + μx₀)∕(1 — x₀). Unsurprisingly, x(t) goes to one as time progresses.

The reason for this limiting behavior is clear. No firms leave the market, and eventually all firms belong to group A. To remedy this, we must allow some firms to leave the market or go bankrupt. We need to abandon the assumption of a fixed number of firms unless we artificially allow entry to keep the number fixed. See Chapter 10 for this alternative model.

7.3.4 A stochastic dynamic model

Next, we set up the master equation for the joint processes to model market shares of group A. We assume that n is fixed. We take the transition rate of the number of firms in group A from k to k + 1 to be

Denote the probabilitywhere k_A is the number of

firms of group A. It is governed by

for k = 1,..., n. There is an obvious boudary equation for k = 1. and denote

Then, the left-hand side of the master equation becomes

This change of variable was introduced earlier in Section 5.2. See also Aoki (1996a, p. 123) or Aoki (1995). As it turns out, to match orders of magnitude of terms on both sides, we need to change the time scale as well by

7.4