Spence Job-Market Signaling Separating Threshold Formula

Enforce Spence’s single-crossing condition to separate High from Low productivity

In Michael Spence’s (1973) signaling model, even when a credential adds zero direct productivity, High-productivity workers (output y_H, credential unit cost c_H > 0) can credibly separate from Low-productivity workers (output y_L < y_H, higher unit cost c_L > c_H) iff credential threshold e* satisfies two incentive constraints: Low types refuse to mimic (y_H - c_L * e* <= y_L -> e* >= e_min = (y_H - y_L) / c_L) and High types prefer signaling over y_L (y_H - c_H * e* >= y_L -> e* <= e_max = (y_H - y_L) / c_H)! Under the Riley / Cho-Kreps Intuitive Criterion, competition selects the least-cost separating threshold e_Riley* = e_min, leaving High-type net payoff y_H - c_H * ((y_H - y_L) / c_L).

Verify with y_H = 120 (B2), y_L = 60 (B3), High cost c_H = 10 (B4), Low cost c_L = 20 (B5), and High-type population share lambda_H = 0.40 (B6): separating credentials span [3.0000, 6.0000] units, Riley least-cost signaling requires e_min = 3.0000 units, High-type Riley net payoff is 120 - 30 = 90.00, and pooling zero-education wage is 0.40 * 120 + 0.60 * 60 = 84.00 (< 90.00, so High types prefer separating!). Now raise lambda_H to 0.60 (> 1 - c_H / c_L = 0.50): pooling wage reaches 96.00 > 90.00, making signaling Pareto-inferior for BOTH worker types!

Riley Least-Cost Separating Credential e_min = (y_H - y_L) / c_L Formula

Enforce y_H > y_L >= 0 and strict Spence-Mirrlees single-crossing 0 < c_H < c_L (if c_H >= c_L, Low types can mimic any credential High types find profitable, so separation is impossible!).

=LET(y_hi, B2, y_lo, B3, c_hi, B4, c_lo, B5, IF(OR(y_lo < 0, y_hi <= y_lo, c_hi <= 0, c_lo <= c_hi), "Single-Crossing Violated (Require 0 < c_H < c_L)", ROUND((y_hi - y_lo) / c_lo, 4)))

High-Type Riley Separating Net Payoff vs Zero-Signal Pooling Wage Formula

Compare High-type net payoff y_H - c_H * e_min against pooling wage W_pool to audit whether signaling improves High-type welfare or destroys pure deadweight cost.

=LET(y_hi, B2, y_lo, B3, c_hi, B4, c_lo, B5, IF(OR(y_lo < 0, y_hi <= y_lo, c_hi <= 0, c_lo <= c_hi), "Single-Crossing Violated (Require 0 < c_H < c_L)", ROUND(y_hi - c_hi * ((y_hi - y_lo) / c_lo), 4)))

Further reading

• Arte Sheets Lesson — Akerlof Market for Lemons Adverse-Selection Cutoff Formula
• Microsoft Support — LET function