econ.studio
Solow–Swan: Equations & Formulas
Section 2 of 3
Formula sheet

Core Equations

Production

Y=F(K,AL)=Kα(AL)1−α,α∈(0,1)Y = F(K, AL) = K^\alpha (AL)^{1-\alpha}, \quad \alpha \in (0,1)
Aggregate Cobb–Douglas with labour-augmenting (Harrod-neutral) technology.
y=f(k)=kα,k≡KAL,y≡YALy = f(k) = k^\alpha, \qquad k \equiv \frac{K}{AL}, \quad y \equiv \frac{Y}{AL}
Intensive form: output and capital per effective worker.
r=f′(k)=αkα−1,w=f(k)−kf′(k)=(1−α)kαr = f'(k) = \alpha k^{\alpha - 1}, \qquad w = f(k) - k f'(k) = (1-\alpha) k^\alpha
Competitive factor prices: each factor earns its marginal product.

Capital accumulation

I=sY,C=(1−s)YI = sY, \qquad C = (1-s)Y
Goods-market clearing: output is split between investment and consumption.
K˙=I−δK=sY−δK\dot{K} = I - \delta K = sY - \delta K
Aggregate capital stock evolves as gross investment less depreciation.
k˙=sf(k)−(n+g+δ) k=skα−(n+g+δ) k\dot{k} = s f(k) - (n + g + \delta)\,k = s k^\alpha - (n + g + \delta)\,k
The fundamental equation of motion
The single equation that governs the entire dynamics of the model.
skα⏟actual investment    vs    (n+g+δ) k⏟break-even investment\underbrace{s k^\alpha}_{\text{actual investment}} \;\;\text{vs}\;\; \underbrace{(n + g + \delta)\,k}_{\text{break-even investment}}
Break-even investment is what is needed to hold kk constant against depreciation, population growth, and technological growth.

Steady state (k˙=0\dot{k} = 0)

k∗=(sn+g+δ)11−αk^* = \left(\frac{s}{n + g + \delta}\right)^{\frac{1}{1-\alpha}}
Steady-state capital per effective worker
y∗=(k∗)α=(sn+g+δ)α1−αy^* = (k^*)^\alpha = \left(\frac{s}{n + g + \delta}\right)^{\frac{\alpha}{1-\alpha}}
Steady-state output per effective worker
c∗=(1−s) y∗,k∗y∗=sn+g+δc^* = (1-s)\,y^*, \qquad \frac{k^*}{y^*} = \frac{s}{n + g + \delta}
Steady-state consumption and the capital–output ratio.

Golden Rule

f′(kGR∗)=n+g+δf'(k^*_{GR}) = n + g + \delta
The savings rate that maximises steady-state consumption sets the net marginal product of capital to n+g+δn + g + \delta.
kGR∗=(αn+g+δ)11−α,sGR=αk^*_{GR} = \left(\frac{\alpha}{n + g + \delta}\right)^{\frac{1}{1-\alpha}}, \qquad s_{GR} = \alpha
For Cobb–Douglas, the Golden-Rule savings rate equals capital's share of income.

Convergence

∣λ∣=(1−α)(n+g+δ)|\lambda| = (1-\alpha)(n + g + \delta)
Speed of convergence
Rate at which the gap to k∗k^* closes (from linearising k˙\dot{k} around k∗k^*).
t1/2=ln⁡2∣λ∣=0.693(1−α)(n+g+δ)t_{1/2} = \frac{\ln 2}{|\lambda|} = \frac{0.693}{(1-\alpha)(n + g + \delta)}
Half-life of convergence
With α=1/3\alpha = 1/3, n=0.01n = 0.01, g=0.02g = 0.02, δ=0.05\delta = 0.05: ∣λ∣≈5.3%|\lambda| \approx 5.3\%/yr and t1/2≈13t_{1/2} \approx 13 years.

Growth rates at the steady state

VariableLong-run growth rate
k,  y,  c,  ik,\; y,\; c,\; i (per effective worker)00
Y/L,  K/L,  C/LY/L,\; K/L,\; C/L (per worker)gg
Y,  K,  CY,\; K,\; C (aggregate)n+gn + g
Only technological progress gg drives long-run growth in output per worker. The savings rate sets the level of the balanced growth path, not its slope.