This notebook solves the model of Krusell and Smith (1998, JPE) and succesfully replicating the result of Maliar, Maliar, and Valli (2010, JEDC)

The solution strategy is as follows

- Solve individual problem by Euler equation method with 2D interpolation
- Agents are boundedly rational. In the code, they take into account the information about the mean of capital
- Aggregate law of motion is approximated by log-linear relation, i.e. $\log(K_{t+1})=B1+B2\log(K_{t})$ for good aggregate state and $\log(K_{t+1})=B3+B4\log(K_{t})$ for bad aggregate state
- If specified, Howard's policy iteration is used

- Compute the path of aggregate capital using the policy function obtained by VFI
- Simulation is based on Monte Carlo. That is, aggregate technology shocks and idiosyncratic employment shocks are drawn for many agents and many periods. Then, using the LLN, the aggregate capital is computed by aggregating all agents for all period.

- Update the coefficient of aggregate capital law of motion, $B1$, $B2$, $B3$ and $B4$, by regression
- Check convergence of $B1$, $B2$, $B3$ and $B4$

Additionally, this notebook also includes the result with value function iteration as a solution method for individual utility maximization problem.

NOTE: Regarding interpolation, Krusell and Smith uses various interpolation scheme depending on the purpose, including polynomial interpolation. Maliar, Maliar, and Valli uses spline interpolation in their paper. This notebook only uses linear interpolation because I could not find interpolation package for polynomial and spline 2D-interpolation.

First thing to do is import some packages

In [1]:

```
using Interpolations # to use interpolation
using QuantEcon # to use a function, `gridmake`
using Plots # to plot the result
pyplot()
#plotlyjs()
using Optim # to use minimization routine to maximize RHS of bellman equation
using GLM # to regress
using JLD # to save the result
```

Functions in this cell are prepared for model parameters and initial guess of the solutions

- Types
`TransitionMatrix`

: collection of transition matrix`KSParameter`

: collection of model parameters, functional forms, and grids`KSSolution`

: collection of solution, which is guess at first

- Methods
`create_transition_matrix`

: construct an instance of type TransitionMatrix`KSParameter`

: construct an instance of type KSParameter`place_polynominal_grid`

: create polynominal grid`r`

: compute interest rate ( = marginal productivity of capital)`w`

: compute wage rate ( = marginal productivity of labor)`KSSolution`

: construct an instance of type KSSolution- Guess can be the load of previous result

