ECONOMICS 3012
Notes-II: Symbols and Basic Macroeconomic Algebra
Closed Economy: No Foreign Sector
Parts A and B
INTRODUCTION: Symbols
The symbols used in this course are mostly those used in the textbook: Blanchard and Johnson (BJ)
Macroeconomics, Third Canadian Edition. They are defined in Appendix 4 at the end of the book, on p.
A15. I will add some variables to those. BJ roughly follow the conventions that ITALICIZED UPPERCASE Roman letters are VARIABLES, and italicized lower-case Roman letters are simple parameters.
The latter are often subscripted. BJ 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 BJ don’t, and I define a few variables differently from BJ. 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”;
an asterisk, *, means “foreign value” – here usually the US value.
A convention that I use and which you will find useful, one that is not in the textbook, is to use
subscripts, as in Y0 or Y1 , to denote equilibria. I use the subscript 0 normally to denote the initial
equilibrium; other numbers used as subscripts denote equilibria which occur after changes to the initial
conditions.
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.
Here I define only variables that are not defined in the BJ Appendix. So if a variable is not defined
here, check that Appendix.
AE is Aggregate Expenditure. This is identical to Z in BJ: “Demand for Goods”, so
AE ≡ Z
AE ≡ C + I + G
Y≡C+S+T
There are two things that I do a bit differently, although these end up being the same as the symbols
used in BJM in Chapter 4. I define them now and use them throughout, instead of changing the
definitions of some symbols as we go along, which the textbook does (sloppy).
1) I use just I for all investment. The book differentiates Investment (I) and Inventory Investment
(Is) at the beginning of Chapter 3, then drops the latter. There’s no need for it, we will ignore
Inventory Investment.
2) In Chapter 3, the book puts a bar over I to denote that it is exogenous. Later it drops the bar.
The Study Guide doesn’t use the bar. I do not use the bar. (The book doesn’t put a bar over
other exogenous variables; the use of the bar is unnecessary.)
-----------------------------------------------------------------------------------------------------------------PART A: Goods Markets Alone
Notes-I dealt with the basic macroeconomic identities. Now the model becomes analytical rather than
definitional. To make the transition, we need to add “behavioral” relations (equations) to the definitions.
First (another definition) to simplify the algebra I add the variable A, which is the sum of all exogenous,
or “autonomous”, spending:
A = [c0 + I + G – c1T] in Chapter 3 and
Econ 3012: Notes on Symbols and Algebra
page 2
A = [c0 + bo + G – c1T] in Chapters 5 and 10
I also use AE, which is Aggregate Expenditure:
AE = C + I + G
The basic algebra generates complex parameters: parameters that are a mix of other parameters. It
makes things easier to use the convention that lower-case Greek () letters stand for these
parameters. In problems I assign, I will often give you numerical values for these complex parameters.
The Goods Markets, or The Real Sector Alone, Chapter 3
Disposable Income (definition)
Consumption Function (behavior)
Aggregate Expenditure (definition)
I, G, and T are “exogenous” or “autonomous” (behavior)
so
complex parameter:
note that since we know that c1 is positive and c1
also know that  > 1.
Equilibrium (behavior)
This gives the “reduced form” solution (p 48 in BJM)
or, simplified:
YD = Y – T
C = c0 + c1 YD = c0 + c1Y – c1T;
AE ≡ Z = C + I + G
c1 < 1
A = c0 + I + G – c1T
= 1/(1 – c1)
< 1, we not only know that is positive, we
Y = AE ≡ Z
Y0 = [1/(1 – c1)]  [c0 + I + G – c1T]
Y0 =  A
Finally, a special value of Y, called the “natural rate”
In BJM, but usually called “full employment” output.
