ECONOMICS 3012
Notes II: Symbols and Basic Macroeconomic Algebra
Closed Economy: No Foreign Sector
Parts C and D
PART C: IS and LM
Goods Markets, or the Real Sector, again: IS curve, Chapter 5
To the Goods Markets as developed in Part A above, add:
the Investment Function:
I = b0 + b1Y – b2 i;
b1 << c1 < 1
(Note: Here is the first example of a variable that is not upper-case: the interest rate is i.
Note also that in the investment function we always use in my Problems, b1=0. This doesn’t matter
much because now you are almost always given the value of .)
A is now autonomous-like stuff:
The complex parameter,  is now:
Equilibrium:
A = c0 + bo + G – c1T
= 1/(1 – [c1 + b1])
Y = AE.
The solution is the IS curve:
Y =  A – b2 i
The IS curve is a set of points relating values of the interest rate, i, to corresponding equilibrium values
of output, Y in the Real Sector (Goods Markets).
Financial Markets, or the Financial Sector: LM curve, Chapters 4 and 5
Real Money Demand: Transactions + Speculative:
Md = d1 Y – d2 i
(Note: Because Y is “output”, it is real, so this is the demand for real money.)
M is the nominal money stock, and is exogenous
M = Ms
Real Money Stock
M/P
P is the aggregate price level, and is fixed, or predetermined, at this stage of the model
Equilibrium:
Md = M/P
The solution is the LM curve:
i = (d1/d2)Y – (1/d2)(M/P)
For now,
P = 1.00
The LM curve, then, is a set of points relating values of output, Y, to corresponding equilibrium values
of the interest rate, i, in the Financial Sector (Markets)._______________________________________
NOTE: Both BJ and I write the IS curve as reduced form in Y. This is sensible because in the Goods
Markets it is the level of output that is determined and the interest rate, i, is exogenous. This also
works better for solutions. Both BJ and I write the LM curve as reduced form in i, because in the
Financial Markets it is the interest rate that is determined and the level of output, or income, Y, is
exogenous. This, too, works better for solutions.
The model now has two fundamental equations, the IS curve and the LM curve, two endogenous
variables (“unknowns”), Y and i, three exogenous variables, G, T, and M, and seven parameters: the
three bs, the two cs and the two ds. The two equations can be solved simultaneously for the two
endogenous variables. To do the Problems, I strongly suggest that you solve these as follows: 1)
Substitute the LM curve , which will be just numbers, Y, and i , with i isolated on the left-hand side, into
the IS curve, which is just numbers, Y, and i , with Y isolated on the left-hand side. That will give you a
single equation with just numbers and Y. 2) Solve this for the equilibrium value of output, Y0 , which will
now be a number. 3) Substitute this number into the LM curve and you will get the equilibrium value of
the interest rate, i0, which will be a number.
NOTE: In all problems the interest rate, i, is always expressed as a whole number. So an interest rate
of 5% is 5; an interest rate of 10% is 10, and so on.
Econ 3012: Notes on Symbols and Algebra
page 2
[Note also: the model is really richer than this. Lurking in the background are three other endogenous
variables: C, I, and S. These are functions of either Y or i or both, and their values can be found once
we have Y and i. But finding both Y and i must come first. And often, but not always, we’ll stop there.]
On graphs, simultaneous equilibrium is simple. The IS curve is a set of points of equilibrium in the
Real Sector (Goods Markets) and is negatively sloped. The LM curve is a set of points of equilibrium
in the Financial Sector (Markets) and is positively sloped. Simultaneous equilibrium in both sectors
(markets) is where the IS curve crosses the LM curve. This is shown on Figure 4 below:
Figure 4
i
IS
LM
io
Y0
Y
A
(1/d2)(M/P)
The IS and LM curves are shown in bold. Remember that they are not linear and that the straight lines
are just approximations to the curves over a range. The intercepts of the two curves on the two axes –
the Y-axis for the IS curve, and the i-axis for the LM curve – are shown by extending the two linear
approximations with dashed lines. AGAIN: Remember that the values on the dashed lines are
impossible in the real world; they are just there to simplify the algebra of the linear approximations.
Two special variables are the exogenous policy variables. G and T are fiscal policy, and M is monetary
policy. Since G and T are now included in A, A becomes the variable which is the vehicle for fiscal
policy. A is the variable that determines where the IS curve crosses the Y-axis, and M is the variable
(for now) that determines where the LM curve crosses the i-axis.
POLICY:
The IS curve shifts when one of the two exogenous fiscal policy variables, G or T , change. Changes
in G or T change A. Expansionary fiscal policy is to either increase G or decrease T , either of which
increases A and shifts the IS curve to the right. (The model at this stage is symmetrical; so
contractionary fiscal policy is the opposite, both in sign and in size.) The LM curve shifts (for now,
because P is constant) when the exogenous monetary policy variable, nominal money stock, M,
changes. Expansionary monetary policy is to increase M , which shifts the LM curve down, to the right.
