R. Larry Reynolds (Boise State University)
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Principles of Microeconomics
Slide 1
Math Review
· Mathematics is a very precise language
that is useful to express the
relationships between related variables
· Economics is the study of the
relationships between resources and the
alternative outputs
· Therefore, math is a useful tool to
express economic relationships
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Principles of Microeconomics
Slide 2
Relationships
· A relationship between two or more
variables can be expressed as an
equation, table or graph
· equations & graphs are “continuous”
· tables contain “discrete” information
· tables are less complete than equations
· it is more difficult to see patterns in tabular
data than it is with a graph -- economists
prefer equations and graphs
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Principles of Microeconomics
Slide 3
Equations
· a relationship between two variables can
be expressed as an equation
· the value of the “dependent variable” is
determined by the equation and the value
of the “independent variable.”
· the value of the independent variable is
determined outside the equation, i.e. it
is “exogenous”
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Principles of Microeconomics
Slide 4
Equations [cont . . .]
·
·
An equation is a statement about a relationship between
two or more variables
Y = fi (X) says the value of Y is determined by the value
of X; Y is a “function of X.”
·
·
Y is the dependent variable
X is the independent variable
·
A linear relationship may be specified: Y =
[the function will graph as a straight line]
a  mX
· When X = 0, then Y is “a”
· for every 1 unit change in X, Y changes by “ m”
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Principles of Microeconomics
Slide 5
Y = 6 - 2X
· The relationship between Y and X is
determined; for each value of X there
is one and only one value of Y [function]
· Substitute a value of X into the equation
to determine the value of Y
· Values of X and Y may be positive or
negative, for many uses in economics the
values are positive [we use the NE quadrant]
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Principles of Microeconomics
Slide 6
Equations -- Graphs [Cartesian system]
Y>0
The X axis
[horizontal]
(X,Y) where +3
X<0, Y>0
+2
The North East Quadrant
(NE), where X > 0, Y > 0
{both X and Y are positive
numbers}
+1
X<0
-3
-2
-1
(X,Y)
where X<0 and Y<0
X > 0
+1
-1
-2
-3
+2
+3
(X,Y)
where X>0 and Y<0
(Left click mouse to add material)
The Y axis [vertical]
Y<0
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Principles of Microeconomics
Slide 7
When the values of the independent and dependent variables
are positive, we use the North East quadrant
(Left click mouse to add material)
Go to the right {+3} units and
(X, Y)
up {+5} units!
6
5
(3, 5)
Right {+1} one
and up {+6} six
(1,6)
3
(5, 1)
(2.5, 3.2)
2
Right 5 and up 1
1
1
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3
4
to the right 2.5 units
and up 3.2 units
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Slide 8
Given the relationship, Y = 6 - 2X,
Y
6
sets of (X, Y)
[this is Y-intercept]
5
A line that slopes from
when X = 1 (0, 6)
upper left to lower right then Y = 4
(1, 4)
represents an inverse or
negative relationship, when the
(2, 2)
value of X increases, Y
decreases!
(3, 0)
4
3
When X = 2, then Y = 2
2
The relationship for all positive
values of X and Y can be illustrated
by the line AB
1
B
1
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material)
A when X = 0 then Y = 6
2
3
When X = 3, Y = 0,
[this is X-intercept]
4
5
Principles of Microeconomics
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X
Slide 9
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Y Given a relationship, Y = 6 -(Left
.5X
6
(0,6)
5
(1,5.5)
(2, 5)
4
(4,4)
3
2
1
Y  0, when X  12,
What is the X-intercept?  6

  12
 .5

1
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(6,3)
For every one unit increase in
the value of X, Y decreases by
one half unit. The slope of this
function is -.5! The Y-intercept is 6.
2
3
4
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Principles of Microeconomics
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X
Slide 10
Y
For a relationship, Y = 1 + 2X
When X=0, Y=1 (0,1)
6
When X = 1, Y = 3
slope = +2
5
When X = 2, Y = 5
4
rise
+2
3
This function illustrates a positive relationship
between X and Y. For every one unit increase
in X, Y increases by 2 ! for a relationship
Y = -1 + .5X
This
run function shows that for a 1
unit
+1 increase in X, Y increases
one half unit
rise
2
run
+2
1
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slope = +2
+1
1
-1
(1,3)
(2,5)
2
3
4
5
6
X
(Left click mouse to add material)
Principles of Microeconomics
Slide 11
Problem
· Graph the equation: Y = 9 - 3X
· What is the Y intercept? The slope?
