📈 economics

Big Push and the Spectral Radius of Development Complementarities: A Decentralised General-Equilibrium Theory

This paper presents a decentralized general-equilibrium model of the big push where sectoral growth depends on the growth rates of other sectors, demonstrating that the spectral radius of the complementarity matrix governs balanced growth and identifying a specific subset of "Perron-critical" sectors that must be activated to escape a development trap, while highlighting that market failures can still prevent equilibrium even when a viable configuration exists.

Original authors: Taoufik Rajhi

Published 2026-09-01

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Taoufik Rajhi

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Economic development is often described as a race to build factories, roads, and power grids. But for many nations, the problem is not a lack of ideas or resources; it is a failure of timing. Imagine a textile factory that cannot open because it needs dyes from a chemical plant, which cannot open because it needs cheap transport from a logistics company, which cannot invest because there is no freight to carry. Each business is profitable only if the others exist, yet none can start alone. This is a coordination failure, a situation where an economy gets stuck in a low-level trap, not because opportunities are missing, but because no single actor dares to move first. For decades, economists have argued that the solution is a "big push": a government should temporarily subsidize a large group of industries simultaneously to break the deadlock. However, this advice has remained frustratingly vague. It tells leaders to "push everything," a strategy that is fiscally impossible and politically dangerous, without explaining which specific industries to choose or how long the support should last.

A new study by Taoufik Rajhi at the University of Poitiers offers a precise map for this journey. By treating an economy not as a collection of isolated industries but as a connected network, the research reveals that escaping poverty depends on a specific mathematical property of how these industries link together. The study moves beyond the old idea that all sectors are interchangeable. Instead, it shows that some industries are more critical than others, not because they are the largest, but because of how they transmit growth to their neighbors. The paper proves that a government can engineer a successful escape from a poverty trap by identifying a small, essential group of sectors to activate. Once this specific group is supported, the economy can sustain its own growth, allowing the government to withdraw its support after a finite, predictable period.

The core of the discovery lies in how industries help each other grow. In this model, a factory does not just benefit from the existence of its suppliers; it benefits from the speed at which those suppliers are improving. If a chemical plant is rapidly upgrading its technology, the textile factory using its dyes learns faster. If a logistics firm is becoming more efficient, every manufacturer it serves becomes more productive. These improvements ripple through the economy like a signal passing through a circuit. The researchers found that the strength of this signal depends on the structure of the network. They calculated a single number, which they call the spectral radius, to measure how strongly these growth signals circulate. If the network is too sparse or the connections too weak, the signals fade away before they can build momentum, and the economy remains stagnant. But if the network is dense enough, the signals reinforce one another, creating a self-sustaining cycle of growth.

The challenge for a government is to find the right combination of industries to jump-start this cycle. The paper demonstrates that simply picking the biggest industries is a mistake. In a trapped economy, the largest sectors are often the ones that are already inactive and invisible to standard data. Instead, the researchers developed a new way to rank industries based on "deletion centrality." This measures how much the entire network would weaken if a specific industry were removed. An industry with high deletion centrality is one whose absence would cause the growth signal to collapse, even if that industry is not the most famous or the largest. The study identifies a "Perron-critical" set of sectors: these are the essential nodes that must be activated for any escape to succeed. If a government fails to include even one of these critical sectors in its push, the entire effort will fail, no matter how much money is spent.

The research also clarifies the duration of the intervention. Because the goal is to get the growth signals to circulate fast enough to outpace the rising costs of imported equipment and parts, the government does not need to subsidize industries forever. Once the network reaches a critical density, the growth rates of the active sectors naturally exceed the cost of staying in business. At that point, the industries become self-sustaining. The model provides a formula to calculate exactly when this tipping point will be reached, allowing governments to announce a "sunset date" for their subsidies. This commitment prevents temporary aid from turning into permanent, inefficient handouts. The length of the intervention depends on the shape of the network: in economies dominated by a single hub, like a major port or energy grid, the push should be narrow and deep, focused on that hub. In economies with many equal connections, a broader approach is needed.

To test these ideas, the researchers applied their method to real-world data. They analyzed the industrial networks of eighty different economies over nearly thirty years, using detailed records of who supplies whom. They found that the strength of these networks is a persistent national characteristic, varying significantly between countries. In the United States, they computed the deletion centrality for seventy-one different industries and found that the most critical sectors for growth were not the ones that standard measures of importance would highlight. The ranking of critical sectors was almost completely unrelated to traditional measures of influence, proving that the old way of identifying "key players" misses the mark when the goal is to escape a poverty trap. The study confirms that the path out of poverty is not a matter of luck or waiting, but a calculable engineering problem. By activating the right nodes in the network, a country can clear the threshold required for self-sustaining growth, turning a stagnant economy into a thriving one.

Technical Summary: Big Push and the Spectral Radius of Development Complementarities

Problem Statement
The paper addresses the classic coordination failure in development economics: an economy may remain trapped in a low-level equilibrium not due to a lack of opportunities, but because individual sectors cannot profitably enter without the simultaneous activation of others. While the "Big Push" theory (Rosenstein-Rodan, 1943; Murphy et al., 1989) posits that a coordinated, temporary public intervention can escape this trap, it fails to provide actionable guidance on which sectors to target, in what order, and for how long. Canonical models typically assume symmetric sectors and interchangeable linkages, leading to the impractical advice to "push everything at once." This paper argues that real economies possess an uneven, directed network structure where productivity spillovers vary significantly across sector pairs. The central problem is to derive a targeting criterion for a Big Push that accounts for this heterogeneity and identifies the minimal set of sectors required to trigger self-sustaining growth.

