Growth rates and doubling time
Pick an annual growth rate and watch how the economy doubles over decades.
Exact doubling time
T = ln(2)/ln(1+g)
41.1 years
Rule of 70 (approximation)
T ≈ 70/g
41.2 years
Compound growth over 50 years
Y_t = Y_0 · (1+g)^t. Even small growth rates compound to large gaps over generations.
Famous benchmarks
| Episode | g | Doubling time |
|---|
| U.S. real GDP per capita 1820–2016 | 1.7% | ~41 years |
| U.S. real GDP (total) 1820–2016 | 3.5% | ~20 years |
| China GDP per capita 1980–2014 | 8.8% | ~8 years |
| South Korea GDP per capita 1960–2010 | 6.0% | ~12 years |
| Japan post-WWII 1950–1990 | 5.5% | ~13 years |
Solow Model — three versions
Pick parameters. The tool computes k*, y*, c* and time path. Compare versions 1, 2, 3.
Choose Solow version
y = k^0.50; steady state: σ·y* = δ·k* → k* = (σ/δ)^(1/(1−α))
Steady-state values
| k* (capital per worker/effective worker) | 9.00 |
| y* (output per worker/effective worker) | 3.00 |
| c* (consumption) | 2.10 |
| Saving = Investment | 0.90 |
Steady-state growth rates
| Output growth g_Y | 0% |
| Per-capita growth g_y | 0% |
Version 1: no long-run growth. Switch to v3 to see sustained per-capita growth at rate g.
Transition path (k_1 = half of k*)
Golden Rule explorer
Move σ and find the saving rate that maximizes long-run consumption per capita.
Golden Rule: σ_gold = α (capital share) under Cobb-Douglas
Your saving rate σ vs the Golden Rule
| Golden Rule σ_gold | 0.50 |
| Your σ | 0.30 |
| Regime | Below Golden Rule |
Steady-state outcomes
| k* (your σ) | 3.59 |
| y* (your σ) | 1.90 |
| c* (your σ) | 1.33 |
| c_gold (Golden Rule) | 2.50 |
Consumption vs saving rate
You are BELOW the Golden Rule. Raising σ toward σ_gold would increase your long-run consumption. The U.S. is in this regime.
Growth accounting calculator
Compute the Solow residual (TFP growth) from data on output, capital, and labor growth.
g_A = g_Y − α·g_K − (1 − α)·g_L
TFP growth (Solow residual)
1.43%
Decomposition
| Capital contribution (α·g_K) | 0.99% |
| Labor contribution ((1−α)·g_L) | 0.98% |
| TFP contribution | 1.43% |
| Sum (should = g_Y) | 3.40% |
TFP share of growth
42%
In U.S. 1948-2016, TFP contributed roughly 32% of growth. In 1948-73 it was 44% (rapid TFP). In 1973-2016 it was just 23% (productivity slowdown).
Practice quiz — Section 4 v3
10 questions on Solow + growth accounting. Practice Exam 2 style.
Score: 0/10