This is a PowerPoint presentation on fundamental math

1
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.
ã R. Larry Reynolds
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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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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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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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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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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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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
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(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?
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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
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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 -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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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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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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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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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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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
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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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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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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,
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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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(Left click mouse to add material)
Given the function Y 6 - .5X,
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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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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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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
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FERTILIZER
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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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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
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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 sidesby -5
or, P 44 - .2 Q
The equation P 44 - .2Q is the same as
Q 220 - .5P
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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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DerivativeThere 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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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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Can slope vary along a curve?

• Yes, the slope of a curve can vary along the curve

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Y
4
3
A
2
Y2
1
X
X30
10
20
30
40
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Y
20
15
A
Y
10
-10
5
X
X50
25
50
75
100
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END
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