In [2]:

```
"""
Collection of transition matrix
"""
immutable TransitionMatrix
P::Array{Float64,2} # 4x4
Pz::Array{Float64,2} # 2x2 aggregate shock
Peps_gg::Array{Float64,2} # 2x2 idiosyncratic shock conditional on good to good
Peps_bb::Array{Float64,2} # 2x2 idiosyncratic shock conditional on bad to bad
Peps_gb::Array{Float64,2} # 2x2 idiosyncratic shock conditional on good to bad
Peps_bg::Array{Float64,2} # 2x2 idiosyncratic shock conditional on bad to good
end
"""
Collection of model parameters
"""
immutable KSParameter
u::Function
beta::Float64
alpha::Float64
delta::Float64
theta::Float64
l_bar::Float64
k_grid::Vector{Float64}
K_grid::Vector{Float64}
z_grid::Vector{Float64}
eps_grid::Vector{Float64}
s_grid::Array{Float64,2}
k_size::Int64
K_size::Int64
z_size::Int64
eps_size::Int64
s_size::Int64
ug::Float64
ub::Float64
TransMat::TransitionMatrix # bunch of transition matrix
mu::Float64
end
"""
Create transition matrices for aggregate shock,
idiosyncratic shock, and shock state
##### Arguments
- `ug` : unemployment rate in good state
- `ub` : unemployment rate in bad state
- `zg_ave_dur` : average duration of good state
- `zb_ave_dur` : average duration of bad state
- `ug_ave_dur` : average duration of unemployment in good state
- `ub_ave_dur` : average duration of unemployment in bad state
- `puu_rel_gb2bb` : prob. of u to u cond. on g to b relative to that of b to b
- `puu_rel_bg2gg` : prob. of u to u cond. on b to g relative to that of g to g
"""
function create_transition_matrix(ug::Float64,ub::Float64,
zg_ave_dur::Float64,zb_ave_dur::Float64,
ug_ave_dur::Float64,ub_ave_dur::Float64,
puu_rel_gb2bb::Float64,puu_rel_bg2gg::Float64)
# probability of remaining in good state
pgg = 1-1/zg_ave_dur
# probability of remaining in bad state
pbb = 1-1/zb_ave_dur
# probability of changing from g to b
pgb = 1-pgg
# probability of changing from b to g
pbg = 1-pbb
# prob. of 0 to 0 cond. on g to g
p00_gg = 1-1/ug_ave_dur
# prob. of 0 to 0 cond. on b to b
p00_bb = 1-1/ub_ave_dur
# prob. of 0 to 1 cond. on g to g
p01_gg = 1-p00_gg
# prob. of 0 to 1 cond. on b to b
p01_bb = 1-p00_bb
# prob. of 0 to 0 cond. on g to b
p00_gb=puu_rel_gb2bb*p00_bb
# prob. of 0 to 0 cond. on b to g
p00_bg=puu_rel_bg2gg*p00_gg
# prob. of 0 to 1 cond. on g to b
p01_gb=1-p00_gb
# prob. of 0 to 1 cond. on b to g
p01_bg=1-p00_bg
# prob. of 1 to 0 cond. on g to g
p10_gg=(ug - ug*p00_gg)/(1-ug)
# prob. of 1 to 0 cond. on b to b
p10_bb=(ub - ub*p00_bb)/(1-ub)
# prob. of 1 to 0 cond. on g to b
p10_gb=(ub - ug*p00_gb)/(1-ug)
# prob. of 1 to 0 cond on b to g
p10_bg=(ug - ub*p00_bg)/(1-ub)
# prob. of 1 to 1 cond. on g to g
p11_gg= 1-p10_gg
# prob. of 1 to 1 cond. on b to b
p11_bb= 1-p10_bb
# prob. of 1 to 1 cond. on g to b
p11_gb= 1-p10_gb
# prob. of 1 to 1 cond on b to g
p11_bg= 1-p10_bg
# (g1) (b1) (g0) (b0)
P=[pgg*p11_gg pgb*p11_gb pgg*p10_gg pgb*p10_gb;
pbg*p11_bg pbb*p11_bb pbg*p10_bg pbb*p10_bb;
pgg*p01_gg pgb*p01_gb pgg*p00_gg pgb*p00_gb;
pbg*p01_bg pbb*p01_bb pbg*p00_bg pbb*p00_bb
]
Pz=[pgg pgb;
pbg pbb]
Peps_gg=[p11_gg p10_gg
p01_gg p00_gg]
Peps_bb=[p11_bb p10_bb
p01_bb p00_bb]
Peps_gb=[p11_gb p10_gb
p01_gb p00_gb]
Peps_bg=[p11_bg p10_bg
p01_bg p00_bg]
TransMat=TransitionMatrix(P,Pz,Peps_gg,Peps_bb,Peps_gb,Peps_bg)
return TransMat
end
"""
Creates KSParameter instance
"""
function KSParameter(;
beta::Float64=0.99,
alpha::Float64=0.36,
delta::Float64=0.025,
theta::Float64=1.0,
k_min::Float64=1e-16,
k_max::Float64=1000.0,
k_size::Int64=100,
K_min::Float64=30.0,
K_max::Float64=50.0,
K_size::Int64=4,
z_min::Float64=0.99,
z_max::Float64=1.01,
z_size::Int64=2,
eps_min::Float64=0.0,
eps_max::Float64=1.0,
eps_size::Int64=2,
ug::Float64=0.04,
ub::Float64=0.1,
zg_ave_dur::Float64=8.0,
zb_ave_dur::Float64=8.0,
ug_ave_dur::Float64=1.5,
ub_ave_dur::Float64=2.5,
puu_rel_gb2bb::Float64=1.25,
puu_rel_bg2gg::Float64=0.75,
mu::Float64=0.0
)
if theta == 1.0
u = (c) -> log(c)
else
u = (c) -> (c^(1.0-theta)-1.0)/(1.0-theta)
end
l_bar=1/(1-ub)
k_grid=place_polynominal_grid(k_min,k_max,k_size,degree=7.0) # individual capital grid
K_grid=collect(linspace(K_min,K_max,K_size)) # aggregate capital grid
z_grid=collect(linspace(z_max,z_min,z_size)) # aggregate technology shock
eps_grid=collect(linspace(eps_max,eps_min,eps_size)) # idiosyncratic employment shock
s_grid=gridmake(z_grid,eps_grid) # shock grid
# collection of transition matrices
TransMat=create_transition_matrix(ug,ub,
zg_ave_dur,zb_ave_dur,
ug_ave_dur,ub_ave_dur,
puu_rel_gb2bb,puu_rel_bg2gg)
ksp=KSParameter(u,beta,alpha,delta,theta,l_bar,k_grid,K_grid,z_grid,eps_grid,s_grid,
k_size,K_size,z_size,eps_size,z_size*eps_size,ug,ub,TransMat,mu)
return ksp
end
function place_polynominal_grid(k_min::Float64,k_max::Float64,k_size::Int64;degree::Float64=0.7)
grid=Array{Float64}(k_size)
for (i, x) in enumerate(linspace(0,0.5,k_size))
grid[i]=(x/0.5)^degree*(k_max-k_min)+k_min
end
return grid
end
"""
Compute interest rate given aggregate capital, labor, and productivity
##### Arguments
- `alpha` : capital share
- `z` : aggregate shock
- `K` : aggregate capital
- `L` : aggregate labor
"""
r(alpha::Float64,z::Float64,K::Float64,L::Float64)=alpha*z*K^(alpha-1)*L^(1-alpha)
"""
Compute wage given aggregate capital, labor, and productivity
##### Arguments
- `alpha` : capital share
- `z` : aggregate shock
- `K` : aggregate capital
- `L` : aggregate labor
"""
w(alpha::Float64,z::Float64,K::Float64,L::Float64)=(1-alpha)*z*K^(alpha)*L^(-alpha)
"""
Collection of KS solution
"""
type KSSolution
k_opt::Array{Float64,3}
value::Array{Float64,3}
B::Vector{Float64}
R2::Vector{Float64}
end
"""
Create KSSolution instance
"""
function KSSolution(
ksp::KSParameter;
load_value::Bool=false,
load_B::Bool=false,
filename::String="result.jld")
if load_value || load_B
result=load(filename)
kss_temp=result["kss"]
end
if load_value
k_opt=kss_temp.k_opt
value=kss_temp.value
else
k_opt=ksp.beta*repeat(ksp.k_grid,outer=[1,ksp.K_size,ksp.s_size])
k_opt=0.9*repeat(ksp.k_grid,outer=[1,ksp.K_size,ksp.s_size])
value=ksp.u.(0.1/0.9*k_opt)/(1-ksp.beta)
end
if load_B
B=kss_temp.B
else
B=[0.0, 1.0, 0.0, 1.0]
end
kss=KSSolution(k_opt,value,B,[0.0,0.0])
return kss
end
```

