Spatial Economics & Agricultural Land Use
Von Thünen Land Rent & Crop-Ring Boundary Formula
Calculate Johann Heinrich von Thünen’s (1826) agricultural bid-rent R_i(d) = Y_i*(p_i - c_i - f_i*d), concentric crop-switch boundary d*_12, and zero-rent extensive frontier.
Derive concentric land-use rings from transport cost gradients without arbitrary zoning!
Published in The Isolated State (1826), Johann Heinrich von Thünen created the very first mathematical spatial equilibrium model! Each hectare growing Crop i yields Y_i tons sold at central market price p_i, incurring on-farm unit production cost c_i and per-ton-per-km freight rate f_i. At distance d km from the central market, competitive land bidding drives Location Rent per hectare to R_i(d) = Y_i * (p_i - c_i) - (Y_i * f_i) * d! Perishable / heavy inner-ring Crop 1 (higher central intercept Y_1*(p_1 - c_1) AND steeper freight slope Y_1*f_1) outbids grain Crop 2 near town up to exact Ring-Switch Boundary d*_12 = (Y_1*(p_1 - c_1) - Y_2*(p_2 - c_2)) / (Y_1*f_1 - Y_2*f_2), while Crop 2 extends out to Extensive Frontier d_max2 = (p_2 - c_2) / f_2 where rent hits zero!
Verify on Inner_Dairy_Vegetables (Yield Y_1 = 10 t/ha in B2, Net_Margin p_1 - c_1 = 120 $/t in B3 -> Central Rent = 1,200/ha; Freight f_1 = 4.00 $/t/km in B4 -> Slope = 40 $/ha/km) vs Outer_Wheat (Yield Y_2 = 4 t/ha in C2, Net_Margin p_2 - c_2 = 150 $/t in C3 -> Central Rent = 600/ha; Freight f_2 = 2.50 $/t/km in C4 -> Slope = 10 $/ha/km) evaluated at Farm_Distance d = 12 km (B5): Dairy Bid-Rent at 12 km is 1,200 - 480 = 720.00/ha vs Wheat 600 - 120 = 480.00/ha (Highest-and-Best Rent = 720.00/ha!), exact Dairy-to-Wheat Ring Boundary is (1,200 - 600) / (40 - 10) = 20.00 km, and Wheat Extensive Margin is 150 / 2.50 = 60.00 km!
| Von Thünen Spatial Metric (Dairy vs Wheat, d = 12 km) | Formula | Verified Output |
|---|---|---|
| Inner Crop 1 Rent at d = 12 km | 10 * 120 - (10 * 4) * 12 | 720.00 $/ha (Outbids Wheat 480) |
| Envelope Highest-and-Best Land Rent | MAX(0, R_1(d), R_2(d)) | 720.00 $/ha |
| Crop 1 -> Crop 2 Switch Boundary (d*_12) | (1,200 - 600) / (40 - 10) | 20.00 km from Market |
| Outer Extensive Cultivation Frontier | (p_2 - c_2) / f_2 = 150 / 2.5 | 60.00 km (Beyond = Wilderness) |
Highest-and-Best Envelope Bid-Rent at Distance d Formula
Landowners lease to whichever crop bids highest (or leave land uncultivated at 0 rent beyond the extensive margin): MAX(0, R_1(d), R_2(d)).
=LET(y_1, B2, m_1, B3, f_1, B4, y_2, C2, m_2, C3, f_2, C4, d_km, B5, IF(OR(y_1 <= 0, m_1 <= 0, f_1 <= 0, y_2 <= 0, m_2 <= 0, f_2 <= 0, d_km < 0), "Invalid Spatial Input", ROUND(MAX(0, y_1 * (m_1 - f_1 * d_km), y_2 * (m_2 - f_2 * d_km)), 2)))Concentric Crop-Ring Switch Distance d*_12 Formula
Verify single-crossing ring geometry: inner Crop 1 must have BOTH higher central rent (I_1 > I_2) AND steeper transport slope (S_1 > S_2), with intersection strictly inside extensive margin (0 < d*_12 < d_max2)!
=LET(i_1, B2 * B3, s_1, B2 * B4, i_2, C2 * C3, s_2, C2 * C4, IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, C2 <= 0, C3 <= 0, C4 <= 0), "Invalid Spatial Input", IF(OR(i_1 <= i_2, s_1 <= s_2), "No Inner-Outer Ring Crossing", ROUND((i_1 - i_2) / (s_1 - s_2), 2))))Try the example
Use the original values or change them to test your understanding.