Poverty Trap & the Big Push

Economy Setup

Initial capital / worker k₀ 1.5

Subsistence threshold k_mid 4.0

Credit-market access A 1.4

Depreciation + pop. growth (n+δ) 0.14

Economy State

Initializing…

Capital / worker k

1.50

Output / worker y

0.00

Growth rate dk/dt

0.00

Savings rate s(k)

How it works

This is the classic poverty-trap / "Big Push" model from development economics (Rosenstein-Rodan, Nelson, Azariadis–Drazen). Capital per worker k evolves by the Solow accumulation equation:

dk/dt = s(k)·f(k) − (n+δ)k
f(k)  = A·k^0.35            (Cobb-Douglas output per worker)
s(k)  = s_min + (s_max−s_min) / (1 + e^(−b·(k − k_mid)))

Unlike the textbook Solow model, the savings rate s(k) is S-shaped, not constant: near subsistence, households save almost nothing (all income covers survival), so s(k) stays low; only once income clears the subsistence threshold k_mid does saving/investment rise toward its ceiling. That single non-convexity is enough to create three equilibria where s(k)f(k) crosses the break-even line (n+δ)k: k=0, an unstable tipping point k*, and a stable high-income equilibrium.

Start below k* → income can't outrun depreciation and population growth, so capital sinks back down: the poverty trap.
Start above k* → savings outrun break-even and the economy climbs to the high, prosperous equilibrium.
Big Push — a one-off foreign-aid / infrastructure injection can shove a trapped economy's k past k*, after which its own dynamics carry it the rest of the way to prosperity, with no further aid needed.
k_mid and credit-market access A shift where the tipping point sits and how high the prosperous equilibrium is — sliding them shows why the same aid injection rescues one economy and fails another.

Below poverty threshold Near tipping point Prosperous

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The Poverty Trap: Multiple Equilibria in Economic Development