Equilibrium in the IS-LM-MP model
Pedro Serôdio
July 20, 2016
The IS-LM equilibrium
I
General equilibrium in the IS-LM(MP) framework is attained
when both markets, the goods market and the money (asset
markets), are in equilibrium. Mathematically, the equilibrium
is defined by the system of equations:
Y = C (Y − T , i − π e , A, Y e ) + I (i − π e , K , Y e ) + G
M
= L(Y , i)
P
I
This equilibrium can be represented in either (i, Y ) or (r , Y )
space, and it is defined by the intersection of the IS and LM
(MP) schedules.
IS-LM equilibrium
E
E
E
= Y
E
P
∗
Y
i
Y
∗
i
LM
i
∗
i
∗
IS
L(i , Y )
M
Y
Y
∗
P
M
P
IS-MP equilibrium
E
E
E
= Y
E
P
∗
Y
i
Y
∗
i
MP
i
∗
i
∗
IS
L(i , Y )
M
Y
Y
∗
P
M
P
The IS-LM equilibrium
I
I
We can solve the model explicitly by taking the natural
logarithm of the IS and the LM schedules for the IS-LM
model, and the natural logarithm of the IS schedule and the
policy rule (MP) outlined above. We begin by taking logs of
the IS schedule (for simplicity, we’ll momentarily ignore wealth
and expectations of future income, the interest rate channel
for consumption and the role of capital).
The simplified version of the IS schedule is then:
Y = C (Y − T ) + I (i − π e ) + G
I
Taking logs, we have the following expressions:
IS:
y = a + c(y − t) + b − d(i − π e ) + g
LM: m − p = ky − hi
MP(i):
i = r¯ + π + φπ (π) + φy (y − ȳ )
MP(r):
r = r¯ + φπ (π) + φy (y − ȳ )
The IS-LM equilibrium
I
Recall that in the IS-LM model, prices are assumed to be
constant and, therefore, we can impose the condition that
i = r . Under this assumption, equilibrium output and prices
are given by:
h
d
(a − ct + b + g ) +
(m − p)
(1 − c)h + dk
(1 − c)h + dk
k
1−c
i∗ =
(a − ct + b + g ) −
(m − p) =
(1 − c)h + dk
(1 − c)h + dk
= r∗
y∗ =
The IS-MP equilibrium
I
In this formulation, we can relax the assumption that prices
are fixed. That has no implications for the shape of the IS
schedule (aside from the possible difference between nominal
and real interest rates), and it allows us to write the model in
terms of the logarithm of output and inflation:
1
(a − ct + b + g ) +
1 − c − dφy
1
+
(d(π e − r¯) + dφy ȳ − (1 + φπ π))
1 − c − dφy
y∗ =
i ∗ = r¯ + π + φπ (π) + φy (y ∗ − ȳ )
r ∗ = r¯ + φπ (π) + φy (y ∗ − ȳ )
The IS-MP equilibrium
I
The equilibrium outlined above relies on a Taylor-rule
formulation for the monetary policy rule, where policy makers
respond to both deviations of output from trend as well as
deviations of inflation from target.
I
Alternative specifications for the monetary policy rule have
different implications for the equilibrium values, along with a
very different implication for how policy makers respond to
changes in the macroeconomic environment.
I
Under a strict inflation targeting regime, the central bank is
only concerned with deviations of inflation from its target.
The policy rule is summarised as:
π = πT
The IS-MP equilibrium
I
With a flexible inflation target and without a response to
output, the policy rule becomes:
i = r¯ + π + φπ π
I
Under a strict inflation target, the MP schedule is perfectly
horizontal, meaning that the central bank allows deviations of
output from potential output but does not tolerate deviations
of inflation from its target.
I
Under flexible inflation target regimes, the MP schedule may
be upward sloping in (i, y ) space even if the central bank does
not care about output because output and inflation are likely
to be positively correlated.
Policy
I
An increase in government spending has a positive impact on
both output and the nominal (or real) interest rate, as we can
see by taking the derivative of both y and i, r with respect to
g:
∂y ∗
1
=
∂g
1 − c − dφy
φy
∂i ∗
∂r ∗
=
=
∂g
∂g
1 − c − dφy
I
This multiplier is smaller than the Keynesian cross multiplier
because of the crowding out effect: the monetary authority
increases the nominal interest rate to prevent large deviations
of output from its target and, therefore, limits the
effectiveness of fiscal policy.
Fiscal Policy
I
Graphically, we can see the impact of an increase in
bond-financed government spending below:
r
MP
∗
1
r
∗
0
r
IS0
IS1
Y
∗
0
Y
Y
∗
1
Monetary Policy
I
Graphically, we can see the impact of an increase in the
money supply below:
r
MP0
MP1
∗
0
r
∗
1
r
IS
Y
∗
0
Y
Y
∗
1
Summary
I
We have revised models underlying IS, LM and MP curves.
I
We have combined IS and either LM or MP, to find
equilibrium on the demand side with sticky prices.
I
We have analysed fiscal and monetary policy effects in the
IS-MP model.
I
From now on we focus on the MP curve. Although the
analysis goes through with the LM curve instead, the MP
curve has the advantages of simplicity, correspondence with
current advanced theory, descriptive realism, and leading to an
AD curve with inflation on the vertical axis. It is important to
bear in mind, though, that underlying the ability of
policymakers to target a particular interest rate are agents?
actions in the money and bond markets, which are more
obvious in IS-LM.