A. Production Allocation and Capital Stocks Over 4 Periods
B. Utility in Each Period
Explanation

In each period, the total capital stock \( IW_t = v_p K_p + v_h K_h + v_n K_n \) (where \( v_p, v_h, v_n \) are shadow prices) generates output \( W_t = A \cdot K_t^\alpha \) via the production function. This output is then split into consumption \( C_t \) and savings \( S_t = W_t - C_t \).

The savings arrow flows back into the capital stock for the next period. If savings are large enough to offset depreciation, capital grows and future output expands. If consumption is too high, savings are insufficient, capital shrinks, and the economy contracts over time.

Notice how capital composition changes: with typical investment patterns, produced (blue) and human (purple) capital tend to grow while natural (orange) capital declines — reflecting the substitutability question central to sustainability economics.

The Production–Accumulation System
\[ W_t = A \cdot K_t^\alpha \] \[ C_t = \frac{C}{Y} \cdot W_t, \qquad S_t = W_t - C_t \] \[ K_{t+1} = K_t + S_t - D \cdot K_t = F(K_t) - C_t - D \cdot K_t \]
Sustainability Condition
\( \text{If } S_t > D \cdot K_t \text{ then } K_{t+1} > K_t \text{ — capital grows} \)
Scenarios
Production & Consumption
Consumption rate \( C/Y \)0.45
Depreciation \( D \)0.02
TFP \( A \)2.5
Output elasticity \( \alpha \)0.35
\( W_t = A \cdot K_t^\alpha \)
Initial Capital Stocks
Produced \( v_p K_p \)30
Human \( v_h K_h \)30
Natural \( v_n K_n \)40
\( IW_t = v_p K_p + v_h K_h + v_n K_n \)
Period Summary
Produced \( v_p K_p \)
Human \( v_h K_h \)
Natural \( v_n K_n \)
Savings \( S_t \) → reinvested
Consumption \( C_t \)
Depreciation \( D \cdot K_t \)