ECONOMICS 4012
Symbols and Basic Macroeconomic Algebra
INTRODUCTION: Symbols
The symbols used in this course are mostly those used in the textbook: Blanchard, Johnson and Melino
(BJM) Macroeconomics, Second Canadian Edition. They are defined in Appendix 4 at the end of the
book, on p. A15. I will add some variables to those, and I will define one variable differently. BJM
roughly follow the conventions that ITALICIZED UPPER-CASE Roman letters are VARIABLES, and
italicized lower-case Roman letters are simple parameters. The latter are often subscripted. BJM do
not use these conventions exclusively. I will note where they deviate from convention. I will use those
same conventions for variables that I define. I will define some variables that BJM don’t, and I define a
few variables differently from BJM. I also use another convention: lower-case Greek letters, like , 
and , will be used for combinations of simple parameters into more complex parameters. Other
conventions:
a superscript, d, means “demanded”;
a superscript, e, means “expected”;
BJM, presenting a static model, use subscripts to denote differences in parameters. We, of course, use
subscripts now for time periods. So I will have to rewrite BJM’s equations slightly. This won’t matter to
you because in 4012 you will be given number values for the parameters in those you need to use.
A final convention that is extremely useful is that if a relation is negative, I, and the textbook, put a
minus sign before the parameter. That means that all parameters themselves are positive. This has
the happy quality of making it very easy to “sign” the complex parameters. What this means will
become clear in the development of the model below.
In 4012 there is no foreign sector, so there is no need to denote foreign values of variables.
Here I define only variables that are not defined in the BJM Appendix. So if a variable is not defined
here, check that Appendix.
AE is Aggregate Expenditure. This is the same as Z in BJM: “Demand for Goods”, so
AE ≡ Z
AE ≡ C + I + G
Y≡C+S+T
A is autonomous-like spending:
-----------------------------------------------------------------------------------------------------------------The Goods Markets, or The Real Sector: IS curve, Chapter 5
Disposable Income
YD = Y – T
Consumption Function
C = a + c YD = a + cY – cT; c < 1
Aggregate Expenditure
AE = C + I + G
I, G, and T are “exogenous” or “autonomous”
so
A = a + I + G – cT
complex parameter:
= 1/(1 – c)
note that since we know that c is positive and c < 1, we not only know that is positive, we also
know that  > 1.
Equilibrium
Y = AE
This gives the “reduced form” solution (p 48 in BJM)
Y0 = [1/(1 – c)]  [a + I + G – cT]
or, simplified:
Y0 =  A
NOW add the Investment Function:
I = h– bi
NOTE: Here is the first example of a variable that is not upper-case: the interest rate is i.
NOTE: In 4012 we simplify and do NOT have I as a function of Y.
A is now autonomous-like stuff:
A = (a – cT) + h + G
Econ 4012: Symbols and Algebra
page 2
The complex parameter,  is still:
Equilibrium:
= 1/(1 – c)
The solution is the IS curve:
Y =  A – b i
Y = AE.
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 = d Y – f 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 = (d/f)Y – (1/f)(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 BJM 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 BJM 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 six parameters: a, b, c, d,
f, and h. 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, Ye , 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, ie, 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.
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:
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.
Econ 4012: Symbols and Algebra
page 3
Figure 4
i
IS
LM
io
Y0
Y
A
(1/f)(M/P)
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.
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.
This is shown on Figure 5 below:
Econ 4012: Symbols and Algebra
page 4
Figure 5
i
IS0
IS1
e0
ê
LM
i0
e1
i1
Y1
A1
Y0
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.
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:
Figure 6
i
LM0
IS
LM1
e0
io
e1
i1
ê
Y0
(1/f)(M0/P)
(1/f)(M1/P)
Y1
Y
Econ 4012: Symbols and Algebra
page 5
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 simu
Y:
Y = A - b [d/f)Y – (1/f)(M/P)
The solution is the AD curve:
Y =A + (M/P)
with complex parameter:
=
and complex parameter:
f
f  bd
b
=
= (b/f)
f  bd
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.