Growth with Expanding Product Varieties

Finally, we will briefly discuss the equivalent model in which growth is driven by product innovations, that is, by expanding product varieties rather than expanding varieties of inputs.

where

is the consumption index, which is a CES aggregate of the consumption of different varieties. Here c (ν, t) denotes the consumption of product ν at time t, while N (t) is the total measure of products. We assume throughout that ε > 1. Therefore, we have replaced expanding input varieties with expanding product varieties. The log specification in this utility function is for simplicity, and can be replaced by a CRRA utility function.

The patent to produce each productbelongs to a monopolist, and the

monopolist who invents the blueprints for a new product receives a fully enforced perpetual 491

patent on this product. Each product can be produced with the technology

As in model with knowledge spillovers of Section 13.2, we assume that new products can be produced with the production function

The reader will notice that there is a very close connection between the model here and the models of expanding input variety studied so far, especially the model with knowledge spillovers in Section 13.2. For instance, if y (ν, t)s were interpreted as intermediate goods or inputs instead of products and if C (t) in (13.39) were interpreted as the production function for the final good rather than part of the utility function of the representative consumer, the two models would be essentially identical.

An equilibrium and a balanced growth path are defined similarly to before. The represen­tative household now determines both the allocation of its expenditure on different varieties and the time path of consumption expenditures. We assume that the economy is closed and there is no capital, thus all output must be consumed. Nevertheless, the consumer Euler equation will apply to determine the equilibrium interest rate. Labor market clearing requires that

Let us start with expenditure decisions. Since the representative household has Dixit- Stiglitz preferences, the following consumer demands can be derived (see Exercise 13.24): where p^x (ν, t) is the price of product variety ν at time t, and C (t) is defined in (13.39). The term in the denominator is the ideal price index raised to the power —ε. As before, it is most convenience to set this ideal price index as the numeraire, so that the price of output at every instant is normalized to 1. Thus we impose

With this choice of numeraire, we obtain the consumer Euler equation as (see Exercise 13.25):

With similar arguments to before, the net present discounted value of the monopolist owning the patent for product ν can be written as

where w (t) c(ν, t) is the total expenditure of the firm to produce a total quantity of c(ν, t) (given the production function (13.40) and the wage rate at time t equal to w (t)), while is its revenue, consistent with the demand function (13.43).

Since all firms charge the same price, they will all produce the same amount and employ the same amount of labor. At time t, there are N (t) products, so the labor market clearing condition (13.42) implies that

Consequently, the instantaneous profits of each monopolist at time t can be written as

Since prices, sales and profits are equal for all monopolists, we can simplify notation by letting

In addition, since c(ν, t) = c (t) for all ν,

where the second equality uses (13.46).

Labor demand comes from the research sector as well as from the final good producers. Labor demand from research can again be determined using the free entry condition. As­suming that there is positive research, so that the free entry condition holds as an equality, this takes the form

Combining this equation with (13.47), we see that where we use π (t) to denote the profits of all monopolists at time t, which are equal. In BGP, where the fraction of the workforce working in research is constant, this implies that profits and the net present discounted value of monopolists are also constant. Moreover, in this case we must have

where r* denotes the BGP interest rate.

with.denoting the BGP size of the research sector. The R&D employment level of combined with the R&D sector production function, (13.41) then implies

However, we also know from the consumer Euler equation, (13.45) combined with (13.48) which implies

or

Consequently, the growth rate of consumption expenditure (and utility) is

This establishes:

Proposition 13.6. In the above-described expanding product variety model, there exists a unique BGP, in which aggregate consumption expenditure, C (t), grows at the rate g* given by (13.50).

A couple of features are worth noting about this equilibrium. First, in this equilib­rium, there is growth of “real income,” even though the production function of each good remains unchanged. This is because, while there is no process innovation reducing costs or improving quality, the number of products available to consumers expands because of prod­uct innovations. Since the utility function of the representative household, (13.38), exhibits love-for-variety, the expanding variety of products increases utility. What happens to income 494

depends on what we choose as the numeraire. The natural numeraire is the one setting the ideal price index, (13.44), equal to 1, which amounts to measuring incomes in similar units at different dates.

13.3.