1. Principles of Microeconomics
Fall '97
This is a PowerPoint presentation on fundamental math
tools that are useful in principles of economics.
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Principles of Microeconomics
ã R. Larry Reynolds

2. Principles of Microeconomics
Fall '97
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

3. Principles of Microeconomics
Fall '97
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

4. Principles of Microeconomics
Fall '97
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

5. Principles of Microeconomics
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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 = a ± mX [the function will graph as a straight line]
When X = 0, then Y is “a”
for every 1 unit change in X, Y changes by “± m”
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6. Principles of Microeconomics
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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

7. Principles of Microeconomics
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Equations -- Graphs [Cartesian system]
Y>0 +1 +2 +3
The North East Quadrant
(NE), where X > 0, Y > 0
{both X and Y are positive
numbers}
The X axis
[horizontal]
(X,Y) where
X<0, Y>0
X<0 -3 -2 -1
X > 0 +1 +2 +3
Y<0 -1 -2 -3
(X,Y)
where X<0 and Y<0
(X,Y)
where X>0 and Y<0
(Left click mouse to add material)
The Y axis [vertical]
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Principles of Microeconomics

8. Principles of Microeconomics
Fall '97
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
up {+5} units! 1 2 3 4 5 6
(X, Y)
(3, 5)
Right {+1} one
and up {+6} six
(1,6)
(5, 1)
(2.5, 3.2)
Right 5 and up 1
to the right 2.5 units
and up 3.2 units
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9. Principles of Microeconomics
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Given the relationship, Y = 6 - 2X,
(Left click mouse to add
material) 1 2 3 4 5 6 A B
when X = 0 then Y = 6
[this is Y-intercept] Y
sets of (X, Y)
A line that slopes from
upper left to lower right
represents an inverse or
negative relationship, when the
value of X increases, Y
decreases!
when X = 1
then Y = 4
(0, 6)
(1, 4)
(2, 2)
(3, 0)
When X = 2, then Y = 2
The relationship for all positive
values of X and Y can be illustrated
by the line AB
When X = 3, Y = 0,
[this is X-intercept] X
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10. Principles of Microeconomics
Fall '97 1 2 3 4 5 6 Y X
(Left click mouse to add material)
Given a relationship, Y = X
(0,6)
(1,5.5)
(2, 5)
(4,4)
(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.
What is the X-intercept?
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11. Principles of Microeconomics
Fall '97 1 2 3 4 5 6 Y X
For a relationship, Y = 1 + 2X
When X=0, Y=1 (0,1)
When X = 1, Y = 3
slope = +2
(1,3)
When X = 2, Y = 5
run +1
rise +2
(2,5)
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 = X
This function shows that for a 1
unit increase in X, Y increases
one half unit
slope = + 1 2
run +2
rise +1 -1
(Left click mouse to add material)
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12. Principles of Microeconomics
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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 = X
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13. Equations in Economics
Fall '97
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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14. Principles of Microeconomics
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Relationship [cont ]
Q = fp (P) {Q is a function of P}
Example: Q = P
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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15. Principles of Microeconomics
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Q = P
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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16. Principles of Microeconomics
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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!
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17. Principles of Microeconomics
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PRICE
For the relationship, Q = P,
the relationship can be graphed ... $5 10 15 20 25 30 35 40 45 50 55
When the price is $44, 0 unit will be bought;
at a price of $0, 220 units will be bought. 44
Demand
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! 70
At P=$30,
Q = 70
170
At a price of $10, the
the quantity is 40 80
120
160
200
240
280
QUANTITY
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18. Principles of Microeconomics
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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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19. Slope of a function or “line”
Fall '97
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
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20. Principles of Microeconomics
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Slope of a Line
Y = 6 -.5X
DY= -1
DX = 2
as the value of X
increases from 2 to 4,
the value of Y
decreases from
5 to 4
DY is the rise [or change in Y caused by DX]{in this case, -1}
so, slope is -1/2 or -.5
DX is the run {+2},
slope is
rise
run
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21. Principles of Microeconomics
Fall '97
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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22. Principles of Microeconomics
Shifts
right
an increase in the
function would
represent an
increase in the
intercept [from 6
to a larger number]
Fall '97
(Left click mouse to add material) 1 2 3 4 5 6 Y X
the function shifts and its
slope also changes
Given the function Y = X,
Just the slope
changes
{in this case, an increase in the absolute
value of .5 to -1.8}
Y” = X [x intercept = 3.3]
A decrease in the
function would be
Y’ = X
shifts
left
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23. Principles of Microeconomics
Fall '97
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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24. Principles of Microeconomics
Fall '97
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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25. Principles of Microeconomics
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The maximum output of T possible with all inputs
and existing technology is 10 units with 6 units of F
tons of
tomatoes
TPf
With the 3rd unit of F,
T increases to 9
With 2 units of F, the output of T
increases to 8
With 1 unit of Fertilizer [F], we get
6 tons
The increase in tomatoes [DT] “caused” by DF
is +3, this is the slope
With no fertilizer we get 3 tons of tomatoes
use of more F causes the tomatoes to “burn” and
output declines
(Left click mouse to add material) 1 2 3 4 5 6 7 8 9
FERTILIZER
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26. Slope and Marginal Product
Fall '97
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 DTPf “caused” by the DF
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27. Principles of Microeconomics
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Given: T = f (F, ), MPf = [DTPf/DF]
DTPf = +1, DF = +3; +1/+3 @ .33
[this is an approximation because DF>1]
TPf
DTPf = -1, DF = +2; -1/+2 = -.5
Fertilizer [F] Tomatoes [T]
0 3
MPf [slope] 3
{technically,
this is
between 0
and the first
unit of F} +3 3
run=1
rise = +3
2 8 2
3 9 1
6 10
.33
rise/run =+3
8 9
-.5 [a negative slope!]
DTPf = +3, DF = +1; +3/+1 = 3 [slope = +3]
DTPf = +2, DF = +1; +2/+1 = 2
DTPf = +1, DF = +1; +1/+1 = 1 1 2 3 4 5 6 7 8 9
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28. Principles of Microeconomics
Fall '97
Given a functional relationship such as: Q = P,
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
Q = P
subtract 220 from both sides
-220
Q = -5P
divide every term in both sides by -5 -5
44 - 1 5
Q = 1P
or, P = Q
The equation P = Q
is the same as Q = P
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(Left click mouse to add material)