Methodology
The author constructs a multi-sector, decentralized general-equilibrium model featuring:

Productivity Spillovers on Rates: Unlike standard models where spillovers act on capability levels, this model posits that a sector's productivity growth rate ( $\dot{A}_i/A_i$ ) depends on the growth rates of its connected sectors. This is formalized via a capability accumulation equation where spillover elasticities ( $\phi_{ij}$ ) are derived from input-output cost shares and labor mobility.
Network Structure: The economy is represented as a directed graph where nodes are sectors and edges represent learning spillovers. The strength of these linkages is captured in a "development-feedback matrix" $\Phi(S)$ .
Entry and Exit Dynamics: Sectors face a sunk entry cost and a recurring cost for imported inputs that grows at an exogenous rate $\gamma$ . A sector survives only if its endogenous growth rate $g_i$ exceeds $\gamma$ .
Equilibrium Concepts: The paper distinguishes between a "configuration equilibrium" (where growth rates are determined by a given set of active sectors) and a "decentralized entry equilibrium" (where firms enter only if their private Net Present Value exceeds the sunk cost).

The analysis relies heavily on spectral graph theory, specifically the spectral radius ( $\rho$ ) of the feedback matrix, and the Perron-Frobenius theorem to characterize balanced growth paths (BGP). The survival of a configuration is determined by iterating a monotone operator that removes sectors failing the growth threshold, converging to a "surviving core."

Key Contributions

The Spectral Condition for Growth: The paper demonstrates that balanced growth is governed by the spectral radius of the complementarity matrix rather than simple row sums. Aggregate growth is an explicit, increasing function of $\rho(\Phi)$ . Growth becomes self-sustaining only when $\rho$ crosses a critical threshold relative to the cost trend $\gamma$ .
Perron-Critical Sectors and Deletion Centrality: The paper introduces a new targeting metric, deletion centrality ( $\Delta_i$ ), defined as the drop in the spectral radius of the effective network if a sector is removed. It identifies Perron-critical sectors as those whose removal causes the entire candidate set to collapse (i.e., the surviving core becomes empty). These sectors form a necessary "floor" for any successful Big Push.
Distinction Between Margins: The paper rigorously separates the extensive margin (activating dormant sectors) from the intensive margin (investment intensity within active sectors). It shows that these require distinct policy instruments: a one-time entry subsidy for the former and a permanent capability subsidy for the latter.
The Network Growth Multiplier: On the intensive margin, the paper derives a "network growth multiplier" ( $\kappa$ ) representing the wedge between private and social returns to investment. This multiplier is a Bonacich column centrality, showing that denser networks warrant stronger ongoing subsidies.
Finite Duration of Intervention: Unlike perpetual subsidies, the model proves that a Big Push needs only finite duration. Once the "escape set" is activated, the growth rates of all members eventually exceed the imported cost trend, allowing the government to withdraw support at a calculable "sunset date."

Key Results

Necessity of the Critical Set: No successful Big Push can omit a Perron-critical sector. If such a sector is withheld, the exit cascade will eliminate all other candidates, regardless of the size of the subsidy.
Suboptimality of Greedy Ranking: While deletion centrality screens for critical sectors, the paper demonstrates via a specific counter-example that ranking sectors by this metric and adding them greedily is not an optimal strategy for minimizing cost. The selection problem inherits neither submodularity nor supermodularity; in a simulation of random networks, the minimum growth constraint satisfied the increasing differences property in only 67.7% of cases, meaning standard greedy guarantees do not apply and the natural ranking can be strictly suboptimal.
Multiplicity and Traps: The model derives conditions under which a "trap" (a low-level equilibrium where no single firm enters) and a "viable high configuration" (where a coordinated set survives) coexist. The trap is a Nash equilibrium of the entry game, while the high configuration is viable but not necessarily an entry equilibrium without coordination.
Empirical Applicability: The paper demonstrates that the feedback matrix and deletion centrality can be measured using standard input-output tables. In US industry data, deletion centrality is found to be uncorrelated with standard eigenvector centrality; however, the author explicitly reports this pattern as a consistency check rather than a formal test of the derived mechanism, noting that the pattern was observed prior to its theoretical derivation.

Significance and Claims
The paper claims to fill a critical gap in the Big Push literature by moving from a symmetric, aggregate view to a disaggregated, network-based theory of development. Its primary significance lies in providing a computable criterion for industrial policy targeting in economies where candidate sectors have zero domestic output and thus no observable size or profitability metrics.

The author asserts that the paper does not solve the general combinatorial problem of finding the minimum cost escape set (which remains NP-hard), but it successfully identifies a necessary subset (the Perron-critical set) that must be activated in any successful intervention. Furthermore, it clarifies that the Big Push is not merely a coordination device to align beliefs but a mechanism to overcome a technological complementarity where the "race" between growth and imported costs determines survival. The framework offers a disciplined approach to industrial policy: it dictates that interventions must be targeted at specific network nodes to cross a spectral threshold, and once crossed, the intervention can be temporary, with its duration determined by the network's topology.

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