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Some Simple Math Facts:
A.
Working with Changes in Linear Equations:
As an example of a linear equation consider the simple consumption
function:
C = + mpc (Y  T),
where the mpc is the marginal propensity to consume and is a
constant in the consumption function.
We can say that consumption is
a linear function of, Y and T because these numbers enter the
consumption function additively with no exponents.
Many times we will wish to calculate the numerical values of C.
For example suppose that Y = 1000, T = 150, = 50 and the mpc = .8.
In this case,
C(Y = 1000) = 50 + .8(1000  150) = 730,
where the notation C(Y=1000) is meant to denote the value of
consumption at a particular level of income, in contrast to the
consumption function itself.
Now suppose we wanted to calculate by how much C would change by
if Y increased to 1200 and all the other variables remained
unchanged. We could of course plug the new values of Y into our
original equation;
grind out the new value of C, and then subtract the
old value from the new value.
But watch what happens if we write out
this arithmetic:
C(Y =1200)
=
50 + .8(1200  150)
=
890

C(Y = 1000)
=

{50 + .8(1000  150)} =

730.
∆
C
=
0
+ .8(1200  1000)
=
160
where
∆
is the symbol that stands for "change."
This arithmetic is
unnecessarily tedious because 2 out of the 3 terms cancel. We can
take advantage of this canceling by the following rule:
Math Fact on Changes
:
In a linear relationship y = mx + b, the change
in y (i.e.,
∆
y) is the change in x times the coefficient on x (i.e.,
∆
y =
m
∆
x).
In our numerical example of the consumption function the coefficient
on Y is .8, and our numerical example follows the general rule since
∆
C = 160 = .8(200) = mpc
∆
Y.
The general result in this case is so
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View Full Document important I try to emphasize it by telling students that in 99% of the
cases the change in consumption is equal to the mpc times the
change in disposable income.
My 99% rule works for all changes in C
due to changes in Y or T, the one percent of the cases where it does
not work is when C changes due to a change in
(
although the math
facts rule still works because the coefficient onis an implicit 1).
In this
case,
∆
C =
∆
It is possible to make use of this important tool in more complicated
linear equations.
For example, we will spend a great deal of time this
semester studying factors that cause a change in national saving, S.
National Saving, which is simply the sum of public and private saving
can be written as:
S
≡
S
p
+ (T

G) = (Y  T)  C + (T  G); where
S
P
is private
saving
We can substitute the consumption function for C to write a
convenient equation for S:
S =  + (1mpc)(Y  T) + (T  G), or S =  + mps(Y  T) + (T  G),
where mps
≡
marginal propensity to save = (1mpc).
National saving
is a linear function of, Y, T and G because these numbers all enter the
equation for S additively with no exponents. Now suppose that Y =
1000, G = 170, T = 150, = 50 and the mpc = .8.
From this we can
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This note was uploaded on 03/02/2010 for the course ECON 57 taught by Professor Woglom during the Spring '08 term at UMass (Amherst).
 Spring '08
 Woglom
 Marginal Propensity To Consume

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