Out[2]:

The functions in this cell are used to draw aggregate and idiosyncratic shocks

- Methods
`get_shock`

: translate an index shock combination grid to shock value`generate_shocks`

: draw aggregate and idiosyncratic shocks, which is the main function in this cell`draw_eps_shock`

(local function inside`generate_shocks`

): draw idiosyncratic shocks given aggregate shocks

In [3]:

```
"""
Translate shock index into shock value
##### Arguments
- `s_grid` : shock grid
- `s_i` : shock index
"""
get_shock(s_grid::Array{Float64,2},s_i::Integer) =
s_grid[s_i,1], s_grid[s_i,2]
"""
Generate aggregate and idiosyncratic shock
##### Arguments
- `ksp` : instance of KSParameter type
- `z_shock_size` : size of aggregate shock
- `population` : size idiosyncratic shock in one period
"""
function generate_shocks(ksp::KSParameter;
z_shock_size::Int64=11000,population::Int64=10000)
"""
Draw idiosyncratic shock given previous idiosyncratic shock and
transition matrix.
The transition matrix must be consistent with aggregate shock
##### Arguments
- `eps_shock` : preallocated vector. current period shock is stored in it
- `eps_shock_before` : previous period idiosyncratic shock
- `Peps` : transition matrix of the current period
"""
function draw_eps_shock(eps_shock_before::Vector{Float64},
Peps::Array{Float64,2})
eps_shocks = similar(eps_shock_before)
# loop over entire population
for i=1:length(eps_shocks)
rand_draw=rand()
eps_shocks[i]=ifelse(eps_shock_before[i]==1.0,
Float64(Peps[1,1]>rand_draw), # if employed before
Float64(Peps[2,1]>rand_draw)) # if unemployed before
end
return eps_shocks
end
# unpack parameters
Peps_gg=ksp.TransMat.Peps_gg
Peps_bg=ksp.TransMat.Peps_bg
Peps_gb=ksp.TransMat.Peps_gb
Peps_bb=ksp.TransMat.Peps_bb
# draw aggregate shock
z_shock=simulate(MarkovChain(ksp.TransMat.Pz,ksp.z_grid),z_shock_size)
### Let's draw individual shock ###
eps_shock=Array{Float64}(z_shock_size,population) # preallocation
# first period
rand_draw=rand(population)
if z_shock[1]==ksp.z_grid[1] # if good
eps_shock[1,:].=Int64.(rand_draw.>ksp.ug) # if draw is higher, become employed
elseif z_shock[1]==ksp.z_grid[2] # if bad
eps_shock[1,:].=Int64.(rand_draw.>ksp.ub) # if draw is higher, become employed
else
error("the value of z_shock[1] (=$(z_shock[1])) is strange")
end
# from second period ...
for t=2:z_shock_size
if z_shock[t]==ksp.z_grid[1] && z_shock[t-1]==ksp.z_grid[1] # if g to g
eps_shock[t,:]=draw_eps_shock(eps_shock[t-1,:],Peps_gg)
elseif z_shock[t]==ksp.z_grid[1] && z_shock[t-1]==ksp.z_grid[2] # if b to g
eps_shock[t,:]=draw_eps_shock(eps_shock[t-1,:],Peps_bg)
elseif z_shock[t]==ksp.z_grid[2] && z_shock[t-1]==ksp.z_grid[1] # if g to b
eps_shock[t,:]=draw_eps_shock(eps_shock[t-1,:],Peps_gb)
elseif z_shock[t]==ksp.z_grid[2] && z_shock[t-1]==ksp.z_grid[2] # if b to b
eps_shock[t,:]=draw_eps_shock(eps_shock[t-1,:],Peps_bb)
else
error("the value of z_shock[t] (=$(z_shock[t])) is strange")
end
end
# adjustment
for t=1:z_shock_size
n_e=Int64(sum(eps_shock[t,:]))
er_ideal = ifelse(z_shock[t] == ksp.z_grid[1],
1.0-ksp.ug, 1.0-ksp.ub)
gap = Int64(round(er_ideal*population)) - n_e
if gap > 0
for j=1:gap
become_employed_i =
sample(find(x-> isapprox(x,ksp.eps_grid[2]), eps_shock[t,:]))
eps_shock[t,become_employed_i] = ksp.eps_grid[1]
end
elseif gap < 0
for j=1:(-gap)
become_unemployed_i =
sample(find(x-> isapprox(x,ksp.eps_grid[1]), eps_shock[t,:]))
eps_shock[t,become_unemployed_i] = ksp.eps_grid[2]
end
end
end
return z_shock, eps_shock
end
```

