Poverty Traps and Equilibrium Multiplicity

• Poverty traps are self-reinforcing mechanisms where thresholds and feedback loops lock economies in persistent low-income equilibria.
• Dynamic models incorporating sigmoidal functions and agent-based rules demonstrate how equilibrium multiplicity contributes to the coexistence of poor and rich regimes.
• Empirical analyses, network-based measures, and targeted policy interventions underscore the complex interplay of structural and behavioral factors in sustaining or alleviating poverty traps.

Poverty traps are defined as self-reinforcing mechanisms or thresholds that prevent individuals, regions, or entire economies from escaping persistent low levels of income, wealth, or capability. In modern economic theory, the existence and persistence of such traps are tightly linked to the concept of equilibrium multiplicity: the possibility that a given set of fundamentals (preferences, technology, endowments) supports two or more distinct long-run outcomes, one or more of which are characterized by sustained deprivation.

1. Formal Mechanisms and Dynamic Models of Poverty Traps

Dynamic models of poverty traps typically introduce nonlinearities, thresholds, or feedback loops that hinder the convergence to global optimality. In canonical growth models (e.g., Solow, OLG frameworks), these features can be captured via investment thresholds, sigmoid savings functions, or equilibrium conditions in agent-based settings. For example, in a stochastic Solow model with a sigmoidal savings function,

$$\frac{dk_p}{dt} = As(k_p)k_p^\alpha - (n+\delta)k_p + \sigma \xi(t),$$

the steady-state distribution $P(k_p)$ may become bimodal, with pronounced peaks at both low ("poverty trap") and high ("well-being") capital per capita, separated by an unstable equilibrium. The system's long-run behavior is determined by parameter regimes (notably the technology efficiency $A$) and stochastic perturbations (Bose, 2023).

Relatedly, agent-based macroeconomic models with behaviorally-motivated updating rules (e.g., "imitate-the-best" for savings):

$$s_i^{new} = s_{\operatorname{argmax}_{j \in \mathcal{N}(i)} (C_j)} + \epsilon$$

demonstrate that, for high frequency of social updating (small update time $\tau$), the economy settles in a homogeneous, low-savings equilibrium—a poverty trap. Above a critical threshold, endogenous fluctuations and wealth inequality emerge, leading to equilibrium multiplicity (persistent coexistence of poor and rich classes) (Asano et al., 2019).

Threshold effects and path dependence are further emphasized in geometric Brownian motion with reallocation (RGBM) models:

$$dx_i = x_i (u\,dt + \sigma\,dW_i) - T(x_i - \bar{x}_n)dt.$$

Here, both positive (upward) and negative (downward) transitions across a poverty line coexist, but high poverty persistence and low escape probability highlight trap-like behavior, influenced by the reallocation rate $T$ and income volatility (Sahasranaman, 2020).

2. Aggregation, Measurement, and Empirical Detection

Empirically, the measurement of poverty traps and the assessment of possible multiple equilibria pose substantial challenges. The General Poverty Index (GPI) provides a flexible structure,

$$\operatorname{GPI}_n = \frac{A(Q_n, n, Z)}{n\,B(Q_n, n)} \sum_{j=1}^{Q_n} w(\cdot) d\left(\frac{Z-Y_{j,n}}{Z}\right),$$

where choices of the measurable functions $A$, $w$, and $d$ recover most poverty indices in the literature, including Foster–Greer–Thorbecke (FGT) and weighted variants. Sequential data analysis using GPI allows for statistical inference and hypothesis testing on the persistence and significance of chronic poverty, unveiling whether observed poverty is consistent with a dynamic trap or with single-equilibrium convergence (Lo, 2012).

More structural approaches exploit dynamic distributional analysis: continuous distribution dynamics estimate the time evolution of the income distribution through transition densities

$f_{t+\tau}(y) = \int g_\tau(y|x) f_t(x) dx,$

and identify convergence clubs. In China's case, persistent, self-reinforcing splits between urban and rural incomes are confirmed, with urban steady states near $1.1$ times average income and rural at $0.48$, and the poorest regions converging toward much lower peaks, indicative of geographically concentrated poverty traps (Wu et al., 2016).

Network-based perspectives operationalize "trapness" in terms of a country's position in the product space (PS). High poverty is associated with poor connectivity in PS; countries specialized in low–sophistication products at the PS periphery experience greater difficulty evolving into better equilibria. The Product Poverty Index (PPI) and its long-run Eigenpoverty extension quantify these associations and predict the persistence of low-income regimes (Echeverri et al., 2021).

3. Microfoundations: Behavior, Institutions, and Coordination

Behavioral and institutional frictions are crucial for trap formation. Overlapping generations models with "wariness"—households placing weight $\lambda$ on their minimum period utility—demonstrate that the saving function and thus capital accumulation law become regime-dependent:

$$s(w, R) = \begin{cases} s_{\beta_1}(w, R) & R < \frac{1}{\beta + \gamma}, \ \frac{w}{1+R} & \frac{1}{\beta + \gamma} \le R \le \frac{1+\gamma}{\beta}, \ s_{\beta_2}(w, R) & R > \frac{1+\gamma}{\beta}; \end{cases}$$

where $\gamma = \lambda / (1 - \lambda)$, and the transition points depend on relative returns and degree of wariness. Low productivity and low wariness promote precautionary saving and reduce the poverty trap basin. High wariness may induce multiple fixed points (e.g., both a nontrivial steady-state and collapse at $k=0$), thus generating equilibrium multiplicity (Pham et al., 16 Oct 2025).

