Question

Given the following maro-economic model

Y=C+I+G

C=20+0.07Y

I=12+0.1Y

G=10

(a) Express the model in matrix form

(b) Using Cramer's Rule calculate the equilibrium values of the national income (Y), Consumption (C) and Investment (I)

(c) What is the income multiplier?

Answer 1

$$Solution$$

$given :\\ Y=C+I+G\\ C=20+0.07Y\\ I=12+0.1Y\\ G=10$

[A] The model in matrix form is:

$Y-C-I=10\\ -0.07Y +C=20\\ -0.1Y+I=12\\$

$\begin{pmatrix} 1&-1 & -1\\ -0.07&1 & 0\\ -0.1&0&1 \end{pmatrix}\begin{pmatrix} Y \\C\\I \end{pmatrix}=\begin{pmatrix} 10\\ 20\\12 \end{pmatrix}$

[B]Using cramer's rule to calculate equilibrium values of Y,C & I:

$\begin{pmatrix} 1 & -1&-1 \\ -0.07& 1&0\\ -0.1&0&1 \end{pmatrix}\begin{pmatrix} Y \\ C\\ I \end{pmatrix}=\begin{pmatrix} 10 \\ 20\\ 12 \end{pmatrix}$

i]Write down the main matrix and find its determinant to be

:$\Delta _0= 0.83$

ii]Replace the 1st column of the main matrix with the solution vector and find its determinant to be: $\Delta_1=42$

iii]Replace the 2nd column of the main matrix with the solution vector and find its determinant to be: $\Delta_2\ =19.54$

iv]Replace the 3rd column of the main matrix with the solution vector and find its determinant to be: $\Delta_3=14.16$

therefore:

$Y=\frac{ Δ1 }{ Δ_0} = \frac{42} {0.83} = 50.60\\C= \frac{\Delta_2} { Δ_0} =\frac{ 19.54 }{ 0.83} = 23.54\\I= \frac{Δ_3} { Δ _0}= \frac{14.16 }{ 0.83} = 17.06\\$

[C] income multiplier :

$income \ multiplier[{IM}]=\frac{\Delta GDP_{eq}}{\Delta\ C_{eq}}$

therefore :$\ IM=\frac{50.60}{23.54}=2.15$