Out[3]:

Functions in this cell is used to solve individual problem by Euler equation method

Methods

- compute_Kp_L: compute approximated $K'$ and $L$, which depend on current aggregate shock
- compute_expectation_FOC: compute expectation term in Euler equation
euler_method!: find optimal policy by Euler equation methods, namely, iteration of Euler equation

$$ \left(c\right)^{-\theta}=\beta E\left[\left(c'\right)^{-\theta}(1-\delta+r')\right] $$

In [4]:

```
"""
Compute next period aggregate capital and labor
##### Arguments
- `K` : Current aggregate capital
- `s_i` : current shock index
- `B` : coefficient of ALM for capital
"""
function compute_Kp_L(K::AbstractFloat,s_i::Integer,B::Vector{Float64},ksp::KSParameter)
Kp,L=ifelse(ksp.s_grid[s_i,1]==ksp.z_grid[1],
(exp(B[1]+B[2]*log(K)),ksp.l_bar*(1-ksp.ug)),
(exp(B[3]+B[4]*log(K)),ksp.l_bar*(1-ksp.ub)))
Kp = ifelse(Kp<ksp.K_grid[1],ksp.K_grid[1],Kp)
Kp = ifelse(Kp>ksp.K_grid[end],ksp.K_grid[end],Kp)
return Kp, L
end
"""
Compute expectation term in Euler equation
##### Arguments
- `kp` : next period individual capital holding
- `Kp` : next period aggregate capital
- `s_i` : current state of shock
- `ksp` : KSParameter instance
"""
function compute_expectation_FOC(kp::Float64,
Kp::Float64,
s_i::Int64,
ksp::KSParameter)
alpha, theta, delta, l_bar, mu, P =
ksp.alpha, ksp.theta, ksp.delta, ksp.l_bar, ksp.mu, ksp.TransMat.P
expec = 0.0
for s_n_i = 1:ksp.s_size
zp, epsp = ksp.s_grid[s_n_i,1], ksp.s_grid[s_n_i,2]
Kpp, Lp = compute_Kp_L(Kp,s_n_i,kss.B,ksp)
rn=r(alpha,zp,Kp,Lp)
kpp=interpolate((ksp.k_grid,ksp.K_grid),
kss.k_opt[:,:,s_n_i],Gridded(Linear()))
cp = (rn+1-delta)*kp+
w(alpha,zp,Kp,Lp)*(epsp*l_bar+mu*(1.0-epsp))-kpp[kp,Kp]
expec = expec + P[s_i,s_n_i]*(cp)^(-theta)*(1-delta+rn)
end
return expec
end
"""
Solve individual problem by Euler equation method
##### Arguments
- `ksp` : KSParameter instance
- `kss` : KSSolution instance
- `n_iter` : maximum number of iteration of Euler equation method
- `tol` : tolerance of policy function convergence
- `update_k` : degree of update of policy function
"""
function euler_method!(ksp::KSParameter,
kss::KSSolution;
n_iter::Integer=30000,
tol::AbstractFloat=1e-8,
update_k::AbstractFloat=1e-8)
println("solving individual problem by Euler equation method")
alpha, beta, delta, theta, l_bar, mu =
ksp.alpha, ksp.beta, ksp.delta, ksp.theta, ksp.l_bar, ksp.mu
k_grid, k_size = ksp.k_grid, ksp.k_size
K_grid, K_size = ksp.K_grid, ksp.K_size
s_grid, s_size = ksp.s_grid, ksp.s_size
k_min, k_max = minimum(k_grid), maximum(k_grid)
counter=0
k_opt_n=similar(kss.k_opt)
while true
counter += 1
dif_k=0.0
for s_i = 1:s_size
z, eps = s_grid[s_i,1], s_grid[s_i,2]
for K_i = 1:K_size
K = K_grid[K_i]
Kp, L = compute_Kp_L(K,s_i,kss.B,ksp)
for k_i = 1:k_size
k=k_grid[k_i]
wealth = (r(alpha,z,K,L)+1-delta)*k+
w(alpha,z,K,L)*(eps*l_bar+mu*(1.0-eps))
expec=compute_expectation_FOC(kss.k_opt[k_i,K_i,s_i],Kp,s_i,ksp)
cn=(beta*expec)^(-1.0/theta)
k_opt_n[k_i,K_i,s_i] = wealth-cn
k_opt_n[k_i,K_i,s_i] = ifelse(k_opt_n[k_i,K_i,s_i]>k_max,k_max,k_opt_n[k_i,K_i,s_i])
k_opt_n[k_i,K_i,s_i] = ifelse(k_opt_n[k_i,K_i,s_i]<k_min,k_min,k_opt_n[k_i,K_i,s_i])
end
end
end
dif_k=maxabs(k_opt_n-kss.k_opt)
kss.k_opt.=update_k.*k_opt_n.+(1-update_k).*kss.k_opt
if dif_k<tol
println("Euler method converged with $counter iterations")
break
end
if counter >=n_iter
println("Euler method failed to converge with $counter iterations (dif = $dif_k)")
break
end
end
end
```

