5.1 Growth from Expanding Varieties

The modern theory of endogenous growth is rooted in the idea that technological change is not an accident but a purposeful economic activity driven by the pursuit of profits. The first major class of such theories, pioneered by Paul Romer, are expanding variety models. In these models, long-run growth is driven by the invention of new types of goods—specifically, new varieties of intermediate inputs or capital goods that can be used in production. This type of innovation is often called “horizontal” because it increases the breadth of technology.

In the “lab equipment” version of the model, a single final good is produced competitively using labor and a range of differentiated intermediate machines. The aggregate production function takes the form: Y(t) = (1/(1-β)) [∫N(t)₀ x(ν,t)^(1-β)dν] L^β Here, L is labor, x(ν,t) is the quantity of machine variety ν used at time t, and N(t) is the total number of available machine varieties. The crucial variable N(t) represents the stock of knowledge or the level of technology in the economy. The structure of this production function implies that a larger number of machine varieties makes labor more productive, creating an engine for growth.

The market structure is key. While the final good is produced competitively, each machine variety ν is produced by a single firm that holds a perpetual patent on its design, making it a monopolist. New machine designs are created in an R&D sector, where the cost to invent a new design is 1/η units of the final good.

The equilibrium of this economy unfolds as follows:

1. Final good producers have a downward-sloping demand curve for each machine variety. Because of the production function’s structure, this demand curve is isoelastic.
2. Faced with this isoelastic demand, each monopolist maximizes profit by setting the same price: a constant markup over its marginal cost of production.
3. Because all machines are symmetric and have the same price, they are used in the same quantity. Substituting this back into the aggregate production function reveals a crucial result: aggregate output is linear in the number of varieties, Y(t) = (Constant) * N(t)L. This “AN” structure, analogous to the AK model, is what allows for sustained long-run growth.

The rate of innovation is determined by a free-entry condition in the R&D sector. Firms will invest in creating new designs as long as the cost of invention is less than or equal to the expected reward. In equilibrium, the cost of invention (1/η) must be exactly equal to the present discounted value of the stream of future monopoly profits (V(t)) that a new patent generates. This arbitrage condition pins down the equilibrium interest rate, r*, in the economy.

Finally, we connect this interest rate to the household’s consumption-saving decision. The household’s Euler equation requires g = (1/θ)(r* – ρ). Substituting the equilibrium interest rate derived from the free-entry condition gives the economy’s long-run balanced growth rate: g* = (1/θ)(ηβL – ρ) This equation reveals a key prediction of the model: the scale effect. A larger population (L) leads to a faster rate of economic growth. The intuition is that a larger population represents a larger market for any new invention, which increases the profitability of R&D and spurs faster innovation.

The equilibrium growth rate in this model is typically not socially optimal. This is due to a primary “consumer surplus” externality: innovators capture only the monopoly profit from their invention, not the full social value (consumer surplus) it creates for final good producers. A second, though less pronounced, externality is the “business-stealing” effect; future innovations will eventually erode the profits of current patent-holders, and this destruction of rents is not internalized by R&D firms. The consumer surplus effect generally dominates, leading to an underinvestment in R&D and a growth rate that is too low compared to the social optimum.

These models explain growth through the creation of new types of goods. The next class of models explores a different dimension of innovation: improvements in the quality of existing goods.

5.2 Schumpeterian Growth and Creative Destruction

As an alternative to expanding variety models, Schumpeterian growth models, pioneered by Philippe Aghion and Peter Howitt, focus on “vertical” innovation. In this framework, R&D is aimed at improving the quality of existing products, rather than creating entirely new ones. This process embodies Joseph Schumpeter’s famous concept of “creative destruction,” where each new innovation displaces the existing technology and, with it, the incumbent firm that produced it. Growth arises from a continuous process of old technologies being rendered obsolete by new, superior ones.

In a baseline Schumpeterian model, the final good production function depends on the quality q(ν,t) of the intermediate machines being used: Y(t) = (1/(1-β)) [∫¹₀ q(ν,t)x(ν,t)^(1-β)dν] L^β The key measure of aggregate technology is now the average quality of all machines in the economy, Q(t) = ∫¹₀ q(ν,t)dν.

The innovation process is inherently competitive. R&D is undertaken by potential entrants who are trying to improve upon the current leading-edge technology in a given product line. A successful innovation increases the product’s quality by a fixed proportional step, λ > 1. The successful innovator earns the right to be the new monopolist for that product line, but their reign is temporary; they too will eventually be displaced by a future innovator.

