grow theory

Notes on Growth Theory

Russell Cooper

Outline

1 Non-stochastic Growth Model

Decentralization

2 Stochastic Growth

Fixed Labor Endogenous labor

3 Confronting Data

v(k) =maxk u(f(k) + (1−δ)k−k ) +βv(k ) (1)

for allk ∈κ. Here

v(k) is the unknown value function

k is the state variable

k0 is the control (state tomorrow)

u(·) andf(·) are strictly increasing and strictly concave

Conditions for Optimality First Order Condition

u0(c) =βv0(k0) (2)

Euler Equation

u0(c) =βu0(c0)(f0(k0) + (1−δ)) (3)

to get the Euler equation, calculate v0(k) from (1) and then

use envelope condition.

As v(k) is concave (inherited fromu(·) andf(·)), theng(k) is increasing ink.

Finding a Solution

contraction mapping theorem applies: uniqueness, method of

solving

special cases

u(c) =ln(c),f(k) =kα_{, δ= 1}

Guess: v(k) =A+Blnk for allk

Numerical Analysis: Extra

grow.m is Matlab code for value function iteration program

work through it line by line

study transition dynamics and approximations of policy and value functions

Properties of the Solution

u0(f(k) + (1−δ)k−g(k)) =βv0(g(k)

v0(k) =u0(f(k) + (1−δ)k−g(k))f0(k)

v(k) concave implies (v0(k)−v0(g(k)))(k−g(k))≤0

implies (f0(k)−_β1)(k−g(k))≤0

f0(k∗) = 1_β impliesg(k)>k fork <k∗ andg(k)<k for

k >k∗

Figure :Unique, Locally Stable Steady State with Positive Capital

k

k0 _{45 degree line}

steady state

k∗ k∗

Approach

distinguish households and firms

households supply labor and rent capital and save

firms hire factor of production

competitive markets, representative households and firms

Households

K is own capital; k is aggregate capital

supply labor time inelastically: earn ω(k)

rent capital at R(k)

state:(K,k)

Household optimization:

V(K,k) =maxK0u(C) +βV(K0,k0) (4)

whereC +K0 =KR(k) + (1−δ)K +ω(k)

Firms

Y =F(k,n)

rent capital, returns a rental payment and the undepreciated

capital

hire labor, pays a wage

no dynamic choices

Recursive Competition Equilibrium

value function satisfies (4)

H(K,k) is the policy function from (4)

H(k,k) =h(k) for all k

beliefs about aggregates are consistent every household is optimizing

K =k

RCE is PO Planner

u0(f(k) + (1−δ)k−g(k)) =βv0(g(k))

v0(k) =u0(f(k) + (1−δ)k−g(k))[f0(k) + (1−δ)]

Households

u0_{(ω(k) +R(k)k+ (1−δ)k−h(k)) =βV}

1(h(k),h(k)) V1(k,k) =u0(ω(k) +R(k)k+ (1−δ)k−h(k))[R(k) + (1−δ)]

RCE is PO

The equilibrium and planning allocations coincide if

V(k,k) =v(k) for allk h(k) =g(k) for allk

for this case: V1(K,k) =u0(c)[R(k) + (1−δ)] and

v0(k) =u0(c)[f0(k) + (1−δ)]

as R(k) =f0(k), V1(k,k) =v0(k) for all k, policy functions

are the same

k0+c =ω+Rk+ (1−δ)k =f(k) + (1−δ)k from CRS technology. resource constraint holds in RCE so consumptions

are the same

V(A,k) =maxk0u(Af(k) + (1−δ)k−k0) +βE_A0_|AV(A0,k0) (5)

for all (A,k) where

c =Af(k) + (1−δ)k−k0 (6)

AssumeA is in a finite set and the transition matrix for the conditional distribution ofA0|Ahas all non-zero elements.

Finding a solution

existence of a solution

linearization of Euler equation and resource constraint

solve through value function iteration:

do this by building on grow.m: loop overAor use Kronecker product

follows Prescott-Mehra (Eca., 1980), ”Recursive Competitive

Equilibrium” definition

value function satisfies

V(A,K,k) =maxK0u(C) +βE_A0_|AV(A0,K0,k0) (8)

whereC +K0 =KR(A,k) + (1−δ)K +ω(A,k) and

k0=h(A,k) for all (A,K,k).

H(A,K,k) is the policy function from (8)

Optimization

V(A,k) =maxk0_,nu(Af(k,n) + (1−δ)k−k0,n) +βE_A0_|AV(A0,k0) (9)

for all (A,k) where

c =Af(k,n) + (1−δ)k−k0 (10)

FOCs

uc(c,n) =βEA0_|Au_c(c0,n0)[A0f_k(k0,n0) + (1−δ)] (11)

and

Finding a solution

existence of a solution

linearization of Euler equation, intratemporal FOC and resource constraint

solve through value function iteration:

do this by building on grow.m: loop overAor use Kronecker product

solve forn(A,k,k0) outside of DP loop

find policy function: k0=hk(k,A) andn=hn(k,A)

policy functions depend on parameters: Θ

simulated data depends on Θ

match moments by picking Θ to reproduce them

use regression coefficients as moments, Tony Smith paper.

Simulated Method of Moments

pick Θ to minimize distance between actual moments, Ψd, and simulated moments, Ψ(Θ):

minΘ(Ψd−Ψ(Θ))0×W ×(Ψd−Ψ(Θ)) (13)

whereW is a weighting matrix

Ψd _{would be moments on consumption, output, employment,}

From King, Plosser and Rebello

Moment US Data Calibrated Model

Standard Deviation relative to output

consumption 0.69 0.64

investment 1.35 2.31

hours 0.52 0.48

wages 1.14 0.69

Correlation with output

consumption 0.85 0.82

investment 0.60 0.92

hours 0.07 0.79

other shocks: tastes, capital accumulation needed to do MLE

government policy

multiple sectors