Endogenous Poverty Traps in Continuous Time: A Signaling Approach

Abstract

This paper embeds a signaling friction into the continuous-time heterogeneous agent framework. A continuum of producers operate Cobb-Douglas technologies with regime-specific productivity $A_j \in \{A_L, A_H\}$. Stochastic arrival of signaling opportunities and skill obsolescence risk generate an optimal stopping problem -- when to pay a lump-sum cost $φ$ to upgrade productivity -- whose solution yields an endogenous Skiba threshold $k^*$. Diminishing returns create a stable interior attractor in each regime; the signaling cost separates the two basins, producing a poverty trap that is an interior optimum rather than a corner solution. The stationary distribution exhibits Twin Peaks, but its decomposition by regime reveals that agents in three distinct states -- structurally trapped, waiting to signal, and successfully upgraded -- coexist at the same wealth levels with different consumption behavior and mobility prospects. Capital alone is therefore insufficient to identify an agent's position in the polarization dynamics. We show that the joint observation of a low marginal propensity to consume out of wealth and a high average propensity to consume -- a combination invisible to standard Euler equation tests -- is the diagnostic signature of the structural trap, distinguishing it from both liquidity constraints and transitory shocks.

Figures (3)

• Figure 1: Partition of the capital axis under the baseline calibration. The signaling cost $\phi = 9$ and the endogenous Skiba threshold $k^* = 13.4$ divide the state space into three regions: absolute exclusion ($k < \phi$), the wait zone ($\phi \leq k < k^*$, shaded in yellow), and immediate signaling ($k \geq k^*$). The coupled attractors $k^{\mathrm{ss},c}_{L} = 11.5$ and $k^{\mathrm{ss},c}_{H} = 16.1$ are shown as solid vertical lines. The L-attractor lies inside the wait zone: the typical trapped agent can afford to signal but chooses not to.
• Figure 2: Value functions under the baseline calibration. $V_H > V_W > V_L$ for all $k > 0$ (Proposition \ref{['prop:ordering']}). All three are strictly concave (Proposition \ref{['prop:concavity']}). The shifted function $V_H(k - \phi)$ (dotted) crosses $V_W$ at the Skiba threshold $k^* = 13.4$: below this point, the concavity of $V_H$ makes the capital loss from paying $\phi$ too costly; above it, the agent signals immediately. The green dotted line marks $\phi = 9$, below which signaling is infeasible.
• Figure 3: Stationary densities by regime. The L-density $g_L$ (blue) peaks near $k^{\mathrm{ss},c}_{L} = 11.5$, forming the poverty trap mode; the H-density $g_H$ (red) peaks near $k^{\mathrm{ss},c}_{H} = 16.1$, forming the accumulation mode. The W-density $g_W$ (green dashed) is confined to the wait zone $[\phi, k^*]$ and vanishes above $k^*$, where signaling is immediate. The three densities overlap extensively: in $[4, \phi]$, newly signaled H-agents transit rapidly alongside slowly accumulating L-agents; in $[\phi, k^*]$, all three populations coexist; in $[k^*, 20]$, Decaying Rentiers (L) slide back through the region dominated by H-agents. The valley near $k^*$ reflects the absorbing-barrier effect: agents who reach the threshold exit the W-state and re-enter as H-agents at $k^* - \phi$. Ergodic shares: $\pi_L = 25.2\%$, $\pi_W = 11.8\%$, $\pi_H = 63.0\%$.

Theorems & Definitions (65)

• Remark 1: Why Cobb-Douglas rather than linear income
• Remark 2: Additive noise
• Remark 3: Common obsolescence rate
• Proposition 4: Deterministic steady states
• proof
• Remark 5: Separation from TFP gap
• Proposition 6: Stochastic steady states
• proof : Proof sketch
• Remark 7: Economic interpretation
• Proposition 8: Upward shift of the L-attractor