Aggregate Intertemporal Substitution of Consumption Main ...

Aggregate Intertemporal Substitution of Consumption Main ...

The Neoclassical Growth Model: Solutions and Interpretations Econ 332 Spring 2012 copyright 2012 by Casey B. Mulligan Model Characteristics Margins considered: Two sectors (usually labor vs. leisure) Present versus future (either two period or Bellman eqn) Sometimes private and public sectors are distinguished Sometimes uncertainty about the future is (partly) modeled Supply and demand on these margins assumed to be stable around a trend Usually modeled as a representative agent with stable preferences and facing a stationary intertemporal production set Sometimes supply or demand on one of the margins is assumed to vary over time Productivity shock (real business cycle) Labor wedge/tax shock Investment supply shock shocks on all margins & all time periods the model is tautological Without money, the model says nothing about the price level or inflation Results for factor rental rates IF careful about gaps between supply and demand Actual market may bundle factor rentals with other transactions Stationary Preferences t e u (ct , nt )dt 0 deterministic model. t indexes time. = rate of time preference commodity space is time paths for consumption c and market labor n consumption adds public and private, otherwise express utility as a function of private and public consumption separately Preference changes can be modeled by shifters of u() Affecting the MRS: labor supply shift MRS-neutral: time preference shift For simplicity, we take u() as additively separable

Stationary Interemporal Production Set F (kt , nt ) ct kt kt kt 0 , k0 given dots indicate time derivatives kt is the capital stock at time t, depreciating at rate Intertemporal set obtained by solving the differential equation as a function of the time paths for c and n note conformity with national accounting: aggregate value-added = consumption + gross investment Technology changes can be modeled by Introducing a time-varying shifter of F() Sectors can be introduced by treating c and/or n as a composite commodity (Blanchard-Kiyotaki) Efficient Allocations Described by a System of Differential Equations Add shocks/distortions/specification errors {At,t,t}: Analytical Steps parameters constant steady state: for the right k0, efficient allocation has constant c, k stable manifold: dynamics for arbitrary k0 steady state is needed to calculate stable manifold parameters vary over time, but are constant in the long run stable manifold describes dynamics in the long run Short run dynamics must terminate on the stable manifold Boundary Conditions & Steady State k0 given , lim kt e t uc (ct , nt ) 0 t 0 Ass F (k ss , nss ) k ss css ss Fk (kss , nss ) MRS (css , nss ) (1 ss ) Fn (k ss , nss ) Three equations & three unknowns (css,kss,nss) Algebraically recursive solution with inelastic labor supply (the elasticity of labor supply is a characteristic of the MRS function) homogeneous production function F(k,n) = nf(k/n) Capital-Consumption Phase Diagram for the Stationary System: n given The Figure shows the stationary systems steady state, dynamics, and stable manifold. c nss css nss k ss nss

k nss The Stable and Unstable Manifolds Let c(k) describe the solution for ct for a given kt By definition, c t c(kt )kt and css c(k ss ) Thus, c(k) solves the boundary value ODE f (k / n) c(k ) (c)c Anf (k / n) k c (k ) s.t. c( kss ) css , A, n, , given Notes: Integrating away from the steady state is stable: c(k) is increasing (decreasing) in c for k < (>) kss The ODE has two solutions, which at the steady state correspond with the slopes of the eigenvectors of the system in the time dimension. LHopitals rule gives both solutions one positive and one negative corresponding to the stable and unstable manifolds Only the stable manifold satisfies the transversality condition Steady state slope is the basis for calculating a linear system in the time dimension that approximates the actual systems dynamics The Stable Manifold and Stationary System Dynamics Given c(k), a solution for ct is readily calculated from a solution for kt kt solves a boundary value ODE kt Anf (kt / n) kt c(kt ) , s.t. k0 given Numerically, this is calculated to any desired degree of precision dt: kt dt kt kt dt kt Anf (kt / n) kt c(kt ) dt Model Parameters & Post-Recession Stable Arm parameters (annual) pure depreciation rate pop gr rate pure time pref rate technical progress rate labor's share intertemporal subs elasticity labor subs elasticity distortion adjusted depreciation rate adjusted time preference rate productivity level leisure preference non-prime share of labor income step size Pre-Recession Post-Recession ln chg 6.0% 6.0% 1% 1% 1.4% 1.4% 0.4% 0.4% 0.7

