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Unformatted text preview: Comparative Dynamics Prof. Lutz Hendricks October 7, 2009 L. Hendricks () Comparative Dynamics October 7, 2009 1 / 31 Comparative Dynamics We use phase diagrams to uncover the dynamic response to shocks. We study tax changes in a growth model. L. Hendricks () Comparative Dynamics October 7, 2009 2 / 31 Model The household solves max Z ∞ e & ρ t u ( c t ) dt (1) subject to ˙ k t = r t k t + w t & c t & τ t (2) and k given. Firms produce output using F ( K , L ) . The government uses the tax revenue to &nance goverment spending: G t = τ t . L. Hendricks () Comparative Dynamics October 7, 2009 3 / 31 Competitive Equilibrium A competitive equilibrium consists of functions c ( t ) , k ( t ) , τ ( t ) , w ( t ) , r ( t ) that satisfy: 1 Household: Budget constraint and g ( c ) = r & ρ σ (3) 2 Firms: r = f ( k ) & δ (4) w = f ( k ) & f ( k ) k (5) 3 Government: τ = G (6) 4 Market clearing: ˙ k = f ( k ) & δ k & c & G (7) L. Hendricks () Comparative Dynamics October 7, 2009 4 / 31 Phase Diagram The only change relative to the model without government: G shifts the ˙ k = locus down. k c B A L. Hendricks () Comparative Dynamics October 7, 2009 5 / 31 Permanent Tax Increase Consider a permanent, unannounced increase in G . In the phase diagram ˙ k = locus shifts down by Δ G . k ss remains unchanged because the ˙ c = locus does not shift. Dynamics: c ss drops to the new saddle path, then moves along it. An interesting longrun result: full crowding out of consumption ( Δ c ss = & Δ G ) . L. Hendricks () Comparative Dynamics October 7, 2009 6 / 31 Temporary Tax Increase Consider a temporary , unannounced increase in G . G t = G & + Δ G for ¡ t ¡ T , but G t = G & for t > T . To &nd the dynamics, we work backwards. Start from t = T : the economy looks like one without taxes (on saddle path). Consider < t < T : The phase diagram with taxes applies. But the economy is not on the saddle path (why not?). Key point: consumption cannot jump, except when new info arrives. We need to construct a path that follows the withtax phase diagram and connects with the no tax saddle path at t = T ....
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 '09
 LUTZHENDRICKS
 Solid solution, Phase transition

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