Arrow’s Learning-by-Doing Explained | RBC Model with Spillovers in Dynare

Introduction

In 1962, Kenneth Arrow introduced one of the most influential ideas in modern macroeconomics: learning-by-doing. Unlike the standard Real Business Cycle (RBC) model, which relies entirely on exogenous productivity shocks to drive fluctuations, Arrow’s framework recognizes that productivity can also emerge endogenously through experience and capital accumulation.

This article explains how learning-by-doing modifies the RBC model, what the new parameters mean, and how the mechanism generates knowledge spillovers that amplify and propagate shocks. It also discusses how the model can be implemented in Dynare and what we learn from simulations of impulse response functions (IRFs) and time series.

From Standard RBC to Arrow’s Learning-by-Doing

The standard RBC model explains business cycle fluctuations with households making consumption and labor decisions, firms maximizing profits, and productivity evolving through external shocks. In this setup, technology is taken as given and moves randomly over time.

Arrow’s insight was that productivity can also improve internally. As firms invest in and use capital, workers, managers, and entire industries learn how to use it more effectively. This learning is not confined to individual firms but spreads across the whole economy. The more the economy invests, the more knowledge accumulates.

Firms and the Knowledge Spillover

In Arrow’s framework, firms do not just benefit from their own investment. They also benefit from the overall stock of capital in the economy. When one firm adopts new machines and workers learn to operate them, knowledge spreads through labor mobility, suppliers, industry networks, and even education. This creates a spillover effect where productivity rises for all firms.

This means that even a firm that does not invest heavily still benefits from the collective knowledge generated by the economy’s capital accumulation. The result is higher productivity across the board, compared to a world where each firm’s knowledge is fully isolated.

The Role of the Parameters

Two new parameters capture the strength of learning-by-doing:

• Theta: a scale parameter that determines how much knowledge is produced from capital. A higher theta means more knowledge for the same level of capital, but it mainly changes the scale rather than the shape of the response.

• Phi: the elasticity of spillovers, which measures how sensitive productivity is to capital accumulation. If phi is zero, the model reduces to the standard RBC with no spillovers. If phi is positive, productivity increases with capital and growth becomes partly self-sustained.

In practice, researchers often calibrate theta around 0.3 and phi around 0.1. These values create meaningful spillovers without making the model unstable or explosive.

Model Equations

The equations describe household behavior, firm optimization, government budget, resource allocation, and the productivity process. The key difference from the standard RBC model is that productivity now depends not only on external shocks but also on capital accumulation through learning-by-doing. 

Impulse Response Function (IRF) Analysis 

Impulse response analysis shows how the economy reacts to a productivity shock under two scenarios: one without spillovers and one with learning-by-doing. In the spillover case, output, consumption, wages, and capital respond more strongly and remain elevated for longer. The feedback loop of investment leading to knowledge, and knowledge leading to higher productivity, amplifies the effects of shocks. 

Simulated Time Series 

Simulations highlight the long-run differences between the baseline model and the version with spillovers. Economies with learning-by-doing sustain higher levels of output and capital over time. Consumption and wages also grow faster, reflecting the productivity-enhancing role of knowledge accumulation. Even modest spillovers can significantly shape long-run macroeconomic trajectories.

Conclusion

Arrow’s learning-by-doing model is one of the earliest examples of endogenous growth theory. By linking productivity to the overall capital stock, it demonstrates how knowledge spillovers can amplify shocks and sustain higher growth.

Implementing the model in Dynare allows us to simulate both the baseline RBC and the learning-by-doing extension, making it easy to see the impact of spillovers in impulse responses and time series. The results show that even small amounts of learning-by-doing matter for understanding productivity, growth, and business cycles.