0.7 1.35 1.35 0.75 0.75 41% 47% 7.6% 7.6% 1.0% 1.0% 0.29 0.29 0.373 0.373 0.1 0.1 0.0001 0.0001 steady state (detrended, w/ distortion) Pre-Recession SS Post-Recession SS capital stock 1 0.953 -0.048 labor 1 0.953 -0.048 gross output 0.285 0.272 -0.048 consumption 0.210 0.200 -0.048 consumption/(gross output) 73% 73% consumption/capital 0.210 0.210 0.000 adj consumption 0.223 0.211 -0.054 gross investment 0.076 0.072 -0.048 gross investment/(gross output) 27% 27% MPL 0.20 0.20 0.000 eigenvector ingredient 0 -0.231 -0.231 eigenvector ingredient 1 -0.292 -0.292 eigenvector ingredient 2 1.052 1.052

capital stock relative to ss, log relative to ss 0 1 0.0001 1.000 0.0002 1.000 0.0003 1.000 0.0004 1.000 0.0005 1.001 0.0006 1.001 0.0007 1.001 0.0008 1.001 0.0009 1.001 0.001 1.001 0.0011 1.001 0.0012 1.001 0.0013 1.001 0.0014 1.001 0.0015 1.002 0.0016 1.002 0.0017 1.002 0.0018 1.002 0.0019 1.002 0.002 1.002 0.0021 1.002 0.0022 1.002 0.0023 1.002 0.0024 1.002 0.0025 1.003 0.0026 1.003 0.0027 1.003 0.0028 1.003

level relative to bss 0.953 0.953 0.953 0.953 0.953 0.953 0.953 0.953 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.954 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.955 0.956 0.956 0.956 0.956 0.956 0.956 0.956 0.956 Post-Recession Stable Arm

consumption labor levelratio to bss levelratio to bssgross APK 0.200 0.953 0.953 0.953 0.285 0.200 0.953 0.953 0.953 0.285 0.200 0.953 0.953 0.953 0.285 0.200 0.953 0.953 0.953 0.285 0.200 0.953 0.953 0.953 0.285 0.200 0.953 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953

0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.954 0.953 0.953 0.285 0.200 0.955 0.953 0.953 0.285

0.200 0.955 0.953 0.953 0.285 0.200 0.955 0.953 0.953 0.285 0.200 0.955 0.953 0.953 0.285 0.200 0.955 0.953 0.953 0.285 0.200 0.955 0.953 0.953 0.285 cdot 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 kdot/k 0 0.0000 0.0000 0.0000

-0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0001 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0002 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003 -0.0004 -0.0004 -0.0004 -0.0004 c'(ln k) 0.132 0.119 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 0.126 Phase Diagram 1 0.995 Pre-Recession SS

consumption, ratio to pre-recession 0.99 Post-Recession SS 0.985 Post-Recession Stable Arm 0.98 0.975 0.97 0.965 0.96 0.955 0.95 0.95 0.96 0.97 0.98 0.99 capital, ratio to pre-recession 1 1.01 1.02 Nonstationary System Dynamics Let the system be nonstationary on t [0,T], and stationary thereafter Let c(k) be the stable manifold corresponding to the long run (stationary) parameters ct =c(kt) for all t T Guess kT. Solve the final boundary value system of ODEs: kt At nf (kt / n) kt ct c t (ct )ct At f (kt / n) t s.t. cT c( kT ) , kT ,{ At , t } given Iterate with alternative kT until k0 equals its given value Integrating Backwards during the Nonstationary Time Interval The Figure shows the systems dynamics and stable manifold. The dynamics shown by the red arrows correspond to the dynamics that prevailed before date 0. The new stable manifold (shown as a black path) describes dynamics once the system parameters have reached their long run values. c css T css

k ss k ss k Integrating Backwards during the Nonstationary Time Interval The Figure shows the systems dynamics and stable manifold. The dynamics shown by the red arrows correspond to the dynamics that prevailed before date 0. The new stable manifold (shown as a black path) describes dynamics once the system parameters have reached their long run values. c css T css k ss k ss k Transition Time Paths Associated With Safety Net Distortions Only Path from Dec-07 to Apr-14, safety net distortions only parameters during transition pure depreciation rate pop gr rate pure time pref rate technical progress rate labor's share intertemporal subs elasticity labor subs elasticity distortion adjusted depreciation rate adjusted time preference rate productivity level leisure preference non-prime share of labor income time zero time T T increment nobs 6.0% 1% 1.4% 0.4% 0.7 1.35 0.75 7.6% 1.0% 0.29 0.373 0.1