Out[4]:

- Mathods
- simulate_aggregate_path!: simulating aggregate path of capital using shocks drawn

In [5]:

```
"""
Simulate aggregate capital's path using policy function
and generated aggregate and idiosyncratic shock
##### Arguments
- `ksp` : KSParameter instance
- `z_shocks` : aggregate shocks
- `eps_shocks` : idiosyncratic shocks
- `k_population` : initial capital holding of all agents
"""
function simulate_aggregate_path!(ksp::KSParameter,kss::KSSolution,
z_shocks::Vector{Float64},eps_shocks::Array{Float64,2},
k_population::Vector{Float64},K_ts::Vector{Float64})
println("simulating aggregate path ... please wait ... ")
T=length(z_shocks) # simulated duration
N=size(eps_shocks,2) # number of agents
# loop over T periods
for (t,z) = enumerate(z_shocks)
K_ts[t]=mean(k_population) # current aggrgate capital
# s_i_base takes 1 when good and 2 when bad
s_i_base=ifelse(z==ksp.z_grid[1],1,2)
# loop over individuals
for (i,k) in enumerate(k_population)
eps = eps_shocks[t,i] # idiosyncratic shock
s_i=s_i_base+2*(1-Int64(eps)) # transform (z,eps) to s_i
# obtain next capital holding by interpolation
itp_pol=interpolate((ksp.k_grid,ksp.K_grid),kss.k_opt[:,:,s_i],Gridded(Linear()))
k_population[i]=itp_pol[k,K_ts[t]]
end
end
return nothing
end
```

Out[5]:

The functions in this cell are used to obtain the coefficient of approximate aggregate capital law of motion (ALM)

- Methods
- regress_ALM: regress aggregate capital law of motion and obtain approximate ALM
- find_ALM_coef!: main iteration. update ALM coefficient until convergence

In [6]:

```
"""
Obtain new aggregate law of motion coefficients
using the aggregate capital flaw
##### Arguments
- `ksp` : KSParameter instance
- `z_shock` : aggregate shocks
- `K_ts` : aggregate capital flaw
- `n_discard` : number of discarded samples
"""
function regress_ALM!(ksp::KSParameter,kss::KSSolution,
z_shock::Vector{Float64},K_ts::Vector{Float64};
n_discard::Int64=100)
z_grid=ksp.z_grid
n_g=count(i->(i==z_grid[1]),z_shocks[n_discard+1:end-1])
n_b=count(i->(i==z_grid[2]),z_shocks[n_discard+1:end-1])
B_n=Vector{Float64}(4)
x_g=Vector{Float64}(n_g)
y_g=Vector{Float64}(n_g)
x_b=Vector{Float64}(n_b)
y_b=Vector{Float64}(n_b)
i_g=0
i_b=0
for t = n_discard+1:length(z_shocks)-1
if z_shocks[t]==z_grid[1]
i_g=i_g+1
x_g[i_g]=log(K_ts[t])
y_g[i_g]=log(K_ts[t+1])
else
i_b=i_b+1
x_b[i_b]=log(K_ts[t])
y_b[i_b]=log(K_ts[t+1])
end
end
resg=lm(hcat(ones(n_g,1),x_g),y_g)
resb=lm(hcat(ones(n_b,1),x_b),y_b)
kss.R2=[r2(resg),r2(resb)]
B_n[1],B_n[2]=coef(resg)
B_n[3],B_n[4]=coef(resb)
dif_B=maximum(abs(B_n-kss.B))
println("difference of ALM coefficient is $dif_B and B = $B_n")
return B_n, dif_B
end
function find_ALM_coef!(
ksp::KSParameter,
kss::KSSolution,
z_shocks::Vector{Float64},
eps_shocks::Array{Float64,2};
tol_ump::AbstractFloat=1e-8,
max_iter_ump::Integer=100,
Howard_on::Bool=true,
Howard_n_iter::Integer=20,
tol_B::AbstractFloat=1e-8,
max_iter_B::Integer=20,
update_B::AbstractFloat=0.3,
T_discard::Integer=100,
print_skip_VFI::Integer=10,
method::Symbol=:Euler,
update_k::AbstractFloat=0.3)
K_ts=similar(z_shocks)
# populate initial capital holdings
k_population=37.9893*ones(size(eps_shocks,2))
counter_B=0
while true
counter_B=counter_B+1
println(" --- Iteration over ALM coefficient: $counter_B ---")
# solve individual problem
if method == :VFI
solve_bellman!(ksp,kss,
tol=tol_ump,
max_iter=max_iter_ump,
Howard=Howard_on,
Howard_n_iter=Howard_n_iter,
print_skip=print_skip_VFI)
elseif method == :Euler
euler_method!(ksp,kss,
n_iter=max_iter_ump,
tol=tol_ump,
update_k=update_k)
end
# compute aggregate path of capital
simulate_aggregate_path!(ksp,kss,z_shocks,eps_shocks,k_population,K_ts)
# obtain new ALM coefficient by regression
B_n,dif_B = regress_ALM!(ksp,kss,z_shocks,K_ts,n_discard=T_discard)
# check convergence
if dif_B < tol_B
println("-----------------------------------------------------")
println("ALM coefficient successfully converged : dif = $dif_B")
println("-----------------------------------------------------")
kss.B .= update_B .* B_n .+ (1-update_B) .* kss.B
break
elseif counter_B == max_iter_B
println("----------------------------------------------------------------")
println("Iteration over ALM coefficient reached its maximum ($max_iter_B)")
println("----------------------------------------------------------------")
kss.B .= update_B .* B_n .+ (1-update_B) .* kss.B
break
end
# Update B
kss.B .= update_B .* B_n .+ (1-update_B) .* kss.B
end
return K_ts
end
```

