Financial & Accounting
Solow-Swan Growth Model Formula: Steady-State Capital k* & Output y* per Effective Worker
Solve Nobel laureate Robert Solow & Trevor Swan’s 1956 neoclassical growth model under Cobb-Douglas production y = k^alpha for balanced-growth steady-state capital k* = (s / (n + g + delta))^(1 / (1 - alpha)) and output y* = (k*)^alpha.
Balancing gross investment s*k^alpha against break-even capital dilution (n + g + delta)*k!
Independently formulated in 1956 by Robert M. Solow (Quarterly Journal of Economics) and Trevor W. Swan (Economic Record), the neoclassical growth model tracks capital per effective labor unit k = K / (A * L), where population grows at rate n, labor-augmenting Harrod-neutral technology A grows at rate g, physical capital depreciates at rate delta, and households save fraction s of Cobb-Douglas output y = k^alpha!
Setting capital accumulation dk/dt = s * k^alpha - (n + g + delta) * k = 0 yields the closed-form balanced-growth steady state: k* = (s / (n + g + delta))^(1 / (1 - alpha)) and steady-state output per effective worker y* = (s / (n + g + delta))^(alpha / (1 - alpha))! At s = 0.24, n = 0.015, g = 0.020, delta = 0.045 (break-even rate n+g+delta = 0.080 -> ratio s/0.08 = 3.0), and capital share alpha = 1/3, k* = 3^(1.5) = 5.1962 and y* = 3^(0.5) = 1.7321—rising to k* = 4^(1.5) = 8.0000 if the saving rate rises to s = 0.32 (nearly Golden Rule alpha = 1/3)!
| Macroeconomic Saving Rate (n+g+delta = 8%, alpha = 1/3) | Steady-State Capital k* | Steady-State Output y* = (k*)^alpha |
|---|---|---|
| Saving Rate s = 24% (s / 0.08 = 3.0) | 5.1962 | 1.7321 |
| Higher Saving Rate s = 32% (s / 0.08 = 4.0) | 8.0000 (+54.0% deeper capital) | 2.0000 (+15.5% higher income level) |
| Invalid Capital Share alpha >= 1 or s <= 0 | No Diminishing Returns | "Invalid Solow Growth Parameters" |
Solow-Swan Steady-State Capital per Effective Worker k* = (s / (n + g + delta))^(1 / (1 - alpha))
Evaluates steady-state capital intensity k* from saving rate s (B2), labor growth n (B3), technology growth g (B4), depreciation delta (B5), and Cobb-Douglas capital share alpha (B6 in (0, 1)).
=IF(OR(B2 <= 0, B2 >= 1, B3 + B4 + B5 <= 0, B6 <= 0, B6 >= 1), "Invalid Solow Growth Parameters", ROUND((B2 / (B3 + B4 + B5)) ^ (1 / (1 - B6)), 4))Steady-State Output per Effective Worker y* = (s / (n + g + delta))^(alpha / (1 - alpha))
Computes balanced-growth output intensity y* = (k*)^alpha.
=IF(OR(B2 <= 0, B2 >= 1, B3 + B4 + B5 <= 0, B6 <= 0, B6 >= 1), "Invalid Solow Growth Parameters", ROUND((B2 / (B3 + B4 + B5)) ^ (B6 / (1 - B6)), 4))Try the example
Use the original values or change them to test your understanding.