(The model at this stage is symmetrical; so contractionary monetary policy is the opposite both in sign
and in size.)
In either case, because the model is simultaneous, a change in one sector – an action – causes a
reaction in the other sector. The reaction in turn feeds back to the first sector. When you are thinking
about, or describing, what goes on as the system moves from one equilibrium to another after a change
in an exogenous variable, you need to take all three of these into account: action, reaction, feed-back.
Econ 3012: Notes on Symbols and Algebra
page 3
You also need to take into account that the Financial Sector (Markets) moves to an equilibrium very
quickly, while the Real Sector (Goods Markets) moves to its equilibrium relative slowly.
I will describe two of these changes: one for contractionary fiscal policy, and the other for expansionary
monetary policy. Again, because the model is symmetrical at this stage, what happens with
expansionary fiscal, and contractionary monetary, policy is the exact opposite, both in sign and in size.
Contractionary fiscal policy: Action – Contractionary fiscal policy reduces A, which shifts the IS curve
to the left. As in the Real Sector alone, this begins to reduce output, Y, which is a move horizontally to
the left. Reaction – As output or Income, Y, falls, there is less transactions demand, which, with M
held constant, causes a decrease in the interest rate, i, in the Financial Sector. The LM curve remains
where it was and we move down it. Feed-back – The fall in the interest rate, i, increases private
investment, I, in the Real Sector, which causes an increase in output or income, Y. Note that the feedback effect must be smaller than the initial effect. [In dynamic analysis this quality is known as stability,
which is a nice quality. If the feedback effect is larger than the initial action, the model is unstable.]
This is shown on Figure 5 below:
Figure 5
i
IS0
IS1
e0
ê
i0
LM
e1
i1
Y1
Y0
A1
A0 Y
Given the initial action, the increase in A, the IS curve shifts to IS1 . In the absence of a Financial
Sector reaction the economy wants to move to point ê. But there is a reaction in the Financial Sector:
as Y falls, transaction demand decreases and the interest rate i falls. This is the movement down the
LM curve from e0 to e1 . The fall in the interest rate feeds back to the Real Sector by reducing private
investment. This cause a movement along the IS curve from ê to e1 . The economy converges from
disequilibrium points to the new simultaneous equilibrium at point e1 . Here the new equilibrium value
of output or Income, Y1 is lower than the initial value, and the new equilibrium value of the interest rate
i1 , is lower than the initial value. (However, keep in mind that the new equilibrium value of private
Investment, I1 , will be higher than the initial value. This could be called the “crowding in” effect:
Contractionary fiscal policy reduces the interest rate which increases private Investment.)
Expansionary monetary policy: Action – Expansionary monetary policy increases M, which shifts the
LM curve down and to the right. As in the Financial Sector alone, this reduces the interest rate, i,
which is a vertical move down. Reaction – As the interest rate, i, falls, there is an increase in private
investment, I, and therefore in Y, in the Real Sector. The IS curve remains where it was and we move
down it. Feed-back – The increase in Y increases transactions demand in the Financial Sector, which
causes an increase in the interest rate, i. Note that the feed-back effect must be smaller than the initial
effect because the model is stable. This is shown on Figure 6 below:
Econ 3012: Notes on Symbols and Algebra
page 4
Figure 6
i
LM0
IS
LM1
e0
io
e1
i1
ê
Y0
Y1
Y
(1/d2 )(M0/P)
(1/d2)(M1/P)
Given the initial action, an increase in M, the LM curve shifts down to LM1 . In the absence of a Real
Sector reaction the economy wants to move to point ê. But there is a reaction in the Real Sector;
because of simultaneity: as i falls, private Investment, I increases causing output or Income, Y, to
increase. This is the movement down the IS curve from e0 to e1 . The increase in output or Income
feeds back to the Financial Sector by increasing transactions demand and increasing the interest rate,
i. This cause a movement along the LM curve from ê to e1 . The economy converges from the
disequilibrium points to the new simultaneous equilibrium at point e1 . Here the new equilibrium value
of output or Income, Y1 is higher than the initial value, and the new equilibrium value of the interest rate
i1 , is lower than the initial value.
PART D: Aggregate Demand, AD curve, Chapter 10
The aggregate price level, P, now shifts from being fixed, or predetermined, to being an endogenous
variable.
Solve IS and LM simult
Y:
Y = A - b2 [d1/d2)Y – (1/d2)(M/P)
The solution is the AD curve:
Y =A + (M/P)
with complex parameter:
=
and complex parameter:
d 2
d 2  b2 d1
b2
=
= (b2/d2)
d 2  b2 d1
The AD curve has two endogenous variables, Y and P, but is a single equation. (Or, one can look at
this as having two equations, IS curve and LM curve, and three endogenous variables, Y, I, and P,
since the AD curve is just a solution of the IS curve and the LM curve.) Thus, at this stage, to find
either Y or P you have to have values of the other. The AD curve is not a model; it is one half of a
model. As you can probably guess, the other half is the Aggregate Supply curve, the AS curve. That
is added after the second midterm to make a complete model.