Out[6]:

The function is used to plot the path of true path of aggregate capital and approximated one

- Methods
- plot_ALM: plot the path of true path of aggregate capital and approximated one

In [7]:

```
"""
Plot true and approximated ALM of capital
##### Arguments
- `ksp.z_grid` : aggregate shock grid
- `z_shocks` : aggregate shock
- `kss.B` : ALM coefficient
- `K_ts` : actual path of capital
"""
function plot_ALM(z_grid::Vector{Float64},
z_shocks::Vector{Float64},
B::Vector{Float64},
K_ts::Vector{Float64};
T_discard=1000)
K_ts_approx = similar(K_ts) # preallocation
# compute approximate ALM for capital
K_ts_approx[T_discard]=K_ts[T_discard]
for t=T_discard:length(z_shocks)-1
if z_shocks[t] == z_grid[1]
K_ts_approx[t+1]=exp(B[1]+B[2]*log(K_ts_approx[t]))
elseif z_shocks[t] == z_grid[2]
K_ts_approx[t+1]=exp(B[3]+B[4]*log(K_ts_approx[t]))
end
end
plot(T_discard+1:length(K_ts),K_ts[T_discard+1:end],lab="true",color=:red,line=:solid)
plot!(T_discard+1:length(K_ts),K_ts_approx[T_discard+1:end],lab="approximation",color=:blue,line=:dash)
title!("aggregate law of motion for capital")
end
```

Out[7]:

First, construct model instance `ksp`

and initial guess of the solution inside `kss`

(Grid size inconsistency is also checked, which may return error when exiting result is loaded by `load_value=true`

)

In [8]:

```
# instance of KSParameter
ksp=KSParameter(k_size=100,
k_min=1e-16)
# instance of KSSolution
kss=KSSolution(ksp,load_value=false,load_B=false)
if size(kss.k_opt,1) != length(ksp.k_grid)
error("loaded data is inconsistent with k_size")
end
if size(kss.k_opt,2) != length(ksp.K_grid)
error("loaded data is inconsistent with K_size")
end
```

Let's draw the shock for stochastic simulation of aggregate law of motion

In [9]:

```
# generate shocks
srand(123)
z_shocks,eps_shocks =generate_shocks(ksp;
z_shock_size=11000,population=5000);
```

Now, the following cell solves the model with Euler equation method

In [10]:

```
# find ALM coefficient
@time K_ts = find_ALM_coef!(ksp,kss,z_shocks,eps_shocks,
tol_ump=1e-8,max_iter_ump=10000,
Howard_on=true,Howard_n_iter=20,
tol_B=1e-8, max_iter_B=50,update_B=0.3,
T_discard=1000,print_skip_VFI=10,
method=:Euler, update_k=0.7);
```

Let's compare the true aggreate law of motion for capital and approximated one with figure and regression

In [11]:

```
plot_ALM(ksp.z_grid,z_shocks,
kss.B,K_ts)
```

Out[11]:

In [12]:

```
#kss.B # Regression coefficient
println("Approximated aggregate capital law of motion")
println("log(K_{t+1})=$(kss.B[1])+$(kss.B[2])log(K_{t}) in good time (R2 = $(kss.R2[1]))")
println("log(K_{t+1})=$(kss.B[3])+$(kss.B[4])log(K_{t}) in bad time (R2 = $(kss.R2[2]))")
```

The approximated law of motion of capital is very close to the true one, which implies that assuming agents are partially rational is not bad idea since the difference of their actions are negligible.

Note: The mean of capital, about 40, is sufficiently close to Maliar, Maliar, Valli but higher than Krusell-Smith.

In [13]:

```
save("result_Euler.jld","kss",kss)
```

In [14]:

```
# Compute mean of capital implied by regression
mc=MarkovChain(ksp.TransMat.Pz)
sd=stationary_distributions(mc)[1]
logKg=kss.B[1]/(1-kss.B[2])
logKb=kss.B[3]/(1-kss.B[4])
meanK_reg=exp(sd[1]*logKg+sd[2]*logKb)
meanK_sim=mean(K_ts[1001:end])
println("mean of capital implied by regression is $meanK_reg")
println("mean of capital implied by simulation is $meanK_sim")
```