Dec-07 Apr-14 6.33 0.02 317 Input Measured Distortions Here Time Calendar Norm T-Norm distortion Jan-06 -1.91 8.25 0.409992 Feb-06 -1.83 8.16 0.409982 Mar-06 -1.75 8.08 0.409945 Apr-06 -1.67 8.00 0.409964 May-06 -1.59 7.92 0.409902 Jun-06 -1.50 7.83 0.409857 Jul-06 -1.42 7.75 0.409817 Aug-06 -1.33 7.67 0.409723 Sep-06 -1.25 7.58 0.409772 Oct-06 -1.17 7.50 0.410761 Nov-06 -1.08 7.41 0.410729 Dec-06 -1.00 7.33 0.41084 Jan-07 -0.91 7.25 0.410921 Feb-07 -0.83 7.16 0.41083 Mar-07 -0.75 7.09 0.410733 Apr-07 -0.67 7.00 0.410949 May-07 -0.59 6.92 0.410877 Jun-07

-0.50 6.83 0.411085 Jul-07 -0.42 6.75 0.411588 Aug-07 -0.33 6.67 0.41157 Sep-07 -0.25 6.58 0.411473 Oct-07 -0.17 6.50 0.413667 Nov-07 -0.08 6.41 0.413522 Dec-07 0.00 6.33 0.413441 Jan-08 0.08 6.25 0.41524 Feb-08 0.17 6.16 0.415179 Mar-08 0.25 6.08 0.415475 Apr-08 0.33 6.00 0.416208 May-08 0.42 5.92 0.416096 At time 0 ln k/bss -5.4E-05 kdot +/1 T-Norm 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.36 0.38 0.4

0.42 0.44 0.46 0.48 0.5 0.52 0.54 0.56 Norm distortion rel to ss, ln 6.33 47% 0.026952 6.31 47% 0.0270 6.29 47% 0.0271 6.27 47% 0.0272 6.25 47% 0.0273 6.23 47% 0.0273 6.21 47% 0.0274 6.19 47% 0.0275 6.17 47% 0.0276 6.15 47% 0.0276 6.13 47% 0.0277 6.11 47% 0.0278 6.09 47% 0.0279 6.07 47% 0.0279 6.05 46% 0.0280 6.03 46% 0.0281 6.01 46% 0.0281 5.99 46%

0.0282 5.97 46% 0.0282 5.95 46% 0.0282 5.93 46% 0.0283 5.91 46% 0.0283 5.89 46% 0.0283 5.87 46% 0.0283 5.85 46% 0.0284 5.83 46% 0.0284 5.81 46% 0.0284 5.79 46% 0.0285 5.77 46% 0.0285 capital level rel to bss 0.979 0.979 0.979 0.979 0.979 0.979 0.979 0.979 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980

0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.980 0.981 0.981 0.981 0.981 0.981 0.981 0.981 0.981 0.981 0.981 0.981 0.981 0.981 0.981 0.981 0.981 consumption level rel to bss 0.203 0.969 0.2034 0.969 0.2034 0.970 0.2034 0.970 0.2034 0.970 0.2034 0.970 0.2034 0.970 0.2034 0.970 0.2034 0.970 0.2034 0.970 0.2034 0.970 0.2034 0.970 0.2035 0.970

0.2035 0.970 0.2035 0.970 0.2035 0.970 0.2035 0.970 0.2035 0.970 0.2035 0.970 0.2035 0.970 0.2035 0.970 0.2035 0.970 0.2035 0.970 0.2035 0.970 0.2035 0.970 0.2035 0.970 0.2036 0.970 0.2036 0.970 0.2036 0.971 labor level rel to bss gross APK 0.951 0.951 0.280 0.951 0.951 0.280 0.951 0.951 0.280 0.951 0.951 0.280 0.951 0.951 0.280 0.951 0.951 0.280 0.951 0.951 0.280 0.951 0.951 0.279 0.951 0.951 0.279

