Capital accumulation, diminishing returns, and the steady state, why saving alone can't drive growth forever.
Warming up the engine…
The benchmark growth model: output per worker depends on capital per worker, capital accumulates from saving but suffers depreciation and dilution, and the economy converges to a steady state.
Δk = s·f(k) − (δ + n)·k ; steady state where s·f(k*) = (δ + n)·k*
Piling up capital hits diminishing returns: each extra machine adds less output than the last, so saving alone cannot power growth forever. In steady state, investment just covers wear-and-tear and population growth. Sustained growth in living standards must come from technology, the one input that never runs into diminishing returns.
A higher saving rate raises the LEVEL of steady-state income but not the long-run growth RATE. Confusing level effects with growth effects is the classic Solow mistake.
Now prove you have it
Move the curve to where you think it lands, and get told exactly which part you got right.
Real-world scenarios on this model
Equilibrium at k 44, y 4.0.
Equilibrium: k* = 44.4, y* = 4.0
Current equations
Why do economies stop growing from saving alone? Investment (blue) adds capital; depreciationdepreciationThe wearing out of capital over time, or, for currencies, a fall in value against others. and population growth (red line) eat it. Where they cross, capital per worker stands still, the steady statesteady stateThe resting point of a growth model, where capital per worker stops changing because investment exactly covers depreciation and dilution.. Left of it you grow; right of it you shrink. The economy converges to k* and stays.
Source: Mankiw, Macroeconomics, ch. 8-9; Solow (1956)