The Manhattan metaphor

Fixed operating costs draw a sharp distinction between endogenous growth based on horizontal and vertical innovation: a larger number of product lines puts pressure on an economy's resources; greater productivity of existing product lines does not. Consequently, the only plausible engine of endogenous growth is vertical innovation whereby progress along the quality or cost ladder does not require the replication of fixed costs. Is, then, product variety expansion irrelevant? No. The two dimensions of technology are complementary in that using one and the other produces a more comprehensive theory of economic growth. The vertical dimension allows endogenous growth unconstrained by endowments, the horizontal provides the mechanism that translates changes in aggregate variables into changes in product-level variables, which ultimately drive incentives to push the technological frontier in the vertical dimension. We show that the potential for exponential growth due to an externality that makes entry costs fall linearly with the number of products, combined with the limited carrying capacity of the system due to fixed operating costs, yields logistic dynamics for the number of products. This desirable property allows us to provide a closed-form solution for the model's transition path and thereby derive analytically the welfare effects of changes in parameters and policy variables. Our Manhattan Metaphor illustrates conceptually why we obtain this mathematical representation when we simply add fixed operating costs to the standard modeling of variety expansion.