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Generic_1_state_deterministic_3.jl
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Generic_1_state_deterministic_3.jl
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# generic solve_hjb( ) Implicit method.
# TODO: ns states/nc controls
# TODO: add uncertainty, nx shocks...
# fcn NGM() returns struct NGM;
# HJB_options() returns struct HJB_options;
# solve_HJB: (m, opt, v) -> dist, v_err, v, dV_Upwind, c_Upwind, μ_Upwind
# Plot error/value/policy/
# Simulate w/ interp.
# optimal stopping w/ LCPsolve.jl
using LinearAlgebra, SparseArrays, Plots
_m(x) = max(x,1e-10)
_m(x) = max(x,1e-8)
# NGM ###############################
struct NGM
ρ; σ; # preference param
δ; A; α; # tech param
#
r # return fcn. r(s,c)
μ # transition fcn. ṡ=μ(s,c)
dr # derivative of return fcn.
FOC # FOC: solve dr(c)=V_s. c=u'^-1(V_s)
μ_inv # inv transition c=μ_inv(s,ṡ), solve ṡ=μ(s,c) for c
#
s_min; s_max; H; s; ds;
dVf; dVb; dV_Upwind; c_Upwind;
end
function NGM(;
ρ = 0.05, σ= 2.0,
δ = 0.05, A = 1.0, α= 0.3,
r = c -> (c^(1-σ))/(1-σ), # return fcn. make it r(s,c)
μ = (s,c) -> A*(s^α) -δ*s -c, # Transition fcn. ṡ=μ(s,c)
dr = c -> c^(-σ), # derivative of return fcn.
FOC = v -> (max(v,1e-10))^(-1.0/σ), # FOC: solve dr(c)=V_s. c=u'^-1(V_s)
μ_inv = (s,ṡ) -> A*(s^α) - δ*s -ṡ, # Inv Trans c=μ_inv(s,ṡ), solve ṡ=μ(s,c) for c
s_min = 0.001*(α*A/(ρ+δ))^(1/(1-α)),
s_max = 2.000*(α*A/(ρ+δ))^(1/(1-α)),
H = 10_000,
s = collect(LinRange(s_min, s_max, H)),
ds = (s_max-s_min)/(H-1),
dVf = zeros(H,1),
dVb = zeros(H,1),
dV_Upwind = zeros(H,1),
c_Upwind = zeros(H,1),
)
return NGM(ρ,σ,δ,A,α,r,μ,dr,FOC,μ_inv,s_min,s_max,H,s,ds,dVf,dVb,dV_Upwind,c_Upwind)
end
mod = NGM()
mod = NGM(ρ = 0.05)
mod = NGM(A = 0.01)
# con-sav ###############################
struct CS
ρ; σ; # preference param
A; w; # tech param
#
r # return fcn. r(s,c)
μ # transition fcn. ṡ=μ(s,c)
dr # derivative of return fcn.
FOC # FOC: solve dr(c)=V_s. c=u'^-1(V_s)
μ_inv # inv transition c=μ_inv(s,ṡ), solve ṡ=μ(s,c) for c
#
s_min; s_max; H; s; ds;
dVf; dVb; dV_Upwind; c_Upwind;
end
function CS(;
ρ = 0.05, σ= 2.0,
A = 0.045, w = 0.1, # A = interest rate
r = c -> (_m(c)^(1-σ))/(1-σ), # return fcn. make it r(s,c)
μ = (s,c) -> w + A*s - c, # Transition fcn. ṡ=μ(s,c)
dr = c -> _m(c)^(-σ), # derivative of return fcn.
FOC = v -> (_m(v))^(-1.0/σ), # FOC: solve dr(c)=V_s. c=u'^-1(V_s)
μ_inv = (s,ṡ) -> w + A*s -ṡ, # Inv Trans c=μ_inv(s,ṡ), solve ṡ=μ(s,c) for c
s_min = -0.02,
s_max = 1.0,
H = 10_000,
s = collect(LinRange(s_min, s_max, H)),
ds = (s_max-s_min)/(H-1),
dVf = zeros(H,1),
dVb = zeros(H,1),
dV_Upwind = zeros(H,1),
c_Upwind = zeros(H,1),
)
return CS(ρ,σ,A,w,r,μ,dr,FOC,μ_inv,s_min,s_max,H,s,ds,dVf,dVb,dV_Upwind,c_Upwind)
end
# NBC
# μ_inv(a_bar,0) > 0 ⟺ w + A*a_bar >0 ⟺ a_bar > -w/A
#a_bar = -w/A # Natural Borrowing Constraint
#a_bar = -w/A + eps()
mod = CS()
mod = CS(w = 0.1, s_min = -0.1/0.045 + eps())
# verify things are well defined @ corners
mod.μ_inv(mod.s_min,0) > 0
mod.s_max, mod.μ_inv(mod.s_max,0), mod.dr(mod.μ_inv(mod.s_max,0))
mod.s_min, mod.μ_inv(mod.s_min,0), mod.dr(mod.μ_inv(mod.s_min,0))
##################################
# options
struct HJB_options; Δ; maxit; ε; dist; end
HJB_options(;Δ=1_000,maxit=10_000,ε=1e-6,dist=[]) = HJB_options(Δ,maxit,ε,dist)
HJB_options()
##################################
# solve_HJB(m, opt, v)
function solve_HJB(m, opt, v)