· What is the X intercept? Is this a positive
(direct) relationship or negative (inverse)?
· Graph the equation Y = -5 + 2X
· What is the Y intercept? The slope?
· What is the X intercept? Is this a positive
(direct) relationship or negative (inverse)?
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Principles of Microeconomics
Slide 12
Equations in Economics
· The quantity [Q] of a good that a person will
buy is determined partly by the price [P] of the
good. [Note that there are other factors that
determine Q.]
· Q is a function of P,
given a Price the
quantity of goods purchased is determined.
Q = fp (P)
· A function is relationship between two sets in
which there is one and only one element in the
second set determined by each element in the
first set.
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Principles of Microeconomics
Slide 13
Relationship [cont . . . ]
·
·
·
·
·
Q = fp (P)
{Q is a function of P}
Example:
Q = 220 - 5P
If P = 0, then Q = 220
If P = 1, then Q = 215
for each one unit increase in the
value of P, the value of Q
decreases by 5
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Slide 14
Q = 220 - 5P
· This is an inverse or negative relationship
· as the value of P increases, the value of Q decreases
· the “Y intercept” is 220, this is the value of Q
when; P = 0
· the “X intercept” is 44, this is the value of P
when Q = 0
· This is a “linear function,” i.e. a straight line
· The “slope” of the function is -5
· for every 1 unit change in P, Q changes by 5
in the opposite direction
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Principles of Microeconomics
Slide 15
The equation provides the information to construct a table.
However, it is not possible to make a table to include every
possible value of P. The table contains “discrete” data and does
not show all possible values!
Q = 220 -5 P
Value of P
Value of Q
combinat ion A
0
220
combinat ion B
1
2 15
combinat ion C
2
2 10
combinat ion D
3
20 5
combinat ion E
4
20 0
combinat ion F
10
17 0
combinat ion G
44
0
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Principles of Microeconomics
Slide 16
PRICE
55
50
44
45
40
35
For the relationship, Q = 220 - 5P,
the relationship can be graphed ...
When the price is $44, 0 unit will be bought;
at a price of $0, 220 units will be bought.
Notice that we have drawn the
graph “backwards,” P{independent}
variable is placed on the Y-axis.
This is done because we eventually
want to put supply on the same
graph and one or the other must be
reversed! Sorry!
30
At P=$30,
25 Q = 70
20
15
10
At a price of $10, the
$5
the quantity is
70
40
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80
120
160
Demand
170
200
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of Microeconomics
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280
QUANTITY
Slide 17
Slopes and Shifts
· Economists are interested in how one
variable {the independent} “causes” changes in
another variable {the dependent}
· this is measured by the slope of the function
· Economists are also interested in changes
in the relationship between the variables
· this is measured by “shifts” of the function
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Principles of Microeconomics
Slide 18
Slope of a function or “line”
· The slope measures the change in the
dependent variable that will be “caused”
by a change in the independent variable
· When, Y = a  m X; m is the slope
Y
rise
m

X
run
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Principles of Microeconomics
Slide 19
Y
6
5
4
3
Slope of a Line
Y = 6 -.5X
as the value of X
increases from 2 to 4,
Y= -1
2
the value of Y
decreases from
5 to 4
1 2 3 4 5 6 X
Y is the rise [or change in Y caused by X]{in this case, -1}
1
X = 2
X is the run {+2}, so, slope is -1/2 or -.5
rise
slope is run
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Principles of Microeconomics
Slide 20
Shifts of function
· When the relationship between two
variables changes, the function or line
“shifts”
· This shift is caused by a change in some
variable not included in the equation
· [the equation is a polynomial]
· A shift of the function will change the
intercepts [and in some cases the slope]
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Principles of Microeconomics
Slide 21
Y
6
5
(Left click mouse to add material)
Given the function Y = 6 - .5X,
Shifts
right
4
3
2
1
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shifts
left
an increase in the
A decrease in the
function would
function would be
represent an
increase in the
Y’ = 4 - .5X
intercept [from 6
to a larger number]
1 2 3 4 5 6 X
the function shifts and its
Just the slope
slope also changes
changes
{in this case, an increase in the absolute
value of .5 to -1.8}
Y” = 6 - 1.8X [x intercept = 3.3]
Principles of Microeconomics
Slide 22
Shifts in functions
· In Principles of Economics most functions are
graphed in 2-dimensions, this means we have 2
variables. [The dependent and independent]
· Most dependent variables are determined by
several or many variables, this requires
polynomials to express the relationships
· a change in one of these variables which is not
shown on a 2-D graph causes the function to
“shift”
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Principles of Microeconomics
Slide 23
Slope and Production
· The output of a good is determined by
the amounts of inputs and technology
used in production
· example of a case where land is fixed
and fertilizer is added to the production
of tomatoes.
