1 working with graphs. 2 graphs in general: a graph is a visual representation of the relationship...

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Working With GraphsWorking With Graphs

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Graphs In General:Graphs In General:

A graph is a visual A graph is a visual representation of the representation of the relationship between two orrelationship between two or

more variables.more variables.

We will deal with just two We will deal with just two variables at a time.variables at a time.

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1. 1. Independent variable: Independent variable: This is This is the variable that influences the the variable that influences the dependent variable. dependent variable. (X variable)(X variable)

2. 2. Dependent variable: Dependent variable: Its value Its value is determined by the is determined by the independent independent variable. variable. (Y (Y variable)variable)

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3.3. We say that the We say that the dependent dependent variable variable is ais a

function of the function of the independent independent variablevariable::

Y = f(X)Y = f(X)

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The Axis of a Graph:The Axis of a Graph:

Dependent VariableDependent Variable

(Y-axis)(Y-axis)

Independent Variable (Independent Variable (X-axis)X-axis)

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Direct Relationships:Direct Relationships:

A person's weight and height are A person's weight and height are often related. often related.

If we sample 1000 people and If we sample 1000 people and measure their weight and height measure their weight and height we would we would probably find that as probably find that as weight increases so does height.weight increases so does height.

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HeightHeight

WeightWeight

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There is a direct relationship between There is a direct relationship between height and weight.height and weight.

Have a direct relationship when:Have a direct relationship when:

indep. variable indep. variable dep. variable dep. variable

indep. variable indep. variable dep. variable dep. variable

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Inverse Relationships:Inverse Relationships:

There is strong evidence There is strong evidence indicating that as price rises indicating that as price rises for a specific commodity, the for a specific commodity, the amount purchased decreases.amount purchased decreases.

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Price per UnitPrice per Unit

DemandDemand

CurveCurve

Quantity Purchase per Unit TimeQuantity Purchase per Unit Time

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There is an inverse relationship There is an inverse relationship between price per unit and the quantity between price per unit and the quantity purchased per unit of time.purchased per unit of time.

Have an inverse relationship when:Have an inverse relationship when:

(1) indep. variable (1) indep. variable dep. variable dep. variable

(2) indep. variable (2) indep. variable dep. variabledep. variable

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Complex Relationships:Complex Relationships:

Evidence suggests that income Evidence suggests that income from wages increases up to a from wages increases up to a certain age, and then decreases certain age, and then decreases until death.until death.

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Income from Wages ($)Income from Wages ($)

AgeAge

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There is a direct relationship There is a direct relationship between wage income and age between wage income and age up to a certain point known as up to a certain point known as retirement,retirement,

then an inverse relationship then an inverse relationship exists from retirement to the exists from retirement to the individuals expiration date.individuals expiration date.

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Income from All Sources ($)Income from All Sources ($)

AgeAge

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AgeAge

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AgeAge

What should the slope of this line be equal to at the minimum?

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Social Security Issues:Social Security Issues:

True/False:True/False:

Social Security was NEVER Social Security was NEVER intended to provide benefits intended to provide benefits sufficient to be the sole source sufficient to be the sole source of retirement income.of retirement income.

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Current “Social Security” tax rate: Current “Social Security” tax rate:

7.65 percent7.65 percent

OASDI TAX:OASDI TAX: Old Age, Survivors, and Old Age, Survivors, and Disability Insurance. Disability Insurance.

The 2000The 2000 rate of tax is rate of tax is 6.26.2 percent to a percent to a taxable wage limit of taxable wage limit of $76,200$76,200..

The maximum tax an employee may The maximum tax an employee may pay is pay is $4,501.20$4,501.20

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HI TAX:HI TAX: Federal Hospital Insurance. Federal Hospital Insurance.

The 2000 rate of tax is The 2000 rate of tax is 1.45 1.45 percent percent without a wage limitwithout a wage limit. .

The maximum tax is therefore The maximum tax is therefore unlimitedunlimited

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Social Security covered 58 percent Social Security covered 58 percent of the work force in 1940, covered of the work force in 1940, covered over 90 percent in 1990.over 90 percent in 1990.

