Two simple models for the emergence of new goods
Two models are introduced to explain how new goods appear. In the first model, called the Poisson model, firms or sectors independently invent new goods or improve on the existing ones.
The numbers of new goods are then functions of the number of firms, or the size of the economy. The growth rate of this model eventually converges to that of the rate at which new goods are being introduced into the economy. In the second, called the urn model, the number of goods that are introduced is independent of the numbers of the existing goods, and the rate of introduction of new goods decreases as the number of goods increases.8.1.1 Poisson growth model
Let Qk(t) be the probability that there are k goods or sectors in the economy at time t. Each firm (sector) independently has probability λ Δ t in a small time interval ∆t of introducing one new good (sector). Thus the total numbers of the final goods (sectors) go from k to k + 1 in time interval of ∆t with probability 
where N(t) is the number of goods being produced at time t, and λk is the overall rate of new goods being introduced, on the assumption that sectors act independently The probability Qk (t) is governed by the differential equation 
with the initial condition
We assume goods once introduced
are not withdrawn from the market. A different model is later discussed, which has different effects on goods to be introduced to the markets in the future. For simplicity take k0 to be one.
Solving this differential equation, the probability is given by
Suppose that output at time t of a good that was introduced at time s, s ≤ t, is given by
For definiteness assume that μ > υ.
This expression implies that the output changes monotonically from 1 at the time of introduction of the good, and incorporates the assumption that the output eventually levels off at μ∕ν as time goes to infinity
The second term is the output of the original firm that exists in the economy at time zero. Using the generating function to express the sum, it is straightforward to show that the rate of growth of the total output converges to λ, that is, the rate of entry of new goods (sectors) as time goes to infinity:
since d ln y(t, 0)/dt → 0.
In this subsection, we have used a constant λ. More realistically, λ may be a decreasing function of N. For example, by interpreting N as proxy for the stock of R&D, diminishing returns to R&D due to congestion in research, increasing difficulties, and such may cause the growth rate to approach zero, as in Jones (1995), Jones and Williams (1998), Segerstrom (1998), and Young (1998) among others. In the next model, we present a different take on this aspect.
8.1.2 An urn model for growth
The model of this subsection incorporates the idea that goods/sectors that will emerge are not directly linked to R&D, so that their rate of emergence is not tied to the birth rate of the Poisson process, but is strongly conditioned on some opportunities for innovations, which are independent of stock or flows of R&D, such as advances in basic scientific knowledge. We assume that a new good or sector is introduced at time t with probability
for some positive ω, and t = 1, 2,.... We use a discrete-time description for brevity.
Here, the rate of innovation is simply a descreasing function of time.
This probability may be regarded as a probability of drawing a black ball from an urn that initially contains ω black balls. After each drawing, the drawn ball is returned, and one white ball is added to the urn. At time t, then, the urn contains ω black balls and t white balls. Urns to which one or more balls of different colors are added belong to a class of urn models called Polya urns. Such models are extensively used in population genetics models. See Hoppe (1984, 1987), or Appendix A.2.Polya urns are used also in the standard R&D-based total factor productivity models such as Jones (1995), Jones and Williams (1998), Segerstrom (1998), and Young (1998). Their rate of innovation is a decreasing function of the R&D capital stock. Here, the rate of innovation is simply a decreasing function of time.
The probability of k goods being available in the market at time t is denoted as before by Qk(t), but is now governed by the difference equation
with the boundary conditions
and
The first is the probability that no new goods are introduced up to time t, and the second is the probability that new goods are introduced at each and every period up to time t.
We introduce a notation for the ascending factorial
The difference equation has the solution
Here, we have introduced also an important number, the (unsigned) Stirling number of the first kind. It satisfies the recursion relation
This number is also defined by
for some positive integer m, i.e., the coefficient of xj in the expansion of the ascending factorial x[m].
See Appendix A.5 on Stirling numbers, as well as Abramovitz and Stegun (1968, p. 825). Aoki (1997, p. 279) has some nongenetic applications.The total output is now given by 
where y(t - l) is the production at time t of the final good that emerged at time l, l < t.
For simplicity take ω to be a positive integer. Then
Thus, approximately,
Yt = ln(ω + t).
The growth rate goes asymptotically to zero.
Alternatively, suppose that P(t) depends on the number of existing goods. We define the probability that the kth good is invented during period t by
In this case, the rate of growth is given by
When opportunities for innovation declines, sustained growth is not possible. Solow (1994) makes a similar point about endogenous innovations. He points out that if R&D does not produce a proportional increase in the (Hicks-neutral) technical progress factor A in the production function AF(K, L), but only an absolute increase in A, then greater allocation of resources to R&D buys a one-time jump in productivity, but no faster productivity growth. The model in the first Subsection 8.1.1 corresponds to proportionate growth in A, and the second to an absolute increase in A.
8.2