Closed-Form of Two-Agent New Keynesian Model with Price and Wage Rigidities

This paper argues and analytically demonstrates that, in a fully analytical Two-Agent New Keynesian model with Rotemberg-type nominal rigidities, monetary transmission is amplified if and only if two conditions hold: first, the heterogeneity-induced IS-slope effect dominates; second, the price-stickiness channel is active. We also show when amplification weakens or disappears, most notably under pure wage stickiness, where the price channel shuts down and the heterogeneity-driven term vanishes. The framework features household heterogeneity between savers and hand-to-mouth households and adheres strictly to microeconomic foundations while avoiding restrictive assumptions on relative wages or labor supply across types that are common in prior analytical work. The closed-form solution makes transparent how price stickiness, wage stickiness, and the share of hand-to-mouth households jointly shape amplification. We further derive a modified aggregate welfare loss function that quantifies how heterogeneity, operating through distributional effects from firm profits, re-weights the relative importance of stabilizing inflation. Overall, the tractable yet micro-founded analytical framework clarifies the interaction between household heterogeneity and nominal rigidities and pinpoints the precise conditions under which monetary policy gains or loses traction.