A. Capital Stocks Over Time
↗ Open
B. Inclusive Wealth Composition Over Time
Explanation

Inclusive Wealth measures the sum of all capital stocks — produced capital \( K_p \), human capital \( K_h \), and natural capital \( K_n \). Society's present value of wellbeing is \( V_t = \sum_{s=t}^{\infty} \beta^s u(c_s) \), where \( \beta \) is the discount factor and \( u(c_s) \) is per-period utility from consumption.

Production follows \( K_{t+1} = F(K_t) - C_t - DK_t \), where \( F(K_t) \) is output, \( C_t \) is consumption, and \( D \) is the depreciation rate. If you consume too much, savings won't replace lost capital — your welfare will fall over time.

Sustainability requires \( \frac{dV}{dt} \geq 0 \): non-declining human wellbeing. Under weak sustainability, the total inclusive wealth \( IW = v_p K_p + v_h K_h + v_n K_n \) must be non-declining, allowing substitution between capital types (where \( v_p, v_h, v_n \) are the shadow prices of each capital). Under strong sustainability, natural capital itself must be non-declining: \( \frac{dK_n}{dt} \geq 0 \).

Use the sliders to explore how consumption choices, depreciation, and capital composition affect sustainability outcomes.

The Sustainability Problem
\[ V_t = \sum_{s=t}^{\infty} \beta^s \, u(c_s) \] \[ K_{t+1} = F(K_t) - C_t - D \cdot K_t \] \[ \text{Sustainability:} \quad \frac{dV}{dt} \geq 0 \]
Weak Sustainability
\( \frac{d}{dt}(v_p K_p + v_h K_h + v_n K_n) \geq 0 \quad \Longleftrightarrow \quad \frac{dIW}{dt} \geq 0 \)
Strong Sustainability
\( \frac{dK_n}{dt} \geq 0 \qquad \text{(natural capital must be maintained directly)} \)
Substitutability

A key question is whether different types of capital can substitute for one another. As an economy develops, capital composition changes — produced (manufactured) capital tends to grow while natural capital may decline.

Weak Sustainability
\( \frac{d}{dt}(v_p K_p + v_h K_h + v_n K_n) \geq 0 \)
Total wealth can grow even if natural capital declines, as long as other capital compensates.
Strong Sustainability
\( \frac{dK_n^{critical}}{dt} \geq 0 \)
Critical natural capital must be maintained directly — no substitution allowed for essentials like clean air, water, and biodiversity.
Scenarios
Production & Consumption
Consumption rate \( C/Y \)0.70
Depreciation \( D \)0.05
TFP \( A \)1.50
\( K_{t+1} = F(K_t) - C_t - D \cdot K_t \)
Initial Capital Stocks
Produced \( v_p K_p \)30
Human \( v_h K_h \)30
Natural \( v_n K_n \)40
\( IW = v_p K_p + v_h K_h + v_n K_n \)
Welfare & Discount
Discount factor \( \beta \)0.95
Output elasticity \( \alpha \)0.35
Periods \( T \)20
\( V_t = \sum_{s=t}^{\infty} \beta^s \, u(c_s) \)
Sustainability Assessment
Initial \( IW_0 \)100.0
Final \( IW_T \)100.0
\( \Delta IW \)0.0
\( \Delta K_n \)0.0
Weak sust.?—
Strong sust.?—
Produced capital \( v_p K_p \)
Human capital \( v_h K_h \)
Natural capital \( v_n K_n \)
Inclusive Wealth \( IW \)