Over this 50 yr. period, REAL Over this 50 yr. period, REAL benefits have increased and benefits have increased and coverage has been extended to coverage has been extended to spouses, widows, and dependents.spouses, widows, and dependents.

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Today, elderly as a group have Today, elderly as a group have lower poverty rates than the lower poverty rates than the general population, and about general population, and about the same per capita income.the same per capita income.

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Greater than 90 percent of all persons Greater than 90 percent of all persons 65 or older receive Social Security.65 or older receive Social Security.

On average, SS equals 38 percent of On average, SS equals 38 percent of total income received by elderly total income received by elderly

households.households.

For 25 percent of older households, For 25 percent of older households, SS equals 90 percent of family SS equals 90 percent of family income.income.

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For 15 percent of older For 15 percent of older households, SS equals 100 percent households, SS equals 100 percent of family income.of family income.

To maintain pre-retirement living To maintain pre-retirement living standards, middle and upper standards, middle and upper income households must have income households must have additional income from employer additional income from employer pensions or private savingspensions or private savings

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Historical Social Security Historical Social Security Tax Rates:Tax Rates:

Before 1950, SS rate = Before 1950, SS rate = 1.01.0 percent paid percent paid by both the employer and employee. by both the employer and employee.

This covered retirement only, no This covered retirement only, no disability or disability or Medicare.Medicare.

Maximum earnings taxed prior to 1950 Maximum earnings taxed prior to 1950 = = $3,000$3,000

Maximum tax paid prior to 1950 = Maximum tax paid prior to 1950 = $30 $30 per employee, $30 per employerper employee, $30 per employer

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In 1970, the maximum retirement tax paid In 1970, the maximum retirement tax paid was was $284.70$284.70. Matched by employer.. Matched by employer.

In 1990, SS retirement tax rate = In 1990, SS retirement tax rate = 5.60 5.60 percentpercent, Maximum earnings taxed in 1991 , Maximum earnings taxed in 1991 = = $51,300$51,300

Maximum tax paid in 1991 = Maximum tax paid in 1991 = $2,872.80 per $2,872.80 per employee. employee. Matched by employerMatched by employer

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In 2000, SS retirement tax rate = In 2000, SS retirement tax rate = 5.30 percent5.30 percent, Maximum earnings , Maximum earnings taxed in 2000 = taxed in 2000 = $76,200$76,200

Maximum tax paid in 2000 = Maximum tax paid in 2000 = $4,038.60 per employee. $4,038.60 per employee. Matched Matched by employerby employer

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Future of Social SecurityFuture of Social Security

Funds in the SS trust fund will Funds in the SS trust fund will peak in 2030 at peak in 2030 at $12 trillion (no $12 trillion (no cash, all gov. bonds!)cash, all gov. bonds!)

I will be 73 years old.I will be 73 years old.

You will be ? years old.You will be ? years old.

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After After 20302030, these trust fund assets , these trust fund assets decrease rapidly, and will be equal decrease rapidly, and will be equal to zero in to zero in 20462046 at the current SS at the current SS tax rates.tax rates.

In 2046, I probably will be in a state In 2046, I probably will be in a state of mind such that I won't care!of mind such that I won't care!

You will be ? years old.You will be ? years old.

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As the population of our country As the population of our country continues to age, the ratio of continues to age, the ratio of (workers / beneficiaries) will (workers / beneficiaries) will decrease.decrease.

The W/B ratio is expected to The W/B ratio is expected to remain stable between 1989 and remain stable between 1989 and 2010 but will then decrease.2010 but will then decrease.

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YearYear W/BW/B

"Baby Boomers" "Baby Boomers" startstart

20102010 3.53.5 hitting 65 in hitting 65 in 20102010

20202020 2.72.7

> 2020> 2020 2.12.1

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Building up the trust fund Building up the trust fund NOWNOW will reduce the expected tax will reduce the expected tax burden on future workers burden on future workers (YOU) (YOU) by making Baby by making Baby BoomersBoomers (ME) (ME) pay higher taxes pay higher taxes to partially finance their own to partially finance their own retirement benefits.retirement benefits.

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Constructing A GraphConstructing A Graph

We start with a horizontal number We start with a horizontal number line:line:

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1.1. The points on the line divide the The points on the line divide the line into segments.line into segments.

2.2. All the line segments are equally All the line segments are equally spacedspaced

3.3. Numbers associated with the points Numbers associated with the points increase in value from left to right.increase in value from left to right.

4.4. Use a distance, so many points, to Use a distance, so many points, to represent a quantity.represent a quantity.

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Add a Vertical Number Add a Vertical Number Line:Line:

1. Construct a vertical number line.1. Construct a vertical number line.

2. Points divide the line into equal2. Points divide the line into equal

segments.segments.

3. Numbers associated with points3. Numbers associated with points

increase in value from the bottomincrease in value from the bottom

to top.to top.

4. The scale can be different from the4. The scale can be different from the

horizontal number line.horizontal number line.

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Add a Vertical Number Add a Vertical Number Line:Line:

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To Make A Graph:To Make A Graph:

1. The vertical and horizontal 1. The vertical and horizontal number lines must intersect at number lines must intersect at each others zero point. each others zero point.

2. They must be perpendicular.2. They must be perpendicular.

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The vertical and horizontal number The vertical and horizontal number lines should look like the lines should look like the illustration below:illustration below:

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3.3. Result: Result: We get a set of We get a set of coordinate axis, or a coordinate axis, or a coordinate number system. coordinate number system.

e.g. Sighting in a rifle scope on e.g. Sighting in a rifle scope on the range.the range.

How would you callout the location

of this three shot group?

X-Axis

Y-Axis

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4. With a graph, you need two 4. With a graph, you need two numbers to specify a single numbers to specify a single point point

OROR

When you see a point on a When you see a point on a graph, you know that point graph, you know that point represents two numbers !represents two numbers !

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BASICS YOU NEED TO KNOW BASICS YOU NEED TO KNOW ABOUT GRAPHING AND THE ABOUT GRAPHING AND THE

COORDINATE NUMBER SYSTEMCOORDINATE NUMBER SYSTEM

Axis defined:Axis defined: The vertical number line is reserved The vertical number line is reserved

for the for the DependentDependent variable and variable and is is referred to as thereferred to as the Y AXIS Y AXIS..

The horizontal number line is referred The horizontal number line is referred

to as the to as the X AXIS X AXIS and is and is reserved for reserved for the the IndependentIndependent variable. variable.

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The origin and points on The origin and points on the graphthe graph The point of intersection of the The point of intersection of the

two number lines is referred to two number lines is referred to asas the the ORIGINORIGIN..

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Point A represents two numbers: A Point A represents two numbers: A value for x and a value for y.value for x and a value for y.

Point A

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The origin and points on The origin and points on the graphthe graph Every point on a graph Every point on a graph

represents a pair of observations represents a pair of observations of x and y.of x and y.

(x,y)(x,y)

In this class, y will often In this class, y will often represent price and x will often represent price and x will often represent quantity.represent quantity.

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The SlopeThe Slope

1. Slope = change in Y values / change in 1. Slope = change in Y values / change in X valuesX values

= (y= (y11 - y - y00) / (x) / (x11 - x - x00) )

= RISE / RUN= RISE / RUN

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The SlopeThe Slope

PricePrice

8 8

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22 3 3 Quantity demanded per Quantity demanded per unit timeunit time

x1, y1)

x0, y0)

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The SlopeThe Slope

2. As2. As X X goes from 2 to 3,goes from 2 to 3,

Y Y goes from 8 to 6.goes from 8 to 6.

3. 3. Y = RISE = (TO - FROM) = 6 - 8 = Y = RISE = (TO - FROM) = 6 - 8 = -2 -2

X = RUN = (TO - FROM) = 3 - 2 = X = RUN = (TO - FROM) = 3 - 2 = 11

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The SlopeThe Slope

4. SLOPE = 4. SLOPE = Y / Y / X = -2 / 1 = -2X = -2 / 1 = -2

5. The slope of a straight line is 5. The slope of a straight line is CONSTANT.CONSTANT.

Class ExerciseClass Exercise

1 working with graphs. 2 graphs in general: a graph is a visual representation of the relationship...

Documents

y variable slide

variable indep

age slide

x variable

unlimited slide

fx slide

heightweight slide

unit time slide