Let's prepare some functions for figures in Krusell-Smith

In [15]:

```
function plot_Fig1(ksp,kss,K_ts)
B=kss.B
K_min, K_max = minimum(K_ts), maximum(K_ts)
K_lim=collect(linspace(K_min,K_max,100))
Kp_g=exp(B[1]+B[2]*log(K_lim))
Kp_b=exp(B[3]+B[4]*log(K_lim))
p=plot(K_lim,Kp_g,linestyle=:solid,lab="Good")
plot!(p,K_lim,Kp_b,linestyle=:solid,lab="Bad")
plot!(p,K_lim,K_lim,color=:black,linestyle=:dash,lab="45 degree",width=0.5)
title!(p,"FIG1: Tomorrow's vs. today's aggregate capital")
p
end
function plot_Fig2(ksp,kss,K_eval_point)
k_lim=collect(linspace(0,80,1000))
itp_e=interpolate((ksp.k_grid,ksp.K_grid),kss.k_opt[:,:,1],Gridded(Linear()))
itp_u=interpolate((ksp.k_grid,ksp.K_grid),kss.k_opt[:,:,3],Gridded(Linear()))
kp_e(k)=itp_e[k,K_eval_point]
kp_u(k)=itp_u[k,K_eval_point]
p=plot(k_lim,kp_e.(k_lim),linestyle=:solid,lab="employed")
plot!(p,k_lim,kp_u.(k_lim),linestyle=:solid,lab="unemployed")
plot!(p,k_lim,k_lim,color=:black,linestyle=:dash,lab="45 degree",width=0.5)
title!(p,"FIG2: Individual policy function \n at K=$K_eval_point when good state")
p
end
```

Out[15]:

Now, plot the replication figure:

In [16]:

```
plot_Fig1(ksp,kss,K_ts)
```

Out[16]:

In [17]:

```
plot_Fig2(ksp,kss,40)
```

Out[17]:

Both figures are replicated well.

Note that the mean of capital is approximately 40 in this replication, which is different from Krusell-Smith but same as Maliar, Maliar, Valli. Therefore, Figure 1 is plotted around 40 and policy function for Figure 2 is evaluated at K=40

In this section, each agent's utility maximization problem is solved by value function iteration.

Let's prepare functions for value function iteration

The functions in this cell are used to solve individual household problem by VFI

- Methods
- rhs_bellman: evaluate RHS of bellman equation given the guess of value function
- compute_expectation_VFI: compute expectation term of Bellman equation given current shock state
- solve_bellman_once!: maximize RHS of Bellman equation for an agent of state (k_i,K_i,s_i)
- solve_bellman!: maximize RHS of Bellman equation for all state until convergence
- iterate_policy!: iterating policy and compute values under the policy. used for Howard

