Overview
IS-LM Components
Following the example given in Anthony
{% Y = C + I + G %}
Then
consumption function
is assumed to be a linear function of {% Y %}
{% C = C_0 + \beta_0 Y %}
Likewise, the
investment function
is a linear function of the interest rate {% r %}
{% I = I_0 - \alpha_0 r %}
{% (1 - \beta_0) Y + \alpha_0 r = C_0 + I_0 + G %}
Money Demand {% M_d %}
(see LM function)
is assumed to be a linear function of the income {% Y %} and the interest rate {% r %}
{% M_d = M_0 + \beta_1 Y - \alpha_2 r %}
{% \beta_1 Y - \alpha_1 r = M_s - M_0 %}
Stating these equations in
matrix
terms:
{%
\begin{bmatrix}
\beta_1 & - \alpha_1 \\
1-\beta_0 & \alpha_0 \\
\end{bmatrix}
\begin{bmatrix}
Y \\
r \\
\end{bmatrix}
=
\begin{bmatrix}
M_s - M_0 \\
C_0 + I_0 + G \\
\end{bmatrix}
%}
{%
\begin{bmatrix}
Y \\
r \\
\end{bmatrix}
=
\begin{bmatrix}
\beta_1 & - \alpha_1 \\
1-\beta_0 & \alpha_0 \\
\end{bmatrix} ^ {-1}
\begin{bmatrix}
M_s - M_0 \\
C_0 + I_0 + G \\
\end{bmatrix}
%}