Urban Economics & Regional Labor Markets
Rosen-Roback Spatial Equilibrium Wage & Rent Formula
Invert Sherwin Rosen (1979) and Jennifer Roback’s (1982) spatial equilibrium conditions to extract unobserved Household Consumption Amenities (s_R*d_ln_r - d_ln_w) and Firm Agglomeration Productivity (theta_L*d_ln_w + theta_R*d_ln_r).
Infer whether high city rents come from coastal quality-of-life or firm productivity!
Why do some expensive cities pay high nominal wages while resort towns pay low wages despite high housing costs? Sherwin Rosen (1979) and Jennifer Roback (1982) showed that when mobile workers equalize indirect utility (d_ln_w - s_R * d_ln_r + A_cons = 0, where s_R is household housing expenditure share) and mobile tradable-sector firms equalize unit production costs (theta_L * d_ln_w + theta_R * d_ln_r - A_prod = 0, where theta_L and theta_R are firm labor and commercial land cost shares), observed log differentials in Wages d_ln_w and Rents d_ln_r uniquely reveal BOTH Household Consumption Amenity Value A_cons = s_R * d_ln_r - d_ln_w AND Firm Productive Agglomeration Advantage A_prod = theta_L * d_ln_w + theta_R * d_ln_r!
Verify for a coastal tech metropolis with Household_Housing_Share s_R = 0.30 (B2), Firm_Labor_Share theta_L = 0.80 (B3), Firm_Land_Share theta_R = 0.10 (B4), Log_Wage_Premium d_ln_w = +0.12 (+12.0% above national mean, B5), and Log_Rent_Premium d_ln_r = +0.50 (+50.0% above national mean, B6): Real Housing-Adjusted Wage Differential is 0.12 - 0.30 * 0.50 = -0.0300, revealing Positive Consumption Amenity A_cons = +0.0300 (+3.00% income equivalent), while Firm Productivity Advantage is A_prod = 0.80 * 0.12 + 0.10 * 0.50 = +0.1460 (+14.60% TFP advantage -> "Superstar Productive & High-Amenity City")! Test a harsh-climate industrial boomtown d_ln_w = +0.20, d_ln_r = +0.10: A_cons flips negative (-0.1700 disamenity compensated by high real wages) while A_prod = +0.1700 ("Industrial Hub (Disamenity Compensated)")!
| Rosen-Roback Spatial Metric (s_R=0.30, th_L=0.80, th_R=0.10, dw=+0.12, dr=+0.50) | Formula | Verified Output |
|---|---|---|
| Real Housing-Adjusted Log Wage | d_ln_w - s_R * d_ln_r | -0.0300 (-3.00%) |
| Consumption Quality-of-Life Amenity (A_cons) | s_R * d_ln_r - d_ln_w | +0.0300 (+3.00% Income Equiv) |
| Productive Agglomeration TFP (A_prod) | theta_L * d_ln_w + theta_R * d_ln_r | +0.1460 (+14.60% TFP) |
| Roback Spatial Quadrant | Sign(A_cons) x Sign(A_prod) | Superstar Productive & High-Amenity City |
Implied Household Consumption Amenity A_cons Formula
Validate 0 < s_R < 1, theta_L > 0, theta_R >= 0, and theta_L + theta_R <= 1.
=LET(sr_hh, B2, tl_fm, B3, tr_fm, B4, dw_lg, B5, dr_lg, B6, IF(OR(sr_hh <= 0, sr_hh >= 1, tl_fm <= 0, tr_fm < 0, tl_fm + tr_fm > 1), "Invalid Spatial Share", ROUND(sr_hh * dr_lg - dw_lg, 4)))Implied Firm Agglomeration Productivity A_prod & Quadrant Formula
Classify all four Roback (1982) urban quadrants automatically by testing the signs of A_cons and A_prod!
=IF(ISTEXT(E2), E2, IF(AND(E2 > 0, E3 > 0), "Superstar Productive & High-Amenity City", IF(AND(E2 <= 0, E3 > 0), "Industrial Hub (Disamenity Compensated)", IF(AND(E2 > 0, E3 <= 0), "Resort Consumption Amenity Town", "Declining Low-Amenity Low-TFP City"))))Try the example
Use the original values or change them to test your understanding.