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6 ISLM Model
M 0 = kY − li
Notice that the
amount of money in
the economy is fixed! The amount of money in the economy will tend to
The
increase when national income increases.
increase
But it will tend to decrease if higher interest
But
rate depresses the economic activity.
rate Collecting the four equations together, this is the linear system that we
Collecting
get:
get: Y
C
I
G
− −=0
( −) − = a
b
C
−
1 tY I
ei
d
+ = kY − = 0
li
M 1
b(1 −t )
The matrix form representation will be: 0 k If you were to compute the
If
determinant of matrix A, what would
be the most rational way to do this?
be −1
−1
0
0 −1
0
1
0 0 Y G0
0 C − a = e I d −l i M 0 7 ISLM Model
Let us apply Cramer’s rule to solve this system.
First, compute the determinant of the coefficient matrix. Its last row has
First,
two zeroes in it, so it makes sense to expand the determinant by the last
row.
row. −
1
A =− −
k1
0 −
1
0
1 0 1 0 −l b(1 −t )
e
0 −
1 −
1 −
1
0 0
1 In fact, we are lucky since in both third-order determinants there is a
In
column or a row with two zeroes. That means we only have to compute
two two-order determinants.
two 1 −1 −1
= −1
−1 0 A = −k × e( −1) −1 b (1 − t ) − 1
3+3 = −1 + b (1 − t ) × ( −1) − l × ( − 1) 3+ 3 × ( b(1 − t ) − 1) A=
ke − ( − ) +
bl 1 t
l 8 ISLM Model
To find the equilibrium level of national income, for example, we first
To
compute the auxiliary determinant by substituting the first column in
matrix A by the right-hand side column vector:
matrix
G0 0 −a −1 0 0 d 0 1 e M0 A1 = −1 −1 0 0 −l A1 = e( −1) Notice that the fourth column contains
Notice
two zeroes, so let’s expand the
determinant by the last column.
determinant G0
3+4 −1 −a
M0 −1
0 −1 0 −l ( −1)
0 −1 −1
G0
= −eM 0 ( −1) − l d
+1
−a −1 0
= M 0 e − l [ d ( −1) +1( − G0 − a ) ] =
M 0 e + l ( G0 + d + a ) G0
4 +4 −1 −a
d −1
0 −1
0=
1 −1 =
−1 9 ISLM Model
By Cramer’s rule in order to obtain the equilibrium value of national
By
income, we need to divide the auxiliary determinant A by the determinant
1
of the coefficient matrix: M 0 e +l (G0 +d +a )
Y=
ke −bl (1 −t ) +l
* We can rearrange this expression in order to express the equilibrium
We
level of national income as a linear combination of the two exogenous
variables, money stock and the government expenditure.
variables,
e
l
a +d
Y* =
M0 +
G0 +
=
ek + l [1 − b(1 − t ) ]
ek + l [l − b(1 − t ) ]
ek + l [l − b(1 − t ) ]
= A × M 0 + B × G0 + C
A is the Keynesian money multiplier, B is the government
is
expenditure multiplier, while C is the structural coefficient.
expenditure
10 Leontief InputOutput Models
Consider the following question:
How much output should each one of the n industries in the
economy produce so there is no shortage of steel anywhere
in the economy?
Let us formalize our economy with n industries by means of an
Let
input-coefficient matrix.
a23...

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