Labor Economics & Search Theory
Mortensen-Pissarides Beveridge Curve & Hosios Formula
Calculate Cobb-Douglas job-finding rate mu * theta^(1 - eta), vacancy-filling rate mu * theta^(-eta), Beveridge steady-state unemployment u*, and Hosios efficiency.
Derive the downward-sloping Beveridge Curve and Hosios externality balance
In the Diamond-Mortensen-Pissarides (DMP) search-and-matching model, unemployed workers U and job vacancies V meet via a constant-returns matching function M = mu_m * (U^eta) * (V^(1 - eta)) (0 < eta < 1). Defining labor-market tightness theta = V / U > 0, each job-seeker finds work at hazard f(theta) = mu_m * theta^(1 - eta), each vacancy fills at hazard q(theta) = mu_m * theta^(-eta), and balancing job separations s * (1 - u*) against hires f(theta) * u* yields steady-state Beveridge unemployment u* = s / (s + mu_m * theta^(1 - eta))! Moreover, Arthur Hosios (1990) proved decentralized Nash wage bargaining share beta_barg internalizes congestion externalities iff beta_barg = eta!
Verify on monthly separation rate s = 0.03 (B2), matching efficiency mu_m = 0.45 (B3), unemployment matching elasticity eta = 0.50 (B4), tightness theta = 1.00 (B5, 1 vacancy per unemployed worker), and worker bargaining share beta_barg = 0.50 (B6): job-finding rate is 0.4500/mo, steady-state unemployment is u* = 0.03 / (0.03 + 0.45) = 0.0625 (6.25%), implied aggregate vacancy rate v* = theta * u* is 0.0625, and beta_barg = eta satisfies the Hosios Condition! Tighten the market to theta = 2.25 (SQRT(2.25) = 1.50 -> f = 0.675) and unemployment slides down the Beveridge curve to 0.03 / 0.705 = 0.0426 (4.26%)!
| DMP Search Metric (s=0.03, mu=0.45, eta=0.50) | theta = 1.00 | theta = 2.25 (Tight Market) |
|---|---|---|
| Worker Job-Finding Rate f(theta) | 0.4500 / mo | 0.6750 / mo |
| Firm Vacancy-Filling Rate q(theta) | 0.4500 / mo | 0.3000 / mo |
| Beveridge Steady-State u* | 0.0625 (6.25%) | 0.0426 (4.26%) |
| Hosios Efficiency Check (beta vs eta) | beta = 0.50 -> Efficient | beta > 0.50 -> Excess u* |
Steady-State Beveridge Curve Unemployment Rate u* Formula
Validate separation rate s > 0, matching TFP mu_m > 0, elasticity 0 < eta < 1, and tightness theta > 0.
=LET(s_sp, B2, mu_m, B3, eta_m, B4, th_vu, B5, IF(OR(s_sp <= 0, mu_m <= 0, eta_m <= 0, eta_m >= 1, th_vu <= 0), "Invalid DMP Input", LET(f_jb, mu_m * POWER(th_vu, 1 - eta_m), ROUND(s_sp / (s_sp + f_jb), 4))))Hosios Congestion-Externality Efficiency Diagnostic Formula
Compare worker Nash bargaining weight beta_barg against unemployment matching elasticity eta: when beta_barg > eta, wages are too high and firms post too few vacancies; when beta_barg < eta, firms over-post congested vacancies!
=LET(eta_m, B4, bt_bg, B6, IF(OR(eta_m <= 0, eta_m >= 1, bt_bg <= 0, bt_bg >= 1), "Invalid DMP Input", IF(ABS(bt_bg - eta_m) < 1E-9, "Efficient (Hosios Condition Met)", IF(bt_bg > eta_m, "Excess Unemployment (beta > eta)", "Over-Vacancy Congestion (beta < eta)"))))Try the example
Use the original values or change them to test your understanding.