System of Difference Equation: Cobweb model in interelated markets

Hi everybody. I am having problems in solving a system of two difference equation (order=1). I do not find the right procedure and i am not able to write the script. The system is this:

x(t)= -0.4x(t-1)

y(t)= 0.4x(t-1) - 0.5y(t-1)

The aim is to graphically show if the solution X(T)=0 is in equlibrium. Thus, the procedure should calculate eigenvector and eigenvalues and allow me to define C1 and C2 in the general solution. Thank you! I attached the exercise file below.

Answer 1

[Vinayak]

Hi Massimiliano,

I understand you want to calculate eigen values and eigen vectors for the given set of equations. I have created a sample code to solve the equations you provided, you can adjust the values of C1 and C2 before calculating the initial conditions using the eigen vectors.

% Define the system matrix A 
A = [-0.4, 0; 
    0.4, -0.5]; 

% Calculate eigenvalues and eigenvectors 
[eigenvectors, eigenvalues] = eig(A); 

% Set the number of iterations (time steps) 
num_iterations = 50; 

% Initialize arrays to store the solutions 
x_vals = zeros(1, num_iterations); 
y_vals = zeros(1, num_iterations); 

% Adjust C1 C2 and initial conditions using eigenvectors 
C1 = 1;  
C2 = 1;  
initial_conditions = C1 * eigenvectors(:,1) + C2 * eigenvectors(:,2); 

% Set initial values 
x_vals(1) = initial_conditions(1); 
y_vals(1) = initial_conditions(2); 

% Iterate and compute the solution 
for t = 2:num_iterations 
    Z = A * [x_vals(t-1); y_vals(t-1)]; 
    x_vals(t) = Z(1); 
    y_vals(t) = Z(2); 
end 

% Plot the solutions 
figure; 
plot(0:num_iterations-1, x_vals); 
figure; 
plot(0:num_iterations-1, y_vals); 

% Check if the system is in equilibrium 
if all(abs(diag(eigenvalues)) < 1) 
    disp('The system is stable and the equilibrium X(T) = 0 is reached.'); 
else 
    disp('The system is unstable and the equilibrium X(T) = 0 is not reached.'); 
end 

You may refer to the following documentation link for more information about “eig” function:

• https://www.mathworks.com/help/matlab/ref/eig.html

I hope this helps!