A stochastic model with innovators and imitators

We examine an open model with two types of firms, called innovators and imita­tors, to discover the market shares of the two types. We incorporate asymmetric interactions between the two types of firms.

Let n = (n₁, n₂) be the state vector. We assume that the transition rates are such that

where type 1 firms grow at the innovation rate f₁ plus a proportional growth rate of c₁ n₁. On the other hand type 2 firms, which are less technically advanced that those of type 1, grow at the rate

where /2 < /1.

The two types go out of business at the rates

j = 1, 2. We assume that type 2 firms fail more often than type 1 firms: d₂ ≥ d₁. By imitating type 1 firms, a type 2 firm may become type 1 at the rate

while a type 1 firm may slip back to type 2 at the rate

With these transition rates, it is easily verified that the steady-state probability distribution exists by imposing the detailed-balance conditions, provided λ₁₂ = λ₂₁. Let g₁ = c₁∕d₁, and g₂ = /₂∕d₂. We define μ := λd₁d₂. Then we can write the transition rates more succinctly by noting that λ₂₁d₂c₁ = μg₁ and ^λ12 /2 ^d1 = μ^g2^:

while a type 1 firm may slip back to type 2 at the rate

The number of type 1 firms is governed by a negative binomial distribution, while the number of type 2 firms has a Poisson distribution,

with

and

We next examine the nonstationary solution of the master equation

and

Then, expanding

we derive coupled ordinary differential equations for the first and second mo­ments. Unfortunately, the ordinary equations are not solvable in closed form.

We mention some special cases by imposing some conditions on the para­meters in the transiton rates.

7.4.1 Case of a finite total number offirms

First, assume that g₂ = 0, that d₂ = d₁ — c₁, and that h ι is much smaller than these parameters. The first assumption means that no type 1 firms become type 2 firms. Once technically advanced, firms remain technically advanced. The second assumption means that the net dropout rate of type 1 firm is the same as that of type 2 firms. The third assumption means that c₁ is much larger than f₁, that is, the rate of growth of type 1 firms comes primarily from existing firms generating new firms of the same type and not from new entries. Then,

hence the sum of the numbers of firms of both types asymptotically approaches

Another consequence of these parameter values is that κ₁₂ is constant. Suppose that it is zero at time 0. Then, it remains zero for all times. With g₂ = 0 and h₁ = 0, we can drop the assumption that d₂ = d₁ — c₁ and solve for κ₁ and κ₂ separately, because

and

In examining the joint dynamics for κ₁ and κ₂, we see that the covariance κ₁₂ acts on the time derivatives of κ₁ and κ₂ with opposite signs and the same magnitude.

7.5