In equilibrium, the successful innovator does not charge the unconstrained monopoly price. Instead, they engage in limit pricing, setting their price just low enough to drive the producer of the previous-generation technology out of the market. This allows the innovator to capture the entire market and earn a flow of profits.

The aggregate rate of innovation in the economy, z*, is determined by a free-entry condition, similar to the expanding variety model. Potential innovators will invest in R&D as long as the expected value of becoming the new monopolist is greater than or equal to the cost of R&D. This arbitrage process pins down the equilibrium level of R&D activity and thus the arrival rate of innovations.

The aggregate growth rate of the economy is then determined by two factors: the arrival rate of innovations (z*) and the size of each quality improvement (λ-1): g* = (λ-1)z*

The welfare properties of the Schumpeterian model are more complex than those of the expanding variety model. In addition to the consumer surplus externality (which pushes for too little innovation), there is a powerful “business-stealing effect.” When an innovator decides to invest in R&D, they do not account for the fact that their success will destroy the profits of the incumbent firm they are replacing. This negative externality pushes towards too much innovation. The interplay between these two opposing forces means that the equilibrium rate of innovation can be either inefficiently low or inefficiently high compared to the social optimum.

Both the expanding variety and Schumpeterian models typically assume that technology improves in a “neutral” way, augmenting all factors of production symmetrically. This simplification motivates our final topic: models where the very direction of technological change is also an endogenous economic choice.

5.3 Directed Technological Change

The models we have studied so far focus on the overall rate of technological progress. However, in reality, technological progress is often biased, meaning it complements some factors of production more than others. For example, many of the innovations of the late 20th century appear to have been “skill-biased,” increasing the productivity and wages of highly educated workers more than those of less-educated workers. Models of directed technological change endogenize this bias by allowing innovators to choose which type of technology to develop based on profit incentives.

The canonical model features two factors of production, such as skilled labor (H) and unskilled labor (L). The final good is produced by combining two specialized intermediate goods: one made with unskilled labor and L-complementary machines, and the other made with skilled labor and H-complementary machines. Innovators can choose whether to invest in creating new L- or H-complementary machines.

The direction of innovation is determined by the interplay of two competing forces:

1. The Price Effect: Innovating for the scarcer factor of production (e.g., H if H/L is low) is more profitable because the intermediate good it produces is also scarce and thus commands a higher price. This effect encourages the development of technologies that are substitutes for scarce factors.
2. The Market Size Effect: Innovating for the more abundant factor is more profitable because there is a larger market of complementary factors to use the new technology. For example, if there are many skilled workers, the market for a new H-complementary machine is large. This effect encourages the development of technologies that are complements to abundant factors. This effect is a direct consequence of the non-rival nature of technology; a new design for an H-complementary machine can be used by all H-workers, making the size of the H-labor force a critical determinant of the innovation’s profitability.

A central result in this literature is that, under general conditions, the market size effect always dominates the price effect. This leads to a powerful conclusion: an increase in the relative supply of a factor will always induce technological change that is biased towards that factor. For example, an increase in the relative supply of skilled labor (H/L) will trigger a wave of skill-biased technological change.

This result has profound implications for the long-run elasticity of substitution between factors. In standard models, an increase in the supply of a factor lowers its price. Here, however, the induced technological change works in the opposite direction. If the market size effect is strong enough, the long-run relative demand curve for factors can become upward-sloping. This provides a compelling explanation for the puzzle of the rising skill premium in the United States and other advanced economies since the 1980s. During this period, a large increase in the relative supply of college-educated workers occurred simultaneously with a sharp rise in their relative wages. The theory of directed technological change suggests that this massive influx of skilled labor induced such a strong wave of skill-biased innovation that it more than offset the usual downward pressure on wages, causing the skill premium to rise.

——————————————————————————–

Our journey through the theory of modern economic growth has taken us from the foundational empirical puzzles of global inequality to the sophisticated mechanics of endogenous innovation. We began with the Solow model, which provided a powerful framework for understanding capital accumulation but treated technology as an exogenous “black box.” We then moved to the Ramsey model, which gave our theory of saving microeconomic foundations but still relied on unexplained technological progress for long-run growth. This led us to the core of modern growth theory: models where technology is endogenous. We explored how sustained growth can arise from the creation of new varieties of goods, from the Schumpeterian process of creative destruction, and finally, from the directed, profit-driven choice of the bias of new technologies. This body of theory, while still evolving, provides a powerful toolkit for understanding the deep mechanics of economic growth and the fundamental sources of the vast income differences that shape our world.