Capability Accumulation and Conditional Convergence: Towards a Dynamic Theory of Economic Complexity (2512.10672v1)

Abstract: We develop a dynamic model of economic complexity that endogenously generates a transition between unconditional and conditional convergence. In this model, convergence turns conditional as the capability intensity of activities rises. We solve the model analytically, deriving closed-form solutions for the boundary separating unconditional from conditional convergence and show that this model also explains the path-dependent diversification process known as the principle of relatedness. This model provides an explanation for transitions between conditional and unconditional convergence and path-dependent diversification.

• The paper introduces a dynamic model that unites micro-level capability accumulation with macro convergence phenomena.
• It demonstrates that a bifurcation parameter in capability intensity shifts growth from unconditional to conditional convergence.
• The analysis reveals path-dependent diversification, linking empirical relatedness metrics to structural economic change.

Dynamic Economic Complexity, Capability Accumulation, and Transitions in Convergence

The paper "Capability Accumulation and Conditional Convergence: Towards a Dynamic Theory of Economic Complexity" (2512.10672) introduces a formal dynamic model that unites the microfoundations of capability-based production with the macro-level phenomena of conditional and unconditional convergence. It provides a rigorous analytical framework explaining the endogenous transitions between these two regimes and substantiates the path-dependent diversification dynamics (relatedness) observed empirically in economic complexity literature.

Theoretical Background: Weak-Link Production, Conditional Convergence, and Relatedness

The model is situated at the intersection of weak-link production theory, endogenous growth, and economic complexity. The authors extend classic O-ring/Leontief production functions by allowing heterogeneity in the capability intensities required by different activities. This generalization enables a structural representation of economy-activity matrices with arbitrary granularity, incorporating both the probability an economy possesses a given capability ($r_{cb}$) and the probability that an activity requires that capability ($q_{pb}$).

A central contribution is formalizing how micro-level complementarities and bottlenecks endogenously shift the growth regime from unconditional convergence, à la Solow, to conditional convergence, as empirically observed in the literature [barro1992convergence, hidalgo_building_2009]. The model also naturally predicts the principle of relatedness: the path-dependent and adjacent possible nature of economic diversification [hidalgo_product_2007, hidalgo_principle_2018].

Model Formulation and Analytical Results

The core production function models sectoral output as:

$Y_{cp} = \prod_b (1 - q_{pb}(1 - r_{cb}))$

which interpolates smoothly between linear (additive) and Leontief-like (strongly complementary) regimes, depending on the intensity parameters $q_{pb}$. The dynamics of capability accumulation are captured via a Riccati-type nonlinear differential equation, where capability growth is a function of endogenous investment (proportional to current output) and a depreciation process:

$$\frac{dr_{cb}}{dt} = \gamma (1 - r_{cb}) \sum_p Q_{pb}Y_{cp} - \delta r_{cb} N_p \langle q_b \rangle$$

where $Q_{pb} = q_{pb} / \sum_b q_{pb}$ distributes investment intensity across capabilities.

A crucial analytical result is the identification of a bifurcation parameter—the average capability intensity $\langle q \rangle$—which determines the regime of convergence. When $\langle q \rangle < \gamma / (2\gamma - \delta)$, less-endowed economies catch up quickly (unconditional convergence), but above this threshold, growth becomes conditional on initial capability levels and divergence can persist (conditional convergence). The transition is mathematically analogous to a second-order phase transition, with the location of maximal growth shifting continuously from the boundary to the interior of the capability interval as $\langle q \rangle$ increases.

Strong numerical characterization is provided by the closed-form Riccati solution for single and multi-capability settings, detailing both equilibrium levels and transitional dynamics. The extension to multiple capabilities preserves the qualitative regime structure but introduces weighted complementarities, such that the speed and scope of convergence depend on the specific structure of input requirement overlap and existing capabilities.

Theoretical and Empirical Implications

Path Dependency, Relatedness, and Structural Change

The model provides a deductive foundation for empirically observed path dependencies in economic development. By formal differentiation, it is shown that the accumulation rate of one capability increases monotonically with the presence of complementary capabilities and that this effect is modulated by the structure of $q_{pb}q_{pb'}$. This result justifies the empirical relatedness metrics used in the product space literature [hidalgo_product_2007, boschma_technological_2012], directly deriving the likelihood of sectoral expansion from the structural overlaps in capability requirements.

Structural Barriers and Divergence

A pronounced implication is that as sectors become more capability-intensive—reflecting increased technological or organizational complexity—capability accumulation and diversification become path dependent and increasingly conditional. This provides a micro-level mechanism for the emergence of persistent development traps and explains historical divergences following major technological transitions, such as the Industrial Revolution.

Endogeneity of Transitions Between Growth Regimes

The endogenous regime switch provides a mechanism for understanding both the onset and reversal of divergence. In environments where technological change reduces capability intensity requirements, the model predicts a natural return to unconditional convergence. Thus, convergence and divergence are dynamic, history-dependent phenomena, not static structural facts.

Extensions and Future Research Directions

This model sets the foundation for several important theoretical and empirical explorations:

• Endogenous Evolution of Capability Space: Allowing for creation/destruction of capabilities, thereby modeling product innovation and creative destruction dynamics [aghion_model_1992].
• Strategic Investment/Policy Optimization: Embedding the framework into competitive settings where agents or governments strategically allocate investments to minimize capability bottlenecks and maximize long-term diversification.
• Coupled Activity Dynamics: Modeling cross-investment between activities and incorporating demand-side feedback, allowing more realistic representations of value chain development and sectoral interdependencies.
• Technological Dynamics: Modeling endogenous evolution of $q_{pb}(t)$ to capture how exogenous technological progress (e.g., codification, automation) can modulate capability intensity and alter the structure of convergence/divergence over time.

These extensions could further bridge the theory to empirically calibrated, policy-relevant applications in economic diversification, innovation ecosystems, and international development.

Conclusion

The theoretical framework presented in "Capability Accumulation and Conditional Convergence" (2512.10672) rigorously demonstrates how micro-level capability complementarities, intensity parameters, and endogenous investment processes mediate the global regime structure of economic growth and convergence. By unifying weak-link production logic, conditional convergence theory, and path-dependent relatedness, it resolves long-standing questions in macroeconomic development and economic geography, and provides a formal, analytic basis for the empirically observed complexity-diversification-growth nexus. This work paves the way for richer dynamic models of structural transformation, integrating the granular evolution of capability spaces with aggregate patterns of development and divergence.