Economics Chapter 2a Study The Four Accompanying Diagrams

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interactive activity

2 Appendix

Graphs in Economics

1. Study the four accompanying diagrams. Consider the following statements and

indicate which diagram matches each statement. Which variable would appear

on the horizontal and which on the vertical axis? In each of these statements, is

the slope positive, negative, zero, or infinity?

Panel (a) Panel (b)

Panel (c) Panel (d)

a. If the price of movies increases, fewer consumers go to see movies.

b. More experienced workers typically have higher incomes than less experienced

workers.

c. Whatever the temperature outside, Americans consume the same number of

hot dogs per day.

1. a. Panel (a) illustrates this relationship. The higher price of movies causes con-

sumers to see fewer movies. The relationship is negative, and the slope is

therefore negative. The price of movies is the independent variable, and the

Solution

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d. Panel (c) illustrates this relationship. When the price of ice cream goes up, this

causes consumers to choose a close alternative, frozen yogurt. The price of ice

cream is the independent variable and the consumption of frozen yogurt is the

2. During the Reagan administration, economist Arthur Laffer argued in favor of

lowering income tax rates in order to increase tax revenues. Like most econo–

mists, he believed that at tax rates above a certain level, tax revenue would fall

because high taxes would discourage some people from working and that people

would refuse to work at all if they received no income after paying taxes. This

a. Which is the independent variable? Which is the dependent variable? On

which axis do you therefore measure the income tax rate? On which axis do

you measure income tax revenue?

b. What would tax revenue be at a 0% income tax rate?

c. The maximum possible income tax rate is 100%. What would tax revenue be at

a 100% income tax rate?

d. Estimates now show that the maximum point on the Laffer curve is (approxi-

mately) at a tax rate of 80%. For tax rates less than 80%, how would you

describe the relationship between the tax rate and tax revenue, and how is this

relationship reflected in the slope? For tax rates higher than 80%, how would

2 APPENDIX Graphs in Economics S-23

b. If the income tax rate is 0% (there is no tax), tax revenue is obviously zero.

c. If the income tax rate is 100% (all your income is taxed away), you will have

zero income left after tax. Since people are unwilling to work if they receive

no income after tax, no income will be earned. As a result, there is no income

tax revenue.

d. For tax rates less than 80%, tax rate and tax revenue are positively related and

3. In the accompanying figures, the numbers on the axes have been lost. All you

know is that the units shