Development Economics & Personnel Theory
Kremer O-Ring Production & Assortative Wage Formula
Calculate Michael Kremer’s (1993) multiplicative O-Ring output, supermodular positive assortative matching surplus Y_max*(q_H - q_L)^2, and convex n-task equilibrium wage ratio (q_H / q_L)^n.
Understand why high-skill workers match together and earn exponentially higher wages!
Inspired by the 1986 Space Shuttle Challenger disaster — where a single failed O-ring destroyed an entire complex spacecraft — Nobel laureate Michael Kremer (1993) modeled production across n mission-critical tasks where worker i executes their task without fatal defect with reliability q_i in (0, 1]: Expected Output Y = Y_max * PRODUCT(q_1, ..., q_n)! Because cross-partial derivative d^2 Y / (dq_i dq_j) = Y_max > 0 is strictly supermodular, the marginal value of a top surgeon, chip designer, or pilot is highest when paired with equally reliable teammates!
Verify on two 2-task firms (Y_max = 1,000 in B4) matching 2 High-Skill workers (q_H = 0.90, B2) and 2 Lower-Skill workers (q_L = 0.50, B3): Positive Assortative Matching (Firm HH + Firm LL) produces 1,000*(0.90^2) + 1,000*(0.50^2) = 810.00 + 250.00 = 1,060.00, strictly beating Cross-Matched Firms (HL + HL) producing 2 * 1,000*(0.90*0.50) = 900.00 by Exact Supermodular Surplus 1,000 * (0.90 - 0.50)^2 = +160.00! Even more dramatically, in an n-task O-Ring chain (n = 4 tasks in B5), competitive symmetric equilibrium wages w(q) = (Y_max / n) * q^n scale with the n-th power of reliability: a 1.80x skill advantage (0.90 / 0.50) generates a (0.90 / 0.50)^4 = 10.4976x wage ratio!
| Kremer (1993) O-Ring Metric (q_H=0.90, q_L=0.50, Y_max=1,000, n=4) | Formula | Verified Output |
|---|---|---|
| Assortative Matching Output (HH + LL) | Y_max * (q_H^2 + q_L^2) | 1,060.00 (810 + 250) |
| Mixed Cross-Matching Output (HL + HL) | 2 * Y_max * q_H * q_L | 900.00 (450 + 450) |
| Positive Assortative Sorting Surplus | Y_max * ((q_H - q_L) ^ 2) | +160.00 |
| n-Task Equilibrium Wage Ratio (w_H / w_L) | (q_H / q_L) ^ n | 10.4976x (for 1.80x Skill) |
Supermodular Assortative Matching Gain Formula
Validate 0 < q_L <= q_H <= 1, Y_max > 0, and integer task complexity n >= 2!
=LET(qh_s, B2, ql_s, B3, ym_x, B4, n_tk, B5, IF(OR(ql_s <= 0, qh_s < ql_s, qh_s > 1, ym_x <= 0, n_tk < 2, INT(n_tk) <> n_tk), "Invalid O-Ring Input", ROUND(ym_x * ((qh_s - ql_s) ^ 2), 2)))n-Task O-Ring Equilibrium Wage Inequality Ratio Formula
Increase supply-chain complexity from n = 4 to n = 6 tasks in cell B5 (with q_H = 0.90, q_L = 0.45 -> 2x skill ratio): equilibrium wage ratio explodes to 2^6 = 64.00x!
=IF(ISTEXT(E4), E4, ROUND((B2 / B3) ^ B5, 4))Try the example
Use the original values or change them to test your understanding.