# unpack
# σ,δ,A,α = m.σ, m.δ, m.A, m.α # Do not need.
ρ = m.ρ
r,μ,dr,FOC,μ_inv = m.r, m.μ, m.dr, m.FOC, m.μ_inv
s_min,s_max,H,s,ds = m.s_min, m.s_max, m.H, m.s, m.ds
dVf,dVb,dV_Upwind,c_Upwind = m.dVf, m.dVb, m.dV_Upwind, m.c_Upwind
#
Δ,maxit,ε,dist = opt.Δ, opt.maxit, opt.ε, opt.dist
# VFI
for n=1:maxit
#println(n)
V=v
dV = (V[2:end] - V[1:end-1])/ds
# dVf = [dV; dr(μ_inv(s_max,0))]
# dVb = [dr(μ_inv(s_min,0)); dV]
# forward difference: if ṡ>0
dVf[1:end-1] = dV
dVf[end] = dr(μ_inv(s_max,0)) # u'(c(s_max)) state const, for stab #dVf[H]=0
cf = FOC.(dVf) # choice with forward difference
μ_f = μ.(s, cf) # ṡ with forward difference
If = μ_f .> 0
# backward difference: if ṡ<0
dVb[2:end] = dV
dVb[1] = dr(μ_inv(s_min,0)) # state const, for stab #dVb[1] = (A*s_min)^(-σ)
cb = FOC.(dVb) # choice with backward difference
μ_b = μ.(s, cb)
Ib = μ_b .< 0
# neither difference: if ṡ=0
c0 = μ_inv.(s,0) # c if ṡ=0
dV0 = dr.(c0)
# μ_0 = μ.(s, c0) # μ_0 == zero(s)
I0 = (1.0 .- If - Ib) # choice betw forward & backward difference
# I_concave = dVb .> dVf
# scatter(I_concave) #1 everywhere EXCEPT @ last point H.
dV_Upwind = dVf.*If + dVb.*Ib + dV0.*I0
c_Upwind = FOC.(dV_Upwind)
μ_Upwind = μ.(s, c_Upwind)
u = r.(c_Upwind)
# plot(dVf, lab="dVf")
# plot!(dVb, lab="dVb")
# plot!(dV0, lab="dV0")
# plot!(dV_Upwind, lab="dV_Upwind")
# plot(cf, lab="dVf")
# plot!(cb, lab="dVb")
# plot!(c0, lab="dV0")
# plot!(c_Upwind, lab="dV_Upwind")
# plot(μ_f, lab="dVf")
# plot!(μ_b, lab="dVb")
# plot!(zero(μ_f), lab="dV0")
# plot!(μ_Upwind, lab="dV_Upwind")
# create the transition matrix
X = -min.(μ_b,0)/ds
Y = -max.(μ_f,0)/ds + min.(μ_b,0)/ds
Z = max.(μ_f,0)/ds
a1 = sparse(Diagonal((Y[:])))
a2 = [zeros(1,H); sparse(Diagonal((X[2:H]))) zeros(H-1,1)]
a3 = [zeros(H-1,1) sparse(Diagonal((Z[1:H-1]))); zeros(1,H)]
AA = a1 + a2 + a3
B = (ρ + 1/Δ)*sparse(I,H,H) - AA
b = u + V./Δ
# Solve: B V = b
# => (ρ + 1/Δ)*sparse(I,H,H)*V - AA*V = u + V./Δ
# => (ρ + 1/Δ)*V = u + V./Δ + AA*V
# => (ρ)*V + V./Δ = u + V./Δ + AA*V
# => (ρ)*V = u + AA*V
V = B \ b
V_change = V-v
v = V
push!(dist,findmax(abs.(V_change))[1])
println(n, " ", dist[n])
if dist[n] .< ε
println("Value Function Converged Iteration=")
println(n)
break
end
end
μ_Upwind = μ.(s, c_Upwind)
v_err = r.(c_Upwind) + dV_Upwind.*μ_Upwind - ρ.*v # approx @ borrowing constraint
