The Solow Growth Model

A discrete-time model of capital accumulation. Adjust parameters to explore how savings, depreciation, and factor shares determine the economy's steady state.

Parameters
Savings rate (s) 0.20
Depreciation (δ) 0.10
Capital share (α) 0.30
Labor (L) 100
Initial K₀ 50
Display Options
Show per-capita values
Show golden rule
Phase arrows
Steady State (per capita)
k*
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y*
—
c*
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k*ᴳᴿ
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▶ Model Setup
1. Production Function
Yt = Ktα · Lt1−α   ⟹  
Cobb-Douglas with constant returns to scale. α is capital's share of income, (1−α) is labor's share.
2. Capital Accumulation
Kt+1 = (1 − δ) · Kt + It
Next period's capital equals undepreciated current capital plus new investment.
3. Investment = Savings
It = s · Yt
A constant fraction s of output is saved and invested. No government, no trade.
Combined: Substituting (1) and (3) into (2)
Kt+1 = (1 − δ) · Kt + s · Ktα · L1−α
This single equation governs the entire dynamics of the economy.
▶ Solving for the Steady State
Per-capita form: divide by L (treating L as constant)
Let kt = Kt/L,   yt = Yt/L.   Then:   yt = ktα
Per-capita law of motion
kt+1 = (1 − δ) · kt + s · ktα
At steady state: kt+1 = kt = k*
k* = (1 − δ) · k* + s · (k*)α
k* − (1 − δ) · k* = s · (k*)α
δ · k* = s · (k*)α
(k*)1−α = s / δ
k* = (s / δ)1/(1−α)  = 
Steady-state output & consumption
y* = (k*)α = (s/δ)α/(1−α)   |   c* = (1 − s) · y*
Golden Rule: max c* w.r.t. s
sGR = α  ⟹  k*GR = (α / δ)1/(1−α) =
Consumption is maximized when the savings rate equals the capital share.
Solow Diagram (per capita)
Transition Dynamics
▶ Intuition

Production: Yt = Ktα L1−α exhibits diminishing returns to capital. Each additional unit of K raises output, but by less and less.

Capital accumulation: Kt+1 = (1−δ)Kt + sYt. Each period, a fraction δ of the capital stock wears out and a fraction s of output is reinvested. Capital grows when investment exceeds depreciation.

Steady state: At k*, the investment curve s·kα crosses the depreciation line δ·k. New investment exactly replaces worn-out capital, so k stops changing. The closed-form k* = (s/δ)1/(1−α) shows higher savings or lower depreciation raise the long-run capital stock.

Convergence: Below k*, investment exceeds depreciation, so k rises. Above k*, depreciation exceeds investment, so k falls. The economy always converges regardless of where it starts.