Institutional features such as economic segregation (homophily in network formation), entry barriers, and project thresholds generate double-equilibrium regimes in agent-based economies: one where a majority remains poor ("All Poor"), and another where a segment escapes poverty ("Some Rich"), yielding high inequality. Sensitivity analyses confirm the importance of both micro (risk, attention, diversification) and macro (financial inclusion, accessibility) interventions for altering system-wide attractor structure (Dupont et al., 9 Dec 2024).

4. Multisectoral and Network Externalities

Inputs and network structure often mediate the persistence—and escape—of poverty traps. In input–output network models with contagious disruptions, agents choose technology (complexity $\tau$), redundancy ($m$ inputs per output), and buffers ($m-\tau$):

$$U(m, \tau, F, \alpha, \beta) = P(m, \tau, F) \cdot \tau^{\beta} - \alpha m,$$

with

$P = \Pr[\text{Binomial}(m, F) \geq \tau].$

A low reliability $F$ induces agents to minimize complexity, locking the system in a low-income, low-complexity state; attempts at large, sudden increases in technological complexity ("big push" strategies) can trigger failure cascades and revert the system to the trap (Brummitt et al., 2017).

In common–pool resource exploitation, individualized Nash best responses in the face of externality-coupled resource degradation yield catastrophic poverty: typical payoffs scale as $(1/N)^2$ rather than $1/N$ as $N$ increases, even though a "fair" equilibrium is feasible under collective action. The presence of a few low-cost "oligarchs" further increases equilibrium multiplicity, and shifts in the cost-function regime (from linear to strongly concave) instigate abrupt transitions and market exits (Gros, 2022).

Systemic interactions in financial networks can also create threshold- and cycle-driven multiple equilibria. The presence of dependency cycles among banks generates solvency multiplicity: a "good" equilibrium where most debts are paid, and a "bad" equilibrium (credit freeze) with avoidable defaults. Bailout policies that specifically break self-fulfilling cycles are essential to coordinate the system toward the desirable equilibrium (Jackson et al., 2020).

5. Policy Interventions, Thresholds, and Escape

Interventions that nudge economies across critical thresholds—capital infusions, aid, direct transfers, or insurance—are frequently modeled as mechanisms for escaping poverty traps. For instance, in coupled SIRS–Solow models, a large enough injection of capital or development aid can shift an economy out of a high-infection, low-capital poverty trap, whereas mere reallocation of existing budgets to healthcare in poor economies exacerbates the trap due to crowding out of investment (Goerg et al., 2013).

Formally, in household or micro-risk models with proportional insurance,

$$Af(x) = r(x-x^*) f'(x) + \lambda \int_0^1 [f(xz) - f(x)] dG(z) = 0,$$

the existence of an "adjustment coefficient" preventing certain trapping depends on precise relations (e.g., $\lambda/r < \alpha$). Proportional insurance extends the "safe" parameter regime, but premium pricing and coverage levels determine efficacy (Henshaw et al., 2023). Likewise, direct cash transfers—modeled as an external drift term when capital falls below a critical barrier $B$—can reduce both poverty and extreme poverty probabilities provided the transfer rate and eligibility threshold are sufficiently high relative to underlying risk and shock dynamics (Flores-Contró et al., 2023).

6. Spatial Heterogeneity, Migration, and Systemic Multiplicity

In spatial equilibrium models, the multiplicity of equilibrium population distributions arises from location preferences, endogenous amenities, and mobility costs. Homotopy continuation methods reveal not only the existence but also the enumeration of large numbers of equilibria, which cannot be ranked solely by amenities or productivity (Ouazad, 18 Jan 2024).

Climate–mobility feedbacks further illustrate trap–like features. Heterogeneous asset levels, adaptive capacity, and repeated shocks create "immobility traps": below a wealth or adaptive threshold, households cannot migrate in response to climate shocks, reinforcing spatial poverty pockets. Causal forest analysis confirms the existence of distinct empirical regimes corresponding to equilibrium multiplicity in observed migration responses (Letta et al., 14 Mar 2024).

7. Synthesis and Theoretical Unification

Recent analytic advances on the uniqueness or multiplicity of competitive equilibrium situate poverty traps as an endogenous outcome of strong income effects, preference curvature, heterogeneous endowments, and nonlinear production technologies. Multiplicity arises when aggregate excess demand admits non-monotonicities, non-collinearity, or S-shaped market response. Sufficient conditions for uniqueness—such as relative risk aversion $\leq 1$ or gross substitutes structure—eliminate path dependence and poverty trap phenomena; failures of these conditions, or the presence of frictions (behavioral, institutional, or network) generally permit persistent, self-reinforcing poverty regimes as robust equilibrium outcomes (Toda et al., 1 Feb 2024).

The overarching implication is that the existence, persistence, and resolution of poverty traps depend fundamentally on the interaction of micro-behavior (risk preferences, savings, diversification), structural features (production function elasticity, economic complexity, network connectivity), and interventions (insurance, transfers, aid), underpinned by formal dynamic and equilibrium analysis. The literature emphasizes the need for targeted, context-sensitive policy approaches that address thresholds, feedbacks, and multiplicity to sustainably reduce poverty and promote mobility across the full distribution of economic outcomes.