· with no fertilizer some tomatoes, too
much fertilizer and it destroys tomatoes
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Principles of Microeconomics
Slide 24
tons of
tomatoes
12
The maximum output of T possible with all inputs
11 and existing technology is 10 units with 6 units of F
10
TPf
With the 3rd unit of F,
T increases to 9
9
With 2 units of F, the output of T
increases to 8
8
7
With 1 unit of Fertilizer [F], we get
6 tons
6
5
The increase in tomatoes [T] “caused” by F
is +3, this is the slope
4
With no fertilizer we get 3 tons of tomatoes
3
2
use of more F causes the tomatoes to “burn” and
output declines
1
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Principles of Microeconomics
FERTILIZER
Slide 25
Slope and Marginal Product
· Since the output of tomatoes [T] is a
function of Fertilizer [F] , the other
inputs and technology we are able to
graph the total product of Fertilizer
[TPf]
· From the TPf, we can calculate the
marginal product of fertilizer [MPf]
· MPf is the TPf “caused” by the F
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Principles of Microeconomics
Slide 26
12
11
10
Given: T = f (F, . . . ), MPf = [TPf/F]
TPf = +1, F = +3; +1/+3 @ .33
[this is an approximation because F>1]
TPf
9
8
7
6
5
4
3
2
TPf = -1, F = +2; -1/+2 = -.5
Fertilizer [F] Tomatoes [T]
0
3
1
6
2
8
3
9
rise = +3 6
10
9
rise/run =+3
+3 8
TPf = +3, F = +1; +3/+1 = 3
MPf [slope]
3
3
2
1
.33
-.5
{technically,
this is
between 0
and the first
unit of F}
[a negative slope!]
[slope = +3]
run=1TPf = +2, F = +1; +2/+1 = 2
TPf = +1, F = +1; +1/+1 = 1
1
1
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4
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6
7
8
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Principles of Microeconomics
Slide 27
Given a functional relationship such as: Q = 220 - 5P,
we can express the equation for P as a function of Q
Think of an equation as a “balance scale,” what you do to one side
of the equation you must do to the other in order to maintain balance
subtract 220 from both sides
divide every term in both sides
by -5
Q = 220 - 5P
-220
-220
-220 + Q
=
-5P
-5
-5
-5
44 -
or,
Q =
1P
P = 44 - .2 Q
The equation
is the same as
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1
5
P = 44 - .2Q
Q = 220 - .5P
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Principles of Microeconomics
Slide 28
How do economists estimate
relationships?
· Humans behavioral relationships are:
· modeled on the basis of theories
· models are verified through empirical
observations and statistical methods
· The relationships are estimates that
represent populations {or distributions}
not specific individuals or elements
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Principles of Microeconomics
Slide 29
An Example
· Hypothesis: the amount of good X [Q]
that Susan purchases is determined by
the price of the good [Px], Susans’s
income [Y], prices of other related goods
[Pr] and Susan’s preferences.
· Q = fi (Px,Y, Pr, preferences, . . .)