return dist, v_err, v, dV_Upwind, c_Upwind, μ_Upwind
end
##################################
# DO IT.
mod = NGM() # mod.s_min =0.0048
mod = NGM(s_max=3.0)
# NGM Closed form. Need: ρ = δ*(α*σ - 1)
mod = NGM(σ= 4.0, δ = 0.05, A = 1.0, α= 0.3, ρ = 0.05*(0.3*4 - 1))
mod = CS()
# verify TVC: ρ-(1-σ)*A >0
# mod.ρ - (1-mod.σ)*mod.A # AZ: always true if σ>1
mod.w, mod.s_min, -mod.w/mod.A
# AZ: lower s_min, the longer it takes to hit constraint. Bigger T_sim
mod = CS(s_min=-.5)
mod = CS(s_min=-1.5)
mod = CS(s_min=-2)
mod = CS(s_min=-0.1/0.045 + eps())
mod = CS(s_min=-0.1/0.045 -.25)
mod = CS(w=0.0, s_min=0.015) # binding
mod = CS(w=0.0, s_min=0.0) # NBC
mod = CS(w=0.0, s_min=-0.01) # loose
#mod = CS(w=0.0, s_min=-0.10) # loose. Inaccurate when s close to 0
# AZ: got errors before bc w=0 & s_min =-.02
#
opt = HJB_options(ε=1e-7)
v0 = @. mod.r(mod.μ_inv(mod.s,0))/mod.ρ #v0 = @. r(μ_inv(s,0))/ρ #initial guess for V
dist, v_err, v, dV_Upwind, c_Upwind, μ_Upwind = solve_HJB(mod, opt, v0)
##################################
# Plot: dist, v_err, v, c_Upwind, μ_Upwind
plot(dist,
xlabel="Iteration", ylabel="||V^{n+1} - V^n||",
legend=false, title="")
plot(mod.s, v_err,
xlabel="s", ylabel="Error in the HJB equation",
legend=false, title="")
plot(mod.s, v,
xlabel="s", ylabel="V(s)",
legend=false, title="")
plot(mod.s, c_Upwind,
xlabel="s", ylabel="c(s)",
legend=false, title="")
plot(mod.s, μ_Upwind,
xlabel="s", ylabel="ṡ(s)",
title="", label="ṡ(s)", legend=false);
plot!([0.0], seriestype = :hline, lab="", color="grey", l=:dash)
##################################
# Simulate: s, c, ṡ
# problem: c(s) only on finite grid s => interpolate
using Interpolations
ĉ = LinearInterpolation(mod.s, c_Upwind[:], extrapolation_bc = Line())
# NGM: converges quickly
Δt = 0.01;
T_sim = 200;
# CS: converges slowly
Δt = 0.01;
T_sim = 3_500;
t_sim = 0.0:Δt:T_sim
s_sim, c_sim, ṡ_sim = [zeros(length(t_sim),1) for i in 1:3]
s_0=mod.s[1] # start below ss
s_0=0.5
#s_0=0.75
s_sim[1] = s_0
for i in 2:length(t_sim)
c_sim[i-1] = ĉ(s_sim[i-1])
ṡ_sim[i-1] = mod.μ(s_sim[i-1], c_sim[i-1])
s_sim[i] = s_sim[i-1] + Δt * ṡ_sim[i-1]
# s_sim[i] = s_sim[i-1] + Δt * μ(s_sim[i-1], c_sim[i-1])
end
#
ix = 1:(length(t_sim)-1)
plot();
plot!(t_sim[ix], s_sim[ix], lab = "s");
plot!(t_sim[ix], c_sim[ix], lab = "c");
plot!(t_sim[ix], ṡ_sim[ix], lab = "ṡ");
plot!([mod.s_min], seriestype = :hline, lab="", color="grey", l=:dash)
# Verify: HACT Fig 2 Cara/F3 CRRA, hit borrowing constraint in finite time
########################################################
# Special case of NGM has closed-form sol. Compare Policy & Sim
# Policy: c = (1-s)(k^α)
cp(s) = (1-((α*δ)/(ρ+δ)))*s^α
s_d(s) = (s^α) - δ*s - cp(s)
plot(legend=:topleft)
plot!(s, cp, lab="c: closed form")
plot!(s, c_Upwind,lab="c: Upwind")
plot!(s, s_d, lab="ṡ: closed form")
plot!(s, s_dot, lab="ṡ: Upwind")
#
s_ss = (α*A/(ρ+δ))^(1/(1-α))
z_ss = (s_ss)^(1-α)
c_ss = (s_ss)^(α) - δ*s_ss
#s_0 = 5;
z_0 = (s_0)^(1-α)
λ=(1-α)*δ
z(t) = z_ss + exp(-λ*t)*(z_0 - z_ss)
kk(t) = z(t)^(1/(1-α))
cc(t) = (1-((α*δ)/(ρ+δ)))*kk(t)^α
#t_sim = 0.01:0.01:100
t_sim = time[ix]
plot(legend=:topleft)
#plot!(t_sim, z, lab="z")
plot!(t_sim, kk, lab="s: closed form")
plot!(t_sim, s_sim[ix], lab="s: Upwind")
plot!(t_sim, cc, lab="c: closed form")
plot!(t_sim, c_sim[ix], lab="c: Upwind")
#plot!([z_ss s_ss c_ss], seriestype = :hline, lab="", color="grey")
plot!([s_ss c_ss], seriestype = :hline, lab="", color="grey")