Comparative Static Analysis
of the Keynesian Model
Macroeconomics I
ECON 309 -- Cunningham
1
Simple IS-LM Analysis
S (Y ) − I (r ) − G = 0
M
L(Y , r ) − = 0
P
Two equations, two endogenous variables (Y and r), and one
exogenous variable G. Real money supply (M/ P) is taken as
constant since nominal money (M) and (P) are exogenous as well.
Take total differentials:
SY dY − I r dr = dG
LY dY + Lr dr = 0
Write in matrix form:
SY
L
 Y
− Ir  dY  dG 
= 



Lr   dr   0 
2
Simple IS-LM, Continued
Applying Cramer’s rule for solution:
1
0
dY
=
SY
dG
LY
− Ir
Lr
Lr
=
>0
− Ir SY Lr + Ir LY
Lr
SY
L
dr
= Y
dG SY
LY
1
0
− LY
=
>0
− I r SY Lr + I r LY
Lr
So, in a Keynesian economy,
under the conditions given,
cet. par. (i.e., prices), an
increase in government
spending increases GDP and
interest rates.
Because, by assumption, the following hold:
Lr < 0, LY > 0
SY > 0, I r < 0
3
Extension
What if prices are flexible? To examine this, we must include the
labor market and real wage computation.
w 
Nd   − N = 0
P
Y − F (N ) = 0
S (Y ) − I (r ) − G = 0
M
L(Y , r ) − − 0
P
Define the following variable as a convenience:
∂N d  w 
X =
−
 >0
∂ (w P )  P 2 
4
Extension (Continued)
0 −1
0 − FN
1 0
0
dY
=
0
dG
1
SY
LY
0
0
− Ir
0
Lr
−1
− FN
0
0
0
− Ir
0
Lr
X
0
0
M
P 2 = − Lr FN X > 0
X
Jac
0
0
M
P2
Similarly:
dN
LX
=− r >0
dG
Jac
dr
>0
dG
dP
>0
dG
w
d  
P <0
dG
(Note that the denominator turns out to be positive.)
5