· [. . . indicates there are other variables
that are not included in the equation]
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Principles of Microeconomics
Slide 30
Model of Relationship
· Q = fi (Px,Y, Pr, preferences, . . .) acts
a a model to represent the relationships
of each independent variable to Q [dependent
variable]
· For simplicity, the relationship is
described as “linear.” If the
relationship were believed not to be
linear, with a bit more effort we might
construct a “nonlinear model.”
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Principles of Microeconomics
Slide 31
Empirical verification
· To test the model, we would like to observe
Susan’s buying pattern.
· If Px,Y, Pr and preferences were all changing
at the same time, we would use a multivariate
analysis called “multiple regression.” For
simplicity we have been lucky enough to find a
period where only Px has changed. Y, Pr and
preferences have remained unchanged over
the period in which we observe Susan’s
purchases
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Principles of Microeconomics
Slide 32
Price of good X
During a 5 week period,
Susan was observed
making the following
purchases
Susan’s purchases each week
week
price of good
during t he week [P]
quant it y Susan
purchased [Q]
1
$10
20 unit s X
2
$15
10 unit s X
3
$11
15 unit s X
18
4
$7
22 unit s X
16
5
$6
22 unit s X
14
12
10
8
6
4
2
Data from these observations
can be plotted on the graph
Clearly there is a pattern, however it is not a perfect relationship.
Through statistical inference we can estimate some general
characteristics about the relationship
2
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Principles of Microeconomics
Quantity per weekSlide 33
Price of good X
Given the observed data about Susan’s purchases:
We can estimate a line that minimizes the square of the
difference that each point [that represents two variables]
lies off the estimated line.
18
16
( Q= 10, P= $15)
14
(15, 11)
12
P = 23 - .75Q may be written
Q = 30.667- 1.333P
No single point may lie
the line, but the line is an
(20,10) of the relationship
estimate
10 P = 23 - .75Q is our estimate
8 of the relationship between the (22,7)
6 price and the quantity that Susan
(22,6)
purchases each week, ceteris paribus or
4
all other things equal
2
2
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8
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Quantity per week
Principles of Microeconomics
Slide 34
Given the observed data about Susan’s purchases:
Price of good X
and our estimated function: P = 23 - .75Q or Q = 30.67 - 1.33P,
we would predict that at a price of $10 Susan would
purchase about 17.37 units, [Q = 30.67 - 1 .33 P,
P = 10 so Q =17.37]
We observed that Susan bought 20 units
when the price was $10 so estimate is off
by a small amount [-2.63 units]
18
16
At a price of $6 our equation predicts
that 22.67 units will be purchased
14
12
10
8
6
4
2
Since we observed that she
purchased 22, we are off
by .67 units
P = 10
P=6
our estimates are not perfect, but they
give an approximation of the relationship
Q = 22.67
Q = 17.37
2
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8
10 12 14 16 18 20 22 24 26
Quantity per week
Principles of Microeconomics
Slide 35
Statistical Estimates
· The estimates are not “perfect” but they
provide reasonable estimates
· There are many statistical tools that
measure the confidence that we have in out
predictions
· these include such things as correlation,
coefficient of determination, standard errors,
t-scores and F-ratios
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Principles of Microeconomics
Slide 36
Slope & Calculus
· In economics we are interested in how a change
in one variable changes another
·
How a change in price changes sales. How a change in an
input changes output. How a change in output changes cost.
etc.
· The rate of change is measured by the slope of
the functional relationship
· by subtraction the slope was calculated as
rise over run where rise = Y = Y1 - Y2 and
run = X = X1 - X2,
slope  Y X
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Principles of Microeconomics
Slide 37
Derivative
There are still more slides on this topic
· When we have a nonlinear function, a
simple derivative can be used to calculate
the slope of the tangent to the function at
any value of the independent variable
· The notation for a derivative is written:
dY
is the change in Y " caused " by a change in X
dX
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Principles of Microeconomics
Slide 38
Summary
· a derivative is the slope of a tangent
at a point on a function
dY
· dX is the rate of change, it
measures the change in Y caused by a
change in X as the change in X
approaches 0
dY
· in economics jargon, dX [the slope or
rate of change] is the “marginal”
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Principles of Microeconomics
Slide 39