Urban Economics & Real Estate Valuation
Alonso-Muth-Mills Monocentric Bid-Rent & Fringe Formula
Model urban floor-space rent decay p_H(d) = p_H(0) * (1 - t*d/Y)^(1/alpha), suburban dwelling expansion, and agricultural city-edge radius d_max.
Derive urban rent gradients and dwelling-size expansion from Muth’s Condition
In the Alonso-Muth-Mills monocentric city model, identical households earning CBD income Y > 0 spend round-trip commuting cost t_comm * d at distance d >= 0 from downtown and maximize Cobb-Douglas utility U = z^(1 - alpha) * h^alpha (0 < alpha < 1 housing budget share). Spatial indifference forces unit housing rent to decay convexly with distance: p_H(d) = p_H(0) * (1 - (t_comm * d) / Y)^(1 / alpha) — satisfying Muth’s Condition -p_H’(d) * h(d) = t_comm (marginal rent savings per km exactly offset marginal commuting cost!) while optimal dwelling size h(d) = alpha * (Y - t_comm * d) / p_H(d) expands toward the suburbs!
Verify with CBD income Y = 80,000 (B2), annual commuting cost t_comm = 1,600 / km (B3), housing budget share alpha = 0.25 (B4 -> exponent 1 / alpha = 4), downtown rent p_H(0) = 1,600 / m^2 (B5), distance d = 10 km (B6), and rural agricultural edge rent p_agr = 100 / m^2 (B7): at d = 10 km, net income ratio is 1 - 16,000 / 80,000 = 0.80, rent drops to 1,600 * 0.80^4 = 655.36 / m^2, dwelling size nearly doubles from 12.50 m^2 downtown to 16,000 / 655.36 = 24.4141 m^2, and the urban fringe where p_H(d_max) = p_agr sits at d_max = (80,000 / 1,600) * (1 - (100 / 1,600)^0.25) = 50 * 0.50 = 25.00 km!
| Urban Spatial Metric (Y=80k, t=1,600, alpha=0.25, p_0=1,600) | Formula | Verified Output |
|---|---|---|
| Bid-Rent at d = 10 km | p_H(0) * (1 - t * d / Y)^(1 / alpha) | 655.36 / m^2 |
| Dwelling Size h(0) vs h(10) | alpha * (Y - t * d) / p_H(d) | 12.50 vs 24.4141 m^2 |
| Urban Fringe Radius (d_max) | (Y / t) * (1 - (p_agr / p_H(0))^alpha) | 25.00 km (at p_agr = 100) |
Alonso-Muth-Mills Bid-Rent p_H(d) Formula
Validate Y > t_comm * d >= 0, housing budget share 0 < alpha < 1, and downtown rent p_H(0) > 0.
=LET(inc_y, B2, t_cm, B3, alp_h, B4, p_cbd, B5, d_km, B6, IF(OR(inc_y <= 0, t_cm <= 0, alp_h <= 0, alp_h >= 1, p_cbd <= 0, d_km < 0, t_cm * d_km >= inc_y), "Invalid Urban Input", ROUND(p_cbd * POWER(1 - (t_cm * d_km) / inc_y, 1 / alp_h), 4)))Equilibrium Urban Fringe Boundary d_max Where p_H(d_max) = p_agr Formula
Lowering commuting cost t_comm (via transit or remote/hybrid work) flattens the rent gradient AND expands d_max outward (urban sprawl)!
=LET(inc_y, B2, t_cm, B3, alp_h, B4, p_cbd, B5, p_ag, B7, IF(OR(inc_y <= 0, t_cm <= 0, alp_h <= 0, alp_h >= 1, p_ag <= 0, p_cbd <= p_ag), "Invalid Fringe Input", ROUND((inc_y / t_cm) * (1 - POWER(p_ag / p_cbd, alp_h)), 4)))Try the example
Use the original values or change them to test your understanding.