29. How do economists estimate relationships?
Fall '97
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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30. Principles of Microeconomics
Fall '97
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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31. Principles of Microeconomics
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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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32. Empirical verification
Fall '97
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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33. Principles of Microeconomics
Fall '97
During a 5 week period,
Susan was observed
making the following
purchases
Quantity per week
Price of good X 2 4 6 8 10 12 14 16 18
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 4 6 8 10 12 14 16 18 20 22
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34. Principles of Microeconomics
Fall '97
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.
Quantity per week
Price of good X 2 4 6 8 10 12 14 16 18
P = Q may be written
Q = P
( Q= 10, P= $15)
No single point may lie
the line, but the line is an
estimate of the relationship
(15, 11)
(20,10)
(22,7)
(22,6)
P = Q is our estimate
of the relationship between the
price and the quantity that Susan
purchases each week, ceteris paribus or
all other things equal
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35. Principles of Microeconomics
Fall '97
Given the observed data about Susan’s purchases:
and our estimated function: P = Q or Q = P,
we would predict that at a price of $10 Susan would
purchase about units, [Q = P,
P = 10 so Q =17.37]
Quantity per week
Price of good X 2 4 6 8 10 12 14 16 18
We observed that Susan bought 20 units
when the price was $10 so estimate is off
by a small amount [-2.63 units]
At a price of $6 our equation predicts
that units will be purchased
Since we observed that she
purchased 22, we are off
by .67 units
Q = 17.37
P = 10
P = 6
Q = 22.67
our estimates are not perfect, but they
give an approximation of the relationship
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36. Statistical Estimates
Fall '97
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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37. Principles of Microeconomics
Fall '97
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 = DY = Y1 - Y2 and run = DX = X1 - X2,
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38. Derivative There are still more slides on this topic
Fall '97
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:
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39. Principles of Microeconomics
Fall '97
Summary
a derivative is the slope of a tangent at a point on a function
is the rate of change, it measures the change in Y caused by a change in X as the change in X approaches 0
in economics jargon, [the slope or rate of change] is the “marginal” dY dX dY dX
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