Chapter 3: Markov Dynamics

inventory_sim.jl

using Distributions, IterTools, QuantEcon

function create_inventory_model(; S=100,  # Order size
                                  s=10,   # Order threshold
                                  p=0.4)  # Demand parameter
    ϕ = Geometric(p)
    h(x, d) = max(x - d, 0) + S*(x <= s)
    return (; S, s, p, ϕ, h)
end

"Simulate the inventory process."
function sim_inventories(model; ts_length=200)
    (; S, s, p, ϕ, h) = model
    X = Vector{Int32}(undef, ts_length)
    X[1] = S  # Initial condition
    for t in 1:(ts_length-1)
        X[t+1] = h(X[t], rand(ϕ))
    end
    return X
end

"Compute the transition probabilities and state."
function compute_mc(model; d_max=100)
    (; S, s, p, ϕ, h) = model
    n = S + s + 1  # Size of state space
    state_vals = collect(0:(S + s))
    P = Matrix{Float64}(undef, n, n)
    for (i, j) in product(1:n, 1:n)
        P[i, j] = sum((h(i-1, d) == j-1)*pdf(ϕ, d) for d in 0:d_max)
    end
    return MarkovChain(P, state_vals)
end

"Compute the stationary distribution of the model."
function compute_stationary_dist(model)
    mc = compute_mc(model)
    return mc.state_values, stationary_distributions(mc)[1]
end


# Plots

using PyPlot
using LaTeXStrings
PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering


function plot_ts(model; fontsize=16, 
                   figname="./figures/inventory_sim_1.pdf",
                   savefig=false)
    (; S, s, p, ϕ, h) = model
    X = sim_inventories(model)
    fig, ax = plt.subplots(figsize=(9, 5.2))
    ax.plot(X, label=L"X_t", lw=3, alpha=0.6)
    ax.set_xlabel(L"t", fontsize=fontsize)
    ax.set_ylabel("inventory", fontsize=fontsize)
    ax.legend(fontsize=fontsize, frameon=false)
    ax.set_ylim(0, S + s + 20)

    if savefig == true
        fig.savefig(figname)
    end
end


function plot_hist(model; fontsize=16, 
                   figname="./figures/inventory_sim_2.pdf",
                   savefig=false)
    (; S, s, p, ϕ, h) = model
    state_values, ψ_star = compute_stationary_dist(model) 
    X = sim_inventories(model; ts_length=1_000_000)
    histogram = [mean(X .== i) for i in state_values]

    fig, ax = plt.subplots(figsize=(9, 5.2))
    ax.plot(state_values, ψ_star, "k-",  lw=3, alpha=0.7, label=L"\psi^*")
    ax.bar(state_values, histogram, alpha=0.7, label="frequency")
    ax.set_xlabel("state", fontsize=fontsize)

    ax.legend(fontsize=fontsize, frameon=false)
    ax.set_ylim(0, 0.015)

    if savefig == true
        fig.savefig(figname)
    end
end

plot_hist (generic function with 1 method)

model = create_inventory_model()

(S = 100, s = 10, p = 0.4, ϕ = Geometric{Float64}(p=0.4), h = var"#h#4"{Int64, Int64}(100, 10))

plot_ts(model; savefig=true)

plot_hist(model; savefig=true)

is_irreducible.jl

using QuantEcon
P = [0.1 0.9;
     0.0 1.0]
mc = MarkovChain(P)
print(is_irreducible(mc))

false

laborer_sim.jl

function create_laborer_model(; α=0.3, β=0.2)
    return (; α, β)
end

function laborer_update(x, model)  # update X from t to t+1
    (; α, β) = model
    if x == 1    
        x′ = rand() < α ? 2 : 1
    else 
        x′ = rand() < β ? 1 : 2
    end
    return x′
end

function sim_chain(k, p, model)
    X = Array{Int32}(undef, k)
    X[1] = rand() < p ? 1 : 2
    for t in 1:(k-1)
        X[t+1] = laborer_update(X[t], model)
    end
    return X
end

function test_convergence(; k=10_000_000, p=0.5)
    model = create_laborer_model()
    (; α, β) = model
    ψ_star = (1/(α + β)) * [β α]

    X = sim_chain(k, p, model)
    ψ_e = (1/k) * [sum(X .== 1)  sum(X .== 2)]
    error = maximum(abs.(ψ_star - ψ_e))
    approx_equal = isapprox(ψ_star, ψ_e, rtol=0.01)
    println("Sup norm deviation is $error")
    println("Approximate equality is $approx_equal")
 end

test_convergence (generic function with 1 method)

markov_js.jl

"""
Infinite-horizon job search with Markov wage draws.

"""

using QuantEcon, LinearAlgebra
include("s_approx.jl")

"Creates an instance of the job search model with Markov wages."
function create_markov_js_model(;
        n=200,       # wage grid size
        ρ=0.9,       # wage persistence
        ν=0.2,       # wage volatility
        β=0.98,      # discount factor
        c=1.0        # unemployment compensation
    )
    mc = tauchen(n, ρ, ν)
    w_vals, P = exp.(mc.state_values), mc.p
    return (; n, w_vals, P, β, c)
end

" The Bellman operator Tv = max{e, c + β P v} with e(w) = w / (1-β)."
function T(v, model)
    (; n, w_vals, P, β, c) = model
    h = c .+ β * P * v
    e = w_vals ./ (1 - β)
    return max.(e, h)
end

" Get a v-greedy policy."
function get_greedy(v, model)
    (; n, w_vals, P, β, c) = model
    σ = w_vals / (1 - β) .>= c .+ β * P * v
    return σ
end

"Solve the infinite-horizon Markov job search model by VFI."
function vfi(model) 
    v_init = zero(model.w_vals)  
    v_star = successive_approx(v -> T(v, model), v_init)
    σ_star = get_greedy(v_star, model)
    return v_star, σ_star
end

        

# == Policy iteration == #

"Get the value of policy σ."
function get_value(σ, model)
    (; n, w_vals, P, β, c) = model
    e = w_vals ./ (1 - β)
    K_σ = β .* (1 .- σ) .* P
    r_σ = σ .* e .+ (1 .- σ) .* c
    return (I - K_σ) \ r_σ
end


    
"Howard policy iteration routine."
function policy_iteration(model)
    σ = Vector{Bool}(undef, model.n)
    i, error = 0, 1.0
    while error > 0
        v_σ = get_value(σ, model)
        σ_new = get_greedy(v_σ, model)
        error = maximum(abs.(σ_new - σ))
        σ = σ_new
        i = i + 1
        println("Concluded loop $i with error $error.")
    end
    return σ
end


# == Plots == #

using PyPlot
using LaTeXStrings
PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering
fontsize=16

default_model = create_markov_js_model()


function plot_main(; model=default_model,
                     method="vfi", 
                     savefig=false, 
                     figname="./figures/markov_js_1.pdf")
    (; n, w_vals, P, β, c) = model


    if method == "vfi"
        v_star, σ_star = vfi(model)
    else
        σ_star = policy_iteration(model)
        v_star = get_value(σ_star, model)
    end

    h_star = c .+ β * P * v_star
    e = w_vals / (1 - β)

    fig, ax = plt.subplots(figsize=(9, 5.2))
    ax.plot(w_vals, h_star, lw=4, ls="--", alpha=0.4, label=L"h^*(w)")
    ax.plot(w_vals, e, lw=4, ls="--", alpha=0.4, label=L"w/(1-\beta)")
    ax.plot(w_vals, max.(e, h_star), "k-", alpha=0.7, label=L"v^*(w)")
    ax.legend(frameon=false, fontsize=fontsize)
    ax.set_xlabel(L"w", fontsize=fontsize)
    if savefig
        fig.savefig(figname)
    end
end

plot_main (generic function with 1 method)

plot_main(savefig=true)

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Terminated successfully in 365 iterations.

markov_js_with_sep.jl

"""
Infinite-horizon job search with Markov wage draws and separation.

"""

include("s_approx.jl")
using QuantEcon, LinearAlgebra

"Creates an instance of the job search model with separation."
function create_js_with_sep_model(;
        n=200,          # wage grid size
        ρ=0.9, ν=0.2,   # wage persistence and volatility
        β=0.98, α=0.1,  # discount factor and separation rate
        c=1.0)          # unemployment compensation
    mc = tauchen(n, ρ, ν)
    w_vals, P = exp.(mc.state_values), mc.p
    return (; n, w_vals, P, β, c, α)
end

" The Bellman operator for the value of being unemployed."
function T(v, model)
    (; n, w_vals, P, β, c, α) = model
    d = 1 / (1 - β * (1 - α))
    accept = d * (w_vals + α * β * P * v)
    reject = c .+ β * P * v
    return max.(accept, reject)
end

" Get a v-greedy policy."
function get_greedy(v, model)
    (; n, w_vals, P, β, c, α) = model
    d = 1 / (1 - β * (1 - α))
    accept = d * (w_vals + α * β * P * v)
    reject = c .+ β * P * v
    σ = accept .>= reject
    return σ
end

"Solve by VFI."
function vfi(model) 
    v_init = zero(model.w_vals)  
    v_star = successive_approx(v -> T(v, model), v_init)
    σ_star = get_greedy(v_star, model)
    return v_star, σ_star
end


# == Plots == #

using PyPlot
using LaTeXStrings
PyPlot.matplotlib[:rc]("text", usetex=true) # allow tex rendering
fontsize=16

default_model = create_js_with_sep_model()


function plot_main(; model=default_model,
                     method="vfi", 
                     savefig=false, 
                     figname="./figures/markov_js_with_sep_1.pdf")
    (; n, w_vals, P, β, c, α) = model
    v_star, σ_star = vfi(model)

    d = 1 / (1 - β * (1 - α))
    accept = d * (w_vals + α * β * P * v_star)
    h_star = c .+ β * P * v_star

    w_star = Inf
    for (i, w) in enumerate(w_vals)
        if accept[i] ≥ h_star[i]
            w_star = w
            break
        end
    end
    @assert w_star < Inf "Agent never accepts"

    fig, ax = plt.subplots(figsize=(9, 5.2))
    ax.plot(w_vals, h_star, lw=4, ls="--", alpha=0.4, label="continuation value")
    ax.plot(w_vals, accept, lw=4, ls="--", alpha=0.4, label="stopping value")
    ax.plot(w_vals, v_star, "k-", alpha=0.7, label=L"v_u^*(w)")
    ax.legend(frameon=false, fontsize=fontsize)
    ax.set_xlabel(L"w", fontsize=fontsize)
    if savefig
        fig.savefig(figname)
    end
end

function plot_w_stars(; α_vals=LinRange(0.0, 1.0, 10),
                        savefig=false, 
                        figname="./figures/markov_js_with_sep_2.pdf")

    w_star_vec = similar(α_vals)
    for (i_α, α) in enumerate(α_vals)
        print(i_α, α)
        model = create_js_with_sep_model(α=α)
        (; n, w_vals, P, β, c, α) = model
        v_star, σ_star = vfi(model)

        d = 1 / (1 - β * (1 - α))
        accept = d * (w_vals + α * β * P * v_star)
        h_star = c .+ β * P * v_star

        w_star = Inf
        for (i_w, w) in enumerate(w_vals)
            if accept[i_w] ≥ h_star[i_w]
                w_star = w
                break
            end
        end
        @assert w_star < Inf "Agent never accepts"
        w_star_vec[i_α] = w_star
    end

    fig, ax = plt.subplots(figsize=(9, 5.2))
    ax.plot(α_vals, w_star_vec, lw=2, alpha=0.6, label="reservation wage")
    ax.legend(frameon=false, fontsize=fontsize)
    ax.set_xlabel(L"\alpha", fontsize=fontsize)
    ax.set_xlabel(L"w", fontsize=fontsize)
    if savefig
        fig.savefig(figname)
    end
end

plot_w_stars (generic function with 1 method)

plot_main(savefig=true)

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plot_w_stars(savefig=true)

10.0Completed iteration 25 with error 0.7358325123867289.
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Completed iteration 625 with error 2.1725140655348696e-6.
Completed iteration 650 with error 1.27298568486367e-6.
Terminated successfully in 663 iterations.
101.0Completed iteration 25 with error 0.8195566251536235.
Completed iteration 50 with error 0.46642498056927906.
Completed iteration 75 with error 0.28037678187634185.
Completed iteration 100 with error 0.16915170563319748.
Completed iteration 125 with error 0.10207516364071978.
Completed iteration 150 with error 0.061598680129037575.
Completed iteration 175 with error 0.03717262745711025.
Completed iteration 200 with error 0.02243236944056548.
Completed iteration 225 with error 0.013537143756735759.
Completed iteration 250 with error 0.008169188798916593.
Completed iteration 275 with error 0.004929817311051465.
Completed iteration 300 with error 0.002974970871491678.
Completed iteration 325 with error 0.0017952899930833155.
Completed iteration 350 with error 0.0010833941905588063.
Completed iteration 375 with error 0.0006537901824827941.
Completed iteration 400 with error 0.0003945393158062416.
Completed iteration 425 with error 0.00023809056163770492.
Completed iteration 450 with error 0.0001436792564533107.
Completed iteration 475 with error 8.670536367105797e-5.
Completed iteration 500 with error 5.2323628878525597e-5.
Completed iteration 525 with error 3.157546458254501e-5.
Completed iteration 550 with error 1.9054679214036696e-5.
Completed iteration 575 with error 1.1498826864908551e-5.
Completed iteration 600 with error 6.9391364547755074e-6.
Completed iteration 625 with error 4.187524119458885e-6.
Completed iteration 650 with error 2.5270231276408595e-6.
Completed iteration 675 with error 1.5249693490204663e-6.
Terminated successfully in 697 iterations.