Yes we can! : factor-augmenting technical change without Cobb-Douglas and despite Uzawa

The standard result in the literature states that balanced growth is only possible with labor-augmenting technological progress. If a more general form of technical progress is considered, the Cobb-Douglas production function is typically required (Uzawa, 1961). However, we show that this result is not universally true. A broader class of functions, known as Karamata functions or regularly varying functions, allows for both labor- and capital-augmenting technical progress. This generalizes the overlooked work of Sato and Beckmann (1970), who first challenged Uzawa's conclusion but in a more restrictive setting. Karamata functions, widely used in mathematical statistics, describe functions with predictable long-term growth while allowing for slow variations. Despite their suitability for scaling behavior in production, they have never been applied in economics. Our approach also aligns with the HouthakkerJones framework and complements solutions to Uzawa's paradox. We demonstrate that factor-augmenting technical progress is fully compatible with balanced growth under general conditions, challenging the conventional view that Cobb-Douglas is the only valid form. We further provide some examples, as well as functions with controlled oscillations, compatible with balanced growth.