Natural Resource & Forestry Economics
Faustmann Forest Rotation & Land Expectation Formula
Calculate Martin Faustmann’s (1849) infinite-rotation Land Expectation Value LEV = (p*V(T) - C*EXP(r*T))/(EXP(r*T) - 1) and contrast optimal Faustmann, single-cycle Wicksell, and biological MAI ages!
Why Paul Samuelson showed nearly every non-Faustmann forestry rule gets optimal harvest age wrong!
In a famous 1976 lecture, Paul Samuelson showed that generations of foresters and textbooks confused three distinct rotation rules for timber volume V(T) at stumpage price p, replanting cost C (paid at t = 0, T, 2T, ...), and continuous real interest rate r > 0: (1) Biological Mean Annual Increment V(T) / T ignores discounting completely; (2) Single-Cycle Wicksell NPV p * V(T) * EXP(-r * T) - C ignores the opportunity cost of tying up bare land from affecting future rotations; and (3) Martin Faustmann’s (1849) exact Infinite-Rotation Land Expectation Value LEV(T) = -C + (p * V(T) - C) / (EXP(r * T) - 1) = (p * V(T) - C * EXP(r * T)) / (EXP(r * T) - 1) harvests EARLIEST because delaying harvest delays every future seedling cycle!
Verify all three peaks on ONE 4-age stand table (p = 2 in H2, Replant_C = 100 in H3, Real_Rate_r = 0.05 in H4) over ages A2:A5 = [20, 30, 40, 50] with timber volumes B2:B5 = [300, 800, 1,350, 1,750]: Biological MAI C2:C5 = [15.00, 26.67, 33.75, 35.00] peaks at T = 50 yr; Single-Cycle PV D2:D5 = [120.73, 257.01, 265.41, 187.30] peaks at T = 40 yr; and Faustmann LEV E2:E5 = [190.99, 330.83, 306.95, 204.05] peaks at T* = 30 yr (LEV* = 330.83)! Now lower r from 5% to 2% to watch Optimal Faustmann Age lengthen from 30 yr to 40 yr!
| Rotation Age T & Volume V(T) | Biological MAI & Single-Cycle PV | Faustmann Infinite-Cycle LEV (r = 5%) |
|---|---|---|
| T = 20 yr (V = 300) | MAI = 15.00 | PV_1 = 120.73 | LEV = 190.99 |
| T = 30 yr (V = 800) | MAI = 26.67 | PV_1 = 257.01 | LEV = 330.83 (Faustmann Optimum!) |
| T = 40 yr (V = 1,350) | MAI = 33.75 | PV_1 = 265.41 (1-Cycle Peak) | LEV = 306.95 |
| T = 50 yr (V = 1,750) | MAI = 35.00 (MAI Peak) | PV_1 = 187.30 | LEV = 204.05 |
Row-Level Faustmann Land Expectation Value (LEV) Formula
Notice the algebra: paying C at t = 0 plus receiving net harvest (p * V(T) - C) every T years in perpetuity equals -C + (p * V(T) - C) / (EXP(r * T) - 1) = (p * V(T) - C * EXP(r * T)) / (EXP(r * T) - 1)!
=IF(OR($H$2 <= 0, $H$3 < 0, $H$4 <= 0, A2 <= 0), "Invalid Stand Input", ROUND(($H$2 * B2 - $H$3 * EXP($H$4 * A2)) / (EXP($H$4 * A2) - 1), 2))Optimal Faustmann Rotation Age Lookup Formula
Select the rotation age maximizing bare-land value via INDEX / MATCH(MAX(E2:E5), E2:E5, 0).
=IF($H$4 <= 0, "Invalid Stand Input", INDEX(A2:A5, MATCH(MAX(E2:E5), E2:E5, 0)))Try the example
Use the original values or change them to test your understanding.