0.951 0.951 0.279 0.951 0.951 0.279 0.951 0.951 0.279 0.951 0.951 0.279 0.952 0.952 0.280 0.955 0.955 0.280 0.958 0.958 0.281 0.961 0.961 0.281 0.962 0.962 0.282 0.962 0.962 0.282 0.962 0.962 0.282 0.962 0.962 0.282 0.962 0.962 0.282 0.962 0.962 0.282 0.962 0.962 0.282 0.962 0.962 0.282 0.962 0.962 0.282 0.962 0.962 0.282 0.962 0.962 0.282 0.962 0.962 0.282 cdot

-0.0005 -0.0005 -0.0005 -0.0005 -0.0005 -0.0005 -0.0005 -0.0005 -0.0005 -0.0005 -0.0005 -0.0005 -0.0005 -0.0005 -0.0004 -0.0004 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003 -0.0003 kdot/k -0.00377 -0.00378 -0.00379 -0.0038 -0.00381 -0.00382 -0.00383 -0.00384 -0.00385 -0.00387 -0.00388 -0.00389 -0.0039 -0.00351 -0.00294 -0.00237 -0.0018 -0.00148 -0.00149 -0.00149 -0.0015 -0.00151 -0.00152 -0.00152 -0.00153 -0.00154 -0.00155 -0.00155 -0.00156 Phase Diagram

1 0.995 consumption, ratio to pre-recession 0.99 Pre-Recession SS Post-Recession SS Post-Recession Stable Arm 0.985 Path from Dec-07 to Apr-14, safety net distortions only 0.98 0.975 0.97 0.965 0.96 0.955 0.95 0.95 0.96 0.97 0.98 0.99 capital, ratio to pre-recession 1 1.01 1.02 Endogenous Labor Supply Same procedure, but take n as a function of c, k, and , rather than a constant: MRS c, n(c, k , ) (1 ) Fn k , n(c, k , ) MRS function is usually restricted so that n is a normal good MPL function is usually restricted so that nFn increases with n Capital-Consumption Phase Diagram for the Stationary System: n chosen The Figure shows the stationary systems steady state, dynamics, and stable manifold. c k 0 css c 0 k ss k

Common Misconceptions NGM: Neoclassical growth model (with labor) SD: Supply and demand model of the labor market NGM/SD assumes perfect competition actually, it just assumes constant markup rates in labor and goods markets. 0 markup is a special case NGM/SD says that all labor market changes come from productivity shocks enough said NGM/SD assumes that labor supply adjusts only on the intensive margin actually, the model makes no distinction between intensive and extensive it just has aggregate hours Common Misconceptions (contd) NGM/SD says that the unemployed are the happiest people in the economy it says that the marginal worker has to be paid to work, and by extension all non-workers must be paid sufficiently more than their productivity it has equilibrium outcomes between zero works hours and zero non-work hours, and says that the hours with the least surplus (if any) are not worked. This is readily testable/refutable it is consistent with the idea that the hours with the least surplus belong to persons having a bad day, week, year, life NGM/SD says that everyone recently out of work is there because they are taking advantage of welfare among other things, dont forget the pre-existing distortions the model says that marginal workers take advantage of (i.e., consider and react to changes in) everything available to them, whether it be offered by the private sector or the public sector. This is readily testable/refutable Common Misconceptions (contd) SD says that UI benefits reduce the ratio of unemployed to vacancies SD really says: UI benefits increase unemployment, which is the numerator of the ratio SD really says: UI benefits increase wages in the short run, which reduces vacancies (in the short run) Corollary: A high ratio of unemployed to vacancies does not prove that its labor demand, and could be entirely due to unemployment benefits NGM/SD says that the unemployment created by UI benefits is chosen by the workers NGM/SD is an equilibrium model, it doesnt have fault Is it the workers fault for not taking or retaining a job, or the employers fault for not paying more to compete with UI? Is it the employers fault for cutting payroll when faced with a payroll tax, or the workers fault for not offering to absorb the employers new costs? Tax incidence theory and evidence: market outcomes do not depend on whether the worker or the employer pays the tax

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