A Numeric Example

The following example uses the same economy — $s = 0.20$, $n = 0.02$ — and passes it through both frameworks. The results illustrate why Solow wrote his 1956 paper.

The two parameters that drive the Harrod-Domar calculation are the savings rate $s = 0.20$ and the incremental capital-output ratio $v = 4$: four units of capital are required to produce one additional unit of output.

The warranted growth rate is 5% per year, but the natural growth rate — the pace at which the labour force expands — is only $G_n = n = 0.02$, or 2% per year. Warranted growth exceeds natural growth by 3 percentage points. The savings rate that would bring them into balance is $s^* = v \cdot n = 4 \times 0.02 = 0.08$, just 8%. Because $s = 0.20 > 0.08$, the economy is saving far more than full employment requires. The model has no built-in mechanism to close this gap: if actual growth drifts below $G_w = 5\%$, the resulting excess capacity discourages investment further, widening the deviation. This is the knife-edge.

The Solow calculation uses the same $s = 0.20$ and $n = 0.02$ but adds technology growth $g = 0.02$, depreciation $\delta = 0.05$, and capital's share $\alpha = 1/3$ in a Cobb-Douglas production function.

The economy converges to $k^* \approx 3.31$ units of capital per effective worker from any positive starting point. At this steady state, output per effective worker is $y^* = (k^*)^{1/3} = (3.31)^{1/3} \approx 1.49$, and per-worker output grows at $g = 2\%$ per year indefinitely — regardless of the savings rate. If $s$ rises from 0.20 to 0.30, the steady-state capital stock climbs to $k^* \approx (3.33)^{1.5} \approx 6.09$ and $y^* \approx (6.09)^{1/3} \approx 1.82$: a permanently higher level of output per worker, but the long-run growth rate remains 2%. Stability is unconditional; no planning intervention is required.