File:Phase diagram of the Ramsey model.svg
Original file (SVG file, nominally 1,313 × 1,875 pixels, file size: 2.14 MB)
Captions
Summary
[edit]DescriptionPhase diagram of the Ramsey model.svg |
English: Model based on Advanced Macroeconomics: An Easy Guide (Campante et al 2021), chapter 3.
Notation: k = capital (stock) per capita c = consumption (flow) per capita n = population growth rate δ = capital depreciation rate ρ = utility discounting rate We have the dynamics in $(k, c)$ space: $$\begin{cases} \dot k = f(k) - c - (n+\delta)k\\ \dot c= (f'(k) - (\delta + \rho))\sigma c \end{cases}$$ with initial conditions $$k(0) = k(0), \quad c(0) = u'^{-1}(\lambda(0))$$ and transversality condition $$\lim_t (ku'(c) e^{(n-\rho) t} )= 0$$ Here, $\sigma$ is the "elasticity of intertemporal substitution in consumption". $$\sigma(c) := - \frac {u'(c)}{u(c)c} > 0$$ In particular, if we let $u(c) = \ln c$, then $\sigma(c) = 1$, a constant. Now plot the diagram with Julia: ```julia using CairoMakie, LinearAlgebra
n, δ, ρ = 1, 1, 0.5
σ = 1
f(k) = √k df(k) = 1/(2√k) invdf(y) = 1/(4 * y^2)
dotkc(k, c) = [f(k)-c-(n + δ)k (df(k)-(δ + ρ)) * σ * c]
dotc0(c) = invdf(δ + ρ)
dotk0(k) = f(k) - (n + δ) * k using Roots invdotk0 = find_zero(dotk0, (0.1, 0.3), Roots.A42()); function plotRamsey(x, y) xs = repeat(x, outer=length(y)) ys = repeat(y, inner=length(x)) xlims!(minimum(x), maximum(x)) ylims!(minimum(y), maximum(y)) # the ̇c=0 curve lines!(dotc0.(y), y, color=:black) # the ̇k=0 curve lines!(x, dotk0.(x), color=:blue) # the equilibrium point scatter!([dotc0(0)],[dotk0(dotc0(0))], color=:black) scatter!([invdotk0],[0], color=:red) # the flow vector field vectors = dotkc.(xs, ys) norms = vec(norm.(vectors)) scaled_norms = tanh.(norms)/10 us = map((x) -> x[1], vectors) vs = map((x) -> x[2], vectors) us ./= norms vs ./= norms us .*= scaled_norms vs .*= scaled_norms arrows!(xs, ys, us, vs, arrowsize = 6, linecolor=scaled_norms, arrowcolor = :black) end
width, height = 1400, 2000 scene = Figure(resolution = (width, height)) fsize = 30 Axis(scene[1:2,1:2], backgroundcolor = "white", xlabel = L"$k$", xlabelsize=fsize, ylabel = L"$c$", ylabelsize=fsize, title = L"Ramsey model with $f(k) = \sqrt{k}$, and $n, δ, ρ, σ = %$(n, δ, ρ, σ)$", titlesize=fsize) xmin, xmax, xres = 0.0001, 0.3, 51 ymin, ymax, yres = -0.02, 0.15, 51 x = range(start = xmin, stop = xmax, length = xres) y = range(start = ymin, stop = ymax, length = yres) plotRamsey(x, y)
Axis(scene[3,1], backgroundcolor = "white", xlabel = L"$k$", xlabelsize=fsize, ylabel = L"$c$", ylabelsize=fsize, title = L"interior equilibrium at $k, c = %$(round(dotc0(0), sigdigits=2)), %$(round(dotk0(dotc0(0)), sigdigits=2))$", titlesize=fsize) ϵ = 0.01 xmin, xmax, xres = dotc0(0)-ϵ, dotc0(0)+ϵ, 31 ymin, ymax, yres = dotk0(dotc0(0))-ϵ, dotk0(dotc0(0))+ϵ, 31 scatter!([dotc0(0)],[dotk0(dotc0(0))], color=:black) x = range(start = xmin, stop = xmax, length = xres) y = range(start = ymin, stop = ymax, length = yres) plotRamsey(x, y)
Axis(scene[3,2], backgroundcolor = "white", xlabel = L"$k$", xlabelsize=fsize, ylabel = L"$c$", ylabelsize=fsize, title = L"all-saving equilibrium at $k = %$(round(invdotk0, sigdigits=2))$", titlesize=fsize) ϵ = 0.01 xmin, xmax, xres = invdotk0-ϵ, invdotk0+ϵ, 31 ymin, ymax, yres = -ϵ, +ϵ, 31 x = range(start = xmin, stop = xmax, length = xres) y = range(start = ymin, stop = ymax, length = yres) plotRamsey(x, y)
scene
|
Date | |
Source | Own work |
Author | Cosmia Nebula |
Licensing
[edit]- You are free:
- to share – to copy, distribute and transmit the work
- to remix – to adapt the work
- Under the following conditions:
- attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
- share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.
File history
Click on a date/time to view the file as it appeared at that time.
Date/Time | Thumbnail | Dimensions | User | Comment | |
---|---|---|---|---|---|
current | 15:12, 8 January 2023 | 1,313 × 1,875 (2.14 MB) | Cosmia Nebula (talk | contribs) | fix caption | |
00:51, 8 January 2023 | 1,313 × 1,875 (2.15 MB) | Cosmia Nebula (talk | contribs) | fixed parameters so that rho > n. | ||
00:18, 8 January 2023 | 1,313 × 1,875 (2.15 MB) | Cosmia Nebula (talk | contribs) | fix caption | ||
23:14, 7 January 2023 | 1,313 × 1,875 (2.15 MB) | Cosmia Nebula (talk | contribs) | Uploaded own work with UploadWizard |
You cannot overwrite this file.
File usage on Commons
There are no pages that use this file.
File usage on other wikis
The following other wikis use this file:
- Usage on en.wikipedia.org
Metadata
This file contains additional information such as Exif metadata which may have been added by the digital camera, scanner, or software program used to create or digitize it. If the file has been modified from its original state, some details such as the timestamp may not fully reflect those of the original file. The timestamp is only as accurate as the clock in the camera, and it may be completely wrong.
Width | 1050pt |
---|---|
Height | 1500pt |