Endogenous Poverty Traps in Continuous Time: A Signaling Approach

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 $\phi$ 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.