In [18]:

```
"""
Compute right hand side of bellman equation
##### Arguments
- `kp` : next period capital
- `ksm` : KSModel instance
- `k` : current individual capital
- `K` : current aggregate capital
- `L` : current labor
- `zeps_i` :
"""
function rhs_bellman(ksp::KSParameter,
kp::AbstractFloat,value::Array{Float64,3},
k::AbstractFloat,K::AbstractFloat,s_i::Integer)
u,s_grid,beta,alpha,l_bar,delta, mu =
ksp.u, ksp.s_grid, ksp.beta, ksp.alpha, ksp.l_bar, ksp.delta, ksp.mu
z, eps = get_shock(s_grid,s_i)
Kp,L = compute_Kp_L(K,s_i,kss.B,ksp) # Next period aggregate capital and current aggregate labor
c = (r(alpha,z,K,L)+1-delta)*k+
w(alpha,z,K,L)*(eps*l_bar+(1.0-eps)*mu)-kp # current consumption
expec = compute_expectation(kp,Kp,value,s_i,ksp)
return u(c)+beta*expec
end
"""
Compute expectation of next period value
##### Arguments
- `kp` : next period individual capital
- `Kp` : next period aggregate capital
- `value` : given value
- `s_i` : current shock state
- `ksp` : KSParameter instance
"""
function compute_expectation(
kp::AbstractFloat, # next period indicidual capital
Kp::AbstractFloat, # next period aggragte capital
value::Array{Float64,3}, # next period value
s_i::Integer, # index of current state,
ksp::KSParameter
)
k_grid, K_grid = ksp.k_grid, ksp.K_grid # unpack grid
beta, P = ksp.beta, ksp.TransMat.P # unpack parameters
# compute expectations by summing up
expec=0.0
for s_n_i=1:ksp.s_size
value_itp=interpolate((k_grid,K_grid),value[:,:,s_n_i],Gridded(Linear()))
expec = expec + P[s_i,s_n_i]*value_itp[kp,Kp]
end
return expec
end
"""
Solve bellman equation for all states once
##### Arguments
- `k` : individual capital
- `K` : aggregate capital
- `s_i` : shock state index
- `ksp` : KSParameter
- `kss` : KSSolution
"""
function solve_bellman_once!(
k_i::Integer,
K_i::Integer,
s_i::Integer,
ksp::KSParameter,
kss::KSSolution,
)
# obtain minimum and maximum of grid
k_min, k_max = ksp.k_grid[1], ksp.k_grid[end]
# unpack parameters
alpha,delta,l_bar, mu =
ksp.alpha, ksp.delta, ksp.l_bar, ksp.mu
# obtain state value
k=ksp.k_grid[k_i] # obtain individual capital value
K=ksp.K_grid[K_i] # obtain aggregate capital value
z, eps = get_shock(ksp.s_grid,s_i) # obtain shock value
Kp,L=compute_Kp_L(K,s_i,kss.B,ksp) # next aggregate capital and current aggregate labor
# if kp>k_c_pos, consumption is negative
k_c_pos=(r(alpha,z,K,L)+1-delta)*k+
w(alpha,z,K,L)*(eps*l_bar+(1-eps)*mu)
obj(kp)=-rhs_bellman(ksp,kp,kss.value,k,K,s_i) # objective function
res=optimize(obj,k_min,min(k_c_pos,k_max)) # maximize value
# obtain result
kss.k_opt[k_i,K_i,s_i]=Optim.minimizer(res)
kss.value[k_i,K_i,s_i]=-Optim.minimum(res)
return nothing
end
"""
Solve bellman equation for all states until convergence
##### Arguments
- `ksp` : KSParameter
- `kss` : KSSolution
- `tol` : tolerance of value function difference
- `max_iter` : maximum number of iteration
"""
function solve_bellman!(
ksp::KSParameter,
kss::KSSolution;
tol::AbstractFloat=1e-8,
max_iter::Integer=100,
Howard::Bool=false,
Howard_n_iter::Integer=20,
print_skip::Integer=10)
counter_VFI=0 # counter
while true
counter_VFI += 1
value_old=copy(kss.value) # guessed value
# maximize value for all state
[solve_bellman_once!(k_i,K_i,s_i,ksp,kss)
for k_i in 1:ksp.k_size, K_i in 1:ksp.K_size, s_i in 1:ksp.s_size]
# Howard's policy iteration
!Howard || iterate_policy!(ksp,kss,n_iter=Howard_n_iter)
# difference of guessed and new value
dif=maxabs(value_old-kss.value)
# print covergence process
!(counter_VFI % print_skip ==0) ||
println("VFI iteration $counter_VFI : dif = $dif")
# if difference is sufficiently small
if dif<tol
println(" ** VFI converged successfully!! dif = $dif")
break
elseif counter_VFI >= max_iter
println("VFI reached its maximum iteration : $max_iter")
break
end
end
end
"""
Iterate the value function fixing the policy function
##### Arguments
- `ksp` : KSParameter instance
- `kss` : KSSolution instance
- `n_iter` : number of iterations
"""
function iterate_policy!(ksp::KSParameter,
kss::KSSolution;n_iter::Int64=20)
value=similar(kss.value)
for i=1:n_iter
# update value using policy
value .=
[rhs_bellman(ksp,
kss.k_opt[k_i,K_i,s_i],kss.value,
ksp.k_grid[k_i],ksp.K_grid[K_i],s_i)
for k_i in 1:ksp.k_size,
K_i in 1:ksp.K_size,
s_i in 1:ksp.s_size]
kss.value.=copy(value)
end
return nothing
end
```

Out[18]:

Let's skip the following steps in this section to save computational time.

- consturuction of
`ksp`

instance since it is same - consturuction of
`kss`

instance to use the previous result as initial guess of the solution - draws of the shocks to use same ones

However, instead of constructing `kss`

again, obtain value from the policy function derived by Euler method:

In [19]:

```
iterate_policy!(ksp,kss,n_iter=30);
```

Now, the model is solved by VFI in the next cell

In [20]:

```
# find ALM coefficient
@time K_ts = find_ALM_coef!(ksp,kss,z_shocks,eps_shocks,
tol_ump=1e-8,max_iter_ump=10000,
Howard_on=true,Howard_n_iter=20,
tol_B=1e-8, max_iter_B=50,update_B=0.3,
T_discard=1000,print_skip_VFI=10,
method=:VFI, update_k=0.7);
```

The following same exercises show that the main result is same as before

In [21]:

```
plot_ALM(ksp.z_grid,z_shocks,
kss.B,K_ts)
```

Out[21]:

In [22]:

```
#kss.B # Regression coefficient
println("Approximated aggregate capital law of motion")
println("log(K_{t+1})=$(kss.B[1])+$(kss.B[2])log(K_{t}) in good time (R2 = $(kss.R2[1]))")
println("log(K_{t+1})=$(kss.B[3])+$(kss.B[4])log(K_{t}) in bad time (R2 = $(kss.R2[2]))")
```

In [23]:

```
save("result_VFI.jld","kss",kss)
```

In [24]:

```
# Compute mean of capital implied by regression
mc=MarkovChain(ksp.TransMat.Pz)
sd=stationary_distributions(mc)[1]
logKg=kss.B[1]/(1-kss.B[2])
logKb=kss.B[3]/(1-kss.B[4])
meanK_reg=exp(sd[1]*logKg+sd[2]*logKb)
meanK_sim=mean(K_ts[1001:end])
println("mean of capital implied by regression is $meanK_reg")
println("mean of capital implied by simulation is $meanK_sim")
```

In [25]:

```
plot_Fig1(ksp,kss,K_ts)
```

Out[25]:

In [26]:

```
plot_Fig2(ksp,kss,40)
```

Out[26]:

Again, the figures are succesfully replicated.