Yn
This is the level of output associated with full employment:
---------------------------------------------------------------------------------------------------------------------PART B: Government Budget
First we make taxes in part a function of the level of Income, Y. This is sensible; most taxes are fixed
rates times a base, and most tax bases (Sales, Personal Income, Corporate Profits) increase or
decrease as the level of Income increases or decreases. The main part of the textbook does not do
this, but it is in Problem 6 at the end of Chapter 3. Add:
Tax Function (behavior)
T = t0 + t1Y; t1 < 1
This has two effects: there is a new equation for  and a new equation for A. You will not need to know
these because I will give you values of  and A in any problems I ask you to do. What is of interest is
that it affects the definition of the Government Budget Deficit. That deficit is BD ≡ (G – T), which is now
the Budget Deficit function:
BD = [G – (t0 + t1Y)] = [(G – t0) – t1Y] .
So the government budget deficit is now (and in reality) a negative function of Income. Again, this is
sensible; if Income falls, Taxes decrease while Government spending remains roughly constant, so the
deficit increases. Since the analysis at this point is perfectly symmetrical, the opposite is true when
Income rises: sizes are the same but the directions are opposite. The BD function can be drawn as on
Figure 1 below:
Econ 3012: Notes on Symbols and Algebra
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Figure 1
$
0
Y
YBB
BD
As Y increases, the Budget Deficit decreases and, obviously, vice versa. For any given values of G
and t0 , both of which are exogenous, there is a unique value of Y, which I label YBB , which gives a
balanced budget. If G increases or t0 decreases, the BD function shifts up; this is shown as the dotted
line above. This in turn shifts YBB to the right – that is, it increases YBB . And vice-versa: if G decreases
or t0 increases, the BD function shifts down; this is shown as the dashed line above. This shifts YBB to
the left – that is, it decreases YBB .
Thus the Budget Deficit increases as three things happen. The first is having G increase. The second
is a tax cut, having t0 decrease. The third is having Y decrease.
However the first two effects are not the whole story. If G increases or t0 decreases, the BD function
shifts up. At the original level of Income, Y, the Budget Deficit increases by the amount of the shift: the
increase in G or the decrease in t0 . But either an increase in G, or a decrease in t0 , cause Y to
increase. So while there is a shift in the BD function, there is also a movement along the BD function.
The movement along reduces the increase in the Budget Deficit. This is shown for a government
spending increase or a tax cut in Figure 2 below:
Figure 2
BD1 : no
Income
change
$
BD1 : after
Income
0
change
BD0
a
b
Y
Y0
BD1 function
Y1 :
After G
or t0
BD0 function
Econ 3012: Notes on Symbols and Algebra
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Initially there is a balanced budget and Income of Y0 . An increase in G or a tax cut shifts the BD
function to the dotted line. If Income stayed at Y0 , then the new Budget Deficit would be BD1. But the
increase in G, or the tax cut, causes Income to increase as the analysis of Part A above demonstrated.
In this case Y increases to Y1 . This causes a movement along the new BD function, from point a to
point b, which reduces the Budget Deficit to BD1 .
Without going into the details, the conditions on the parameters make it impossible for the second effect
to be larger than the first. An increase in spending with no tax increase, and a tax cut with no spending
decrease, always increase the Budget Deficit. And vice-versa.
The “fact” of the BD function lets us define a special value of the budget deficit, known as the structural
deficit, or SD. The SD is the value of the budget deficit when output is at the full employment level, Yn.
The structural deficit function is:
SD = [G – (t0 + t1Yn)] = [(G – t0) – t1Yn] .
This is shown on Figure 3 below:
Figure 3
$
SD1
0
Y
Yn
SD2
BD0
BD1
BD2
With Budget Deficit function, BD0 , the Structural Deficit is zero. If G is higher or t0 is lower, the Budget
Deficit function is BD1 , and the Structural Deficit is positive at SD1 . And if G is lower or t0 is higher,
the Budget Deficit function is BD2 , and the Structural Deficit is negative – that is, it is a budget surplus
– and equal to SD2 .