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PPT – This is a PowerPoint presentation on fundamental math PowerPoint presentation | free to download - id: 6b818b-NDNlO

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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.

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

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

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

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

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

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

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When the values of the independent and dependent

variables are positive, we use the North East

quadrant

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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

Given the relationship, Y 6 - 2X,

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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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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?

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)?

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.

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

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

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!

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

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

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,

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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Given the function Y 6 - .5X,

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

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

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

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

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

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

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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

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

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.

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

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

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

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

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

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,

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

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