Shares of dated final goods among households

This section applies the techniques and elementary building blocks for models, discussed in Chapter 7, to analyze a model in which new final goods become available randomly over time, and they are being adopted or purchased gradu­ally by some fraction of n households.

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The mechanism of growth of the model is basically that of the Ramsey model. Unlike the latter, which relies on the shift of preference of a representative house­hold for the engine of growth, growth in our model is due to diffusion or spread of consumption of the newly available goods for purchase by the households. Spread of consumption of new goods among households creates demand, which This section is based in part on Aoki and Yoshikawa (2000).

in turn induces capital investment and growth. Higher growth rate creates higher income among households, which induces more households to consume.

Because the amount of final goods purchased by households is bounded, growth of production of goods necessarily decelerates. Creation of new goods is the ultimate engine to sustain growth in the model.

8.3.1 Model

For simplicity, assume that there are n households, where n is exogenously fixed. Household i either buys or does not buy good j at time t. See Aoki and Yoshikawa (2000) for full specification of themodel.

The purchase pattern is denoted bywhich is 1 if household i

purchases good j at time or period t and zero otherwise. Actually this depends also on the epoch τ at which good j has appeared on the market. To shorten notation, we sometimes drop this argument. We should and can treat q_ij as stochastic, but for simplicity we stay with the deterministic version.

The total number of households that buy good i is given by

where dj (t, τ) is the number of households that buy good j, which has existed since time τ.

We model the spread of purchase shares among the households by the birth­death process discussed in Chapter 7. As discussed there, we incorporate a nonlinear death rate in the model to reflect the assumption that demands for older goods decline with time.

The (expected) value of the mean of the demand for good j, then, has the S-shaped time profile

This expression is obtained by solving a nonlinear master equation, having μ as the birth rate and jn as the nonlinear exit or disappearance rate. Recall our discussion of a model with a nonlinear exit rate in Chapter 4.

Next, assume that each household has the saving rate s, which is assumed to be the same for all households. Households purchase 1 - s units of any final goods that they consume.

The budget constraints for household i is

where I_i (t) is the income of household i, and m_i,ι is the purchase share of good 1, which is the initially available good at time 0. Here we assume that the number of goods initially available is 1. This budget constraint simplifies to

Recall our discussion of the solution of the master equation

where Q(n, t) is the probability that the number of final goods available to the households is n at time t. We have seen that

Summing the incomes of all the households, we arrive at

where Y (t) is the GDP of this economy

The equilibrium condition of the goods market is s(ρ(t), ψ(t)) = φ(g), where g is the growth rate of the economy, φ(∙)K is the investment in the economy with capital stock K, and ψ is a shift parameter of the saving rate. In the stationary state

where * indicate logarithmic derivativefor example), and ρ is

the instantaneous rate of discount.

8.4