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This is a PowerPoint presentation on fundamental math

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R. Larry Reynolds (Boise State University) This is a PowerPoint presentation on fundamental math tools that are useful in principles of economics. – PowerPoint PPT presentation

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Title: This is a PowerPoint presentation on fundamental math


1
ã R. Larry Reynolds (Boise State University)
This is a PowerPoint presentation on fundamental
math tools that are useful in principles of
economics. A left mouse click or the enter
key will add an element to a slide or move you
to the next slide. The backspace key will take
you back one element or slide. The escape key
will get you out of the presentation.
2
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

3
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

4
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

5
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

6
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

7
Equations -- Graphs Cartesian system
The North East Quadrant (NE), where X gt 0, Y gt 0
both X and Y are positive numbers
(X,Y) where Xlt0, Ygt0
(X,Y) where Xgt0 and Ylt0
(X,Y) where Xlt0 and Ylt0
(Left click mouse to add material)
8
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!
(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
9
Given the relationship, Y 6 - 2X,
(Left click mouse to add material)
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!
(0, 6)
when X 1 then Y 4
(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
10
(Left click mouse to add material)
Given a relationship, Y 6 - .5X
(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?
11
For a relationship, Y 1 2X
When X0, Y1 (0,1)
When X 1, Y 3
slope 2
(1,3)
When X 2, Y 5
(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 -1 .5X
This function shows that for a 1 unit increase
in X, Y increases one half unit
(Left click mouse to add material)
12
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)?

13
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.

14
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

15
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

16
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!
17
For the relationship, Q 220 - 5P, the
relationship can be graphed ...
PRICE
When the price is 44, 0 unit will be bought at
a price of 0, 220 units will be bought.
44
Notice that we have drawn the graph backwards,
Pindependent 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!
QUANTITY
(Left click mouse to add material)
18
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

19
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

20
Slope of a Line
Y 6 -.5X
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 DXin
this case, -1
so, slope is -1/2 or -.5
DX is the run 2,
21
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

22
(Left click mouse to add material)
Given the function Y 6 - .5X,
23
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

24
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

25
The maximum output of T possible with all
inputs and existing technology is 10 units with 6
units of F
tons of tomatoes
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)
FERTILIZER
26
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

27
Given T f (F, . . . ), MPf DTPf/DF
DTPf 1, DF 3 1/3 _at_ .33 this is
an approximation because DFgt1
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
1 6
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
28
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
Q 220 - 5P
subtract 220 from both sides
-220 Q -5P
divide every term in both sides by -5
or, P 44 - .2 Q
The equation P 44 - .2Q is the same as
Q 220 - .5P
(Left click mouse to add material)
29
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

30
An Example
  • Hypothesis the amount of good X Q that Susan
    purchases is determined by the price of the good
    Px, Susanss income Y, prices of other
    related goods Pr and Susans preferences.
  • Q fi (Px,Y, Pr, preferences, . . .)
  • . . . indicates there are other variables that
    are not included in the equation

31
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.

32
Empirical verification
  • To test the model, we would like to observe
    Susans 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 Susans purchases

33
During a 5 week period, Susan was observed making
the following purchases
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
34
We can estimate a line that minimizes the square
of the difference that each point that
represents two variables lies off the estimated
line.
P 23 - .75Q may be written Q 30.667- 1.333P
No single point may lie the line, but the line is
an estimate of the relationship
P 23 - .75Q is our estimate of the
relationship between the price and the quantity
that Susan purchases each week, ceteris paribus
or all other things equal
35
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
At a price of 6 our equation predicts that 22.67
units will be purchased
Since we observed that she purchased 22, we
are off by .67 units
our estimates are not perfect, but they give an
approximation of the relationship
36
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

37
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,

38
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

39
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
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