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example.jl
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### A Pluto.jl notebook ###
# v0.19.42
using Markdown
using InteractiveUtils
# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
quote
local iv = try Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value catch; b -> missing; end
local el = $(esc(element))
global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
el
end
end
# ╔═╡ 09f76afa-5380-4aea-97cb-3a7c8bd8c860
begin
push!(LOAD_PATH, pwd())
using RamseyGrowthModel
end
# ╔═╡ be3a9c76-a806-419f-b688-d060cfed11f4
using PlutoUI, Plots, DataFrames
# ╔═╡ 0fba4001-7e4b-43de-ae53-4b31f9a2e7e2
md"""
# Ramsey Growth Model - examples
"""
# ╔═╡ 0bdca6cf-82ad-4457-9d3c-d66f2d5ad5c6
md"""
In this notebook, we gathered some interesting capital and consumption distributions, along with some interactive examples.
"""
# ╔═╡ 98a54db0-713a-4b0b-8708-3b836bac715b
md"""
For demonstration purposes, two plotting functions have been declared.
"""
# ╔═╡ 500630d6-97ff-4266-ab70-6f3889097314
function plot_capital(allocation::DataFrame, kss::Union{Real,Nothing}=nothing)
capital = plot(allocation.t, allocation.K, xlabel="t", ylabel="Kₜ", label="capital")
if !isnothing(kss)
hline!(capital[1], [kss], label="steady state capital", linestyle=:dash)
end
capital
end
# ╔═╡ 9e450a04-bd36-46a6-95d9-c35252093bb8
function plot_consumption(allocation::DataFrame)
plot(allocation.t, allocation.C, xlabel="t", ylabel="Cₜ", label="consumption")
end
# ╔═╡ bd66627e-2565-4a79-99be-c1d01f48c79c
md"""
-------------------
Before running the simulation with different parameters, an instance of the model needs to be set up.
"""
# ╔═╡ 9acaaaf8-fc92-487b-9c37-40e640b9a6c4
my_model = GrowthModel(0.95, 0.02, 2.0, 0.3, 1.0)
# ╔═╡ b83288f7-0817-4282-aa48-d6c12872dc0c
md"""
#### Example 1: finite ``T``
---
This is the most realistic and commonly analized simulation of RCK model. In this variant, tendencies of the capital and consumption, namely ``$K\rightarrow0$`` and ``$C\rightarrow\infty$`` can be observed very well.
"""
# ╔═╡ 7354f6ed-fb29-4247-90c7-7718a3a395ff
best_allocation = solve(my_model, 20, 0.2)
# ╔═╡ e527fc57-60e0-44dc-9943-bc9095a2166f
plot_capital(best_allocation)
# ╔═╡ 8739cc91-440a-4150-98ac-4ab62614528b
plot_consumption(best_allocation)
# ╔═╡ a44866e1-cccc-42c4-a6ab-a09ed770521c
md"""
#### Example 2: steady state capital as ``K_0``
---
With the increase of time ``T``, a special property of the capital function can be observed. As the time increases, the planner pushes the capital to a fixed value called *steady state capital*. When this value is set as the initial capital, values of capital function are constant and decrease at the final stages of the simulation.
"""
# ╔═╡ 69167b33-034d-480b-9a06-193c11c61c49
kss = steady_state_K(my_model)
# ╔═╡ 77398e1c-55a3-4ae7-868a-cdb7cdf2a8f2
kss_best_allocation = solve(my_model, 100, kss)
# ╔═╡ 47943756-3c8d-4241-ad2a-e02e55965685
plot_capital(kss_best_allocation, kss)
# ╔═╡ a043b75e-3a41-4614-85b5-a752f994cf25
plot_consumption(kss_best_allocation)
# ╔═╡ b12971ba-373b-4abd-b2db-63cb1bd2a407
md"""
#### Example 3: ``T=\infty``
---
For this case, it is not possible to depict the functions of capital and consumption accurately. However, it is possible to predict the behaviour of these functions for a certain finite time period, which is also illustrated in the graphs below.
"""
# ╔═╡ 97b50f22-b883-47fe-a4f1-7c3d8d435d82
infinite_best_allocation = solve(my_model, Inf, 3.1)
# ╔═╡ e67e0207-8d63-4137-9d38-9f60a10091a5
plot_capital(infinite_best_allocation, kss)
# ╔═╡ a2d0128d-79fc-499b-a5fb-1cb204976b8c
plot_consumption(infinite_best_allocation)
# ╔═╡ 0c65bcd5-1636-4aff-a975-156e53e9def4
md"""
#### Example 4: custom example
---
Here we provide a playground, where users can select their own model and simulation parameters and visualize the results.
"""
# ╔═╡ 8eb1e7d2-12ff-4d8d-bee8-71d4c6da11cf
md"""
Model parameters consist of:
- discount factor ``\beta=`` $(@bind β Scrubbable(0.1:0.1:0.9, default=0.9))
- depreciation rate of capital ``\delta=`` $(@bind δ Scrubbable(0.1:0.1:0.9, default=0.1))
- coefficient of relative risk aversion ``\gamma=`` $(@bind γ Scrubbable(0.1:0.1:10, default=2.0))
- return to capital per capita ``\alpha=`` $(@bind α Scrubbable(0.1:0.1:0.9, default=0.3))
- technology ``A=`` $(@bind A Scrubbable(0.1:0.1:10, default=1.0))
"""
# ╔═╡ b2745ea1-a9fc-4671-89b5-756c185321de
custom_model = GrowthModel(β, δ, γ, α, A)
# ╔═╡ 9d0e6b0a-a54a-4af1-9950-8307ef1e25c4
md"""
----
"""
# ╔═╡ 283b6890-89ce-4c57-ac4f-bd5e17ea1329
md"""
``T``:
$(@bind t html"<input type='range' style='width: 80%'/>")
"""
# ╔═╡ e0ca1162-7e30-4d61-b7c1-e8b31dd46cd3
md"""
``T=`` $(t)
"""
# ╔═╡ 9b633282-cbed-44b6-8d6a-70e0e740f1df
md"""
----
"""
# ╔═╡ ac4f0393-b3ba-4b86-8415-12aa2e756995
md"""
``K_0``:
$(@bind k_0 html"<input type='range' style='width: 80%'/>")
"""
# ╔═╡ 83ce4f65-6aae-4484-b11e-73194d5e4271
md"""
``K_0=`` $(k_0/10)
"""
# ╔═╡ 460d0d45-c225-476c-bad5-c8cca105c06e
custom_allocation = solve(custom_model, t, k_0/10)
# ╔═╡ 8214f1e7-f8ad-44ef-9be7-ec5b930e399d
custom_kss = steady_state_K(custom_model)
# ╔═╡ f4e715da-809f-42e5-a948-f8d4de5e297e
plot_capital(custom_allocation, custom_kss)
# ╔═╡ cf74f673-230a-4135-8c67-48d6350346c7
plot_consumption(custom_allocation)
# ╔═╡ 00000000-0000-0000-0000-000000000001
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