Disappearance of goods from markets

In the standard economic literature, diminishing returns to capital stocks es­sentially restrain economic growth. The model of this section is constructed to have its growth impeded by demand saturation, and is led by new goods, which randomly appear on the markets.^[VII] We examine a process of invention of goods and disappearance of goods as a nonlinear birth-death process.

Put simply, we assume that the stochastic process for demand changes is a birth-and-death process with birth rate λ and death rate μn_t. The only non­standard feature is a nonlinear death rate in order to embody an idea that older products have higher probability of dying out. This aspect is somewhat rem- niscent of the old-age effect of Arley referred to in Kendall (1948b). Arley use μt with constant μ to indicate that older particles die faster, probabilistically, than newly formed particles in his study of cosmic showers. We can handle an alternative senario of constant death rate and diminishing birth rate by re­placing λ by λ∕n_t. Basically the same qualitative conclusion follows from this alternative. We do not pursue this alternative further.

8.2.1 Model

Write the probability that output is n (in a suitable unit) as P_n(t). The master equation for this growth process is

n > 1. The boundary condition is ∂P₀(t)∕∂t = μP₁(t). We assume that λ > μ. In this model the birth rate is a constant λ, but the death rate is taken to be μ times the number n. This effect may be congestion effects or old-age effect.

We can solve the master equation for a steady-state (stationary) distribution by setting the left-hand side equal to zero and replacing P_n (t) by π_n.

n > 1. This has the solution

and the cumulant generating function is

where K(θ, t) = G(e ^θ, t).

Unfortunately, the equations for the cumulants do not terminate at any finite moments. By assuming that the steady state of κ₃ is a small bounded number, we can solve for the steady-state values of the first two moments by ignoring this term. We then examine if the linearized differential equations are asymptotically stable. If the answer is yes, then the stationary values are such that the third cumulants are zero. For certain ranges of the ratio λ/μ, there are two steady-state variance values for a positive stationary value of the mean.

8.2.2 Stability analysis

We drop the time argument for simplicity. Assume that μ < λ ≤ 2μ. With this assumption, the linearized equations for x and v about x∞ and v∞ are asymptotically stable, as we show later, and the stationary values are obtained by setting the left-hand side of the differential equation equal to zero, and assuming that the third central moment remains bounded for all time and has a stationary value as well. The stationary values are given by

The eigenvalues of this set of two equations have negative real part if the trace is negative and the determinant is positive of the matrix of the two equations above. At x∞ = λ∕μ, for example, v∞ is zero, and the two eigenvalues of this line­arized equation both have negative real part, under the condition μ ≤ λ ≤ 2μ.

8.3