Barro-Gordon Inflation Bias & Rogoff Central Banker Formula

Model why discretionary output temptations raise inflation without raising average output!

In Barro and Gordon’s (1983) time-inconsistency model, output follows a Lucas surprise supply curve y = y_n + alpha_pc * (pi - pi^e), while society minimizes quadratic loss L = 0.5 * (pi - pi*)^2 + 0.5 * lambda_y * (y - y_n - Delta_y*)^2 with inflation target pi*, output weight lambda_y > 0, and distortionary output-target gap Delta_y* > 0 above natural output y_n. Under discretionary policymaking after wage-setters lock in pi^e, the central bank equates marginal inflation cost (pi - pi*) to marginal surprise-output benefit lambda_y * alpha_pc * Delta_y* — which rational expectations fully anticipate (pi^e = pi), leaving ZERO output gain (y = y_n) and pure discretionary inflation bias Delta_pi = lambda_y * alpha_pc * Delta_y*!

Verify with inflation target pi* = 0.02 (B2), Lucas supply slope alpha_pc = 0.50 (B3), society output weight lambda_y = 1.20 (B4), and output-target wedge Delta_y* = 0.05 (B5): discretionary inflation bias is 1.20 * 0.50 * 0.05 = 0.0300 (+300 bps!), pushing equilibrium inflation to pi_disc = 0.0500 (5.00% vs 2.00% under credible commitment!) and inflicting excess deadweight loss 0.5 * 0.03^2 = 0.000450! Delegating monetary policy to a Rogoff (1985) conservative central banker with output weight lambda_CB = 0.40 (B6) cuts equilibrium inflation to 0.02 + 0.40 * 0.50 * 0.05 = 0.0300 (3.00%)!

Barro-Gordon Discretionary Equilibrium Inflation pi* + lambda_y * alpha_pc * Delta_y* Formula

Validate alpha_pc > 0, lambda_y >= 0, and output distortion wedge Delta_y* >= 0 (notice if central bankers target natural output y_n directly so Delta_y* = 0, discretionary inflation bias disappears!).

=LET(pi_tg, B2, alp_s, B3, lam_y, B4, dy_gp, B5, IF(OR(alp_s <= 0, lam_y < 0, dy_gp < 0), "Invalid Barro-Gordon Input", ROUND(pi_tg + lam_y * alp_s * dy_gp, 4)))

Rogoff Conservative Central Banker Inflation Formula

Evaluate equilibrium inflation under delegated governor preference weight 0 <= lambda_CB <= lambda_y.

=LET(pi_tg, B2, alp_s, B3, lam_y, B4, dy_gp, B5, lam_cb, B6, IF(OR(alp_s <= 0, lam_y < 0, dy_gp < 0, lam_cb < 0, lam_cb > lam_y), "Require 0 <= lambda_CB <= lambda_y", ROUND(pi_tg + lam_cb * alp_s * dy_gp, 4)))

Further reading

• Arte Sheets Lesson — Taylor Rule Central Bank Policy Interest Rate Formula
• Microsoft Support — ROUND function