This is the Lewis dual-sector model (Arthur Lewis, 1954), the founding model of development economics: a "traditional" agricultural sector with surplus labor feeds workers into a growing "modern" industrial sector until wages equalize.
Agriculture (fixed land, diminishing returns):
Y_a = A_a · L_a^0.6 w_a = w̄ (institutional subsistence wage)
Industry (Cobb-Douglas, capital + labor):
Y_i = A_i · K_i^0.4 · L_i^0.6
w_i = MPL_i = 0.6 · A_i · K_i^0.4 · L_i^-0.4 (paid its marginal product)
Capital accumulation: dK_i/dt = s·(Y_i − w_i·L_i) − δ·K_i
Labor migration: dL_i/dt = κ·(w_i − w_a), only while w_i > w_a
- Reinvestment rate s — the share of industrial profit plowed back into new capital each year; higher s builds factories faster.
- Industrial productivity Ai — technology level of the modern sector; raises both output and the wage industry can afford to pay.
- Subsistence wage w̄ — the institutionally fixed wage in traditional agriculture; a higher floor slows migration because industry must clear a higher bar to attract workers.
- Capital depreciation δ — how fast industrial capital wears out, offsetting reinvestment.
Workers keep moving from the green agriculture zone to the blue industry zone as long as the industrial wage exceeds the subsistence wage. As Ki grows, diminishing returns to labor push the industrial wage down toward w̄ — the Lewis turning point, where surplus labor is exhausted and further growth must come from productivity gains, not more migration. This is the classic account of how China, South Korea and other late industrializers moved workers off the land during their growth booms.