Answer to Question #117162 in Linear Algebra for PURBAJYOTI DAS

Question

Solve the following national income models by Cramer’s rule Y=C+I+1000,C=10+0.7(Y-T),I=100+0.2Y,T=0.3Y Where Y , C , I, M and T are national income, consumption, investment imports and taxes.

Answer 1

$$\begin{alignedat}{2} C+ I-Y=-1000 \\ C+0.7T-0.7Y =10 \\ I-0.2Y =100 \\ T-0.3Y =0 \end{alignedat}$$

$$\varDelta=\begin{vmatrix} 1 & 1 & 0 & -1 \\ 1 & 0 & 0.7 & -0.7 \\ 0 & 1 & 0 & -0.2 \\ 0 & 0 & 1 & -0.3 \end{vmatrix}=$$ $$=\begin{vmatrix} 0 & 0.7 & -0.7 \\ 1 & 0 & -0.2 \\ 0 & 1 & -0.3 \end{vmatrix}-\begin{vmatrix} 1 & 0 & -1 \\ 1 & 0 & -0.2 \\ 0 & 1 & -0.3 \end{vmatrix}=$$ $$= -\begin{vmatrix} 0.7 & -0.7 \\ 1 & -0.3 \end{vmatrix}-\begin{vmatrix} 0 & -0.2 \\ 1 & -0.3 \end{vmatrix}+\begin{vmatrix} 0 & -1 \\ 1 & -0.3 \end{vmatrix}=$$ $$=0.21-0.7-0.2+1=0.31\not=0$$

$$\varDelta_C=\begin{vmatrix} -1000 & 1 & 0 & -1 \\ 10 & 0 & 0.7 & -0.7 \\ 100 & 1 & 0 & -0.2 \\ 0 & 0 & 1 & -0.3 \end{vmatrix}=$$ $$=-\begin{vmatrix} 10 & 0.7 & -0.7 \\ 100 & 0 & -0.2 \\ 0 & 1 & -0.3 \end{vmatrix}-\begin{vmatrix} -1000 & 0 & -1 \\ 10 & 0.7 & -0.7 \\ 0 & 1 & -0.3 \end{vmatrix}=$$ $$= -10\begin{vmatrix} 0 & -0.2 \\ 1 & -0.3 \end{vmatrix}+100\begin{vmatrix} 0.7 & -0.7 \\ 1 & -0.3 \end{vmatrix}-$$ $$-0.7\begin{vmatrix} -1000 & -1 \\ 0 & -0.3 \end{vmatrix}+\begin{vmatrix} -1000 & -1 \\ 10 & -0.7 \end{vmatrix}=$$ $$=-2-21+70-210+700+10=547$$

$$\varDelta_I=\begin{vmatrix} 1 & -1000 & 0 & -1 \\ 1 & 10 & 0.7 & -0.7 \\ 0 & 100 & 0 & -0.2 \\ 0 & 0 & 1 & -0.3 \end{vmatrix}=$$ $$=\begin{vmatrix} 10 & 0.7 & -0.7 \\ 100 & 0 & -0.2 \\ 0 & 1 & -0.3 \end{vmatrix}-\begin{vmatrix} -1000 & 0 & -1 \\ 100 & 0 & -0.2 \\ 0 & 1 & -0.3 \end{vmatrix}=$$ $$= 10\begin{vmatrix} 0 & -0.2 \\ 1 & -0.3 \end{vmatrix}-100\begin{vmatrix} 0.7 & -0.7 \\ 1 & -0.3 \end{vmatrix}+$$ $$+1000\begin{vmatrix} 0 & -0.2 \\ 1 & -0.3 \end{vmatrix}+100\begin{vmatrix} 0 & -1 \\ 1 & -0.3 \end{vmatrix}=$$ $$=2+21-70+200+100=253$$

$$\varDelta_T=\begin{vmatrix} 1 & 1 & -1000 & -1 \\ 1 & 0 & 10 & -0.7 \\ 0 & 1 & 100 & -0.2 \\ 0 & 0 & 0 & -0.3 \end{vmatrix}=$$ $$=-0.3\begin{vmatrix} 1 & 1 & -1000 \\ 1 & 0 & 10 \\ 0 & 1 & 100 \end{vmatrix}=$$ $$= -0.3\begin{vmatrix} 0 & 10 \\ 1 & 100 \end{vmatrix}+0.3\begin{vmatrix} 1 & -1000 \\ 1 & 100 \end{vmatrix}=$$ $$=3+30+300=333$$

$$\varDelta_Y=\begin{vmatrix} 1 & 1 & 0 & -1000 \\ 1 & 0 & 0.7 & 10 \\ 0 & 1 & 0 & 100 \\ 0 & 0 & 1 & 0 \end{vmatrix}=$$ $$=-\begin{vmatrix} 1 & 1 & -1000 \\ 1 & 0 & 10 \\ 0 & 1 & 100 \end{vmatrix}=$$ $$= -\begin{vmatrix} 0 & 10 \\ 1 & 100 \end{vmatrix}+\begin{vmatrix} 1 & -1000 \\ 1 & 100 \end{vmatrix}=$$ $$=10+100+1000=1110$$

$$C={\varDelta_C\over \varDelta}={547\over0.31}={54700\over31}$$

$$I={\varDelta_I\over \varDelta}={253\over0.31}={25300\over31}$$

$$T={\varDelta_T\over \varDelta}={333\over0.31}={33300\over31}$$

$$Y={\varDelta_Y\over \varDelta}={1110\over0.31}={111000\over31}$$