How to plot Ramsey Phase Plane in Matlab

Hey,

I am currently working on a script meant to plot the trajectory, saddle paths and the phase plane of the basic RKC model. However I continue to get the following error: Error using streamslice. U,V must all be size 2x2 or greater. Error in Code_Diss (line 96). h = streamslice(K, C, dK, dC, 2, 'noarrows', 'cubic');

I havent worked much in Matlab yet so forgive me if this is a very obvious error.

Thanks!

clear; % clear Workspace

clc; % clear Command Window

close all; % close Figures

disp('----------------------------------');

disp(' Ramsey-Cass-Koopmans model: ');

disp('----------------------------------');

%% [0] define variables

global alpha delta rho n g theta kss css k0

syms t L(t) K(t) C(t) a(t) r(t) w(t) u(t) rho n theta

L(t) = exp(n.*t); % total population at t

n = diff(L(t),t)/L(t); % labor growth rate

c(t) = C(t)/L(t); % per capita consumption

k(t) = K(t)/L(t); % per capita capital

%% [1] define parameters

rho = 0.04; % discount rate

theta = 1.5; % IES

nu = 0.35;

sigma = 1;

delta = 0.05; % depreciation rate

n = 0.01; % population growth rate

alpha = 0.5 ; % capital share

c_0 = 3 ; % Initial per capital consumption

k_0 = 5 ; % Initial per capita capital (wealth)

initial = [3, 5]; % c(0) and k(0)

kmin=0; % smallest capital stock to consider

kmax=100; % largest capital stock to consider

u = ((C(t)/L(t))^(1-theta))/(1-theta) ; % utility at time t

simplify(int(u*exp(-(rho-n)*t),0,inf)); % lifetime utility

%% [2] compute steady state for c and k

syms k l c ;

f = k.^ (alpha) * l.^(1 - alpha) ;

y = f

df_dk = alpha * k.^(alpha - 1) * l.^(1 - alpha) ;

df_dl = (1 - alpha) * k.^(alpha) * l^(- alpha) ;

% Write down the equations for dk/dt and dc/dt.

k_dot = y - c - (delta - n) * k == 0 ;

c_dot = c * (((df_dk - delta - rho)/ theta )+ n) == 0 ;

% Steady state value of k obtained when dc/dt = 0.

kss = df_dk - delta - rho - ( n * theta ) ;

% Compute the steady-state curve c = c*(k) obtained when dk/dt = 0.

css = y - (delta + n) * k ;

%% Define sample values for the x-axis (per-capita wealth).

k_root = (n + delta) ^ (1 / (alpha - 1));

nPoints = 50;

k_init = 0;

k_fine = linspace(k_init, k_root, nPoints);

%% Define sample values for the y-axis (per-capita consumption).

c_fine = linspace(k_init, c_0, nPoints);

%% Create a lattice of (k, c) points for the stream slices.

[K, C] = meshgrid(k_fine, c_fine);

dK = k_dot

dC = c_dot

kss = 4

css = 3

disp('steady-state:')

fprintf('k* = %14.6f \n',kss);

fprintf('c* = %14.6f \n',css);

%plot_ramsey

%% Visualize the phase portrait.

figure('Units', 'Normalized', 'Position', 0.25*[1, 1, 2, 2])

h = streamslice(K, C, dK, dC, 2, 'noarrows', 'cubic');

set(h, 'LineStyle', ':', 'LineWidth', 1.5)

% Hide all but one streamslice from the legend.

set(h(2:end), 'HandleVisibility', 'off')

% Plot the steady-state vertical line and smooth curve.

hold on

plot(k_fine, c_star, 'r', 'LineWidth', 2)

plot([k_steady, k_steady], ylim(), 'm', 'LineWidth', 2)

xlabel('{\it{k}} (capital per capita)')

ylabel('{\it{c}} (consumption per capita)')

title('Ramsey-Cass-Koopmans Capital-Consumption Phase Plane')

grid on

% Increase the font size.

ax = gca;

ax.FontSize = 15;

% Add the legend.

legend({'Stream slices', '$dk/dt = 0$', '$dc/dt = 0$'}, ...

'Interpreter', 'LaTeX', ...

'FontSize', 15)

pause

close

Answer 1

[SAI SRUJAN]

Hi Sabrina Haumann,

I understand that you are attempting to plot the trajectory, saddle paths, and phase plane, for the fundamental RKC model. However, you are encountering an issue when utilizing the streamslice function in MATLAB.

In the line 96,

%streamslice(X,Y,U,V);
h=streamslice(K,C,dK,dC,2,"noarrows","cubic");

The aforementioned code line encounters an error due to the requirement of "dK" and "dC" in streamslice to have a size of 2x2 or greater. The acceptable components, "dK" and "dC", should be either a 2D array or a 3D array.

However, in your code, "dK" and "dC" have a size of 1x1 syms. Consequently, the function "streamslice" fails to execute successfully.

You may find valuable insights in the provided documentation, which includes examples demonstrating the proper usage of streamslice.

Plot streamlines in slice planes - MATLAB streamslice - MathWorks India