Question

your given the following national income model Y=C+I+G ,C=12+0.8Y ,I=100+0.1Y ,G=300

present the model to matrix formula.

Answer 1

To transform to a matrix, the following guideline has to be adhered.

$Y=C+I+G\to$$Y-C=I+G$

$C=a+bY\to$ $C-bY=a$

$T=tY\to$ $T-tY$

$Y=C+I+G\to Y=12+0.8Y+100+0.1Y+300$

$Y-0.9Y=412\to Y=4120\therefore I=112+0.9(4120)$ $=512$

We then replace the values from the given equations where relevant, and placing a coefficient of zero to the missing constants in the guide equation.

$1\times Y-1\times C+0\times T=I+G$

$-0.8\times Y+1\times C + 0\times T=12$

$-t\times Y + 0\times C + 1\times T=0$

The matrix obtained from the variable coefficients become;

$A=$ $\begin{bmatrix} 1 & -1 & 0 \\ -0.8 & 1 & 0 \\ -t & -0 & 1 \\ \end{bmatrix}$ $X=$ $\begin{bmatrix} Y \\ C \\ T \\ \end{bmatrix}$ $Z=$ $\begin{bmatrix} I+G \\ 12 \\ 0 \\ \end{bmatrix}$ but I and G values are known.

$I+G=512+300=812$

so $Z=\begin{bmatrix} 812 \\ 12 \\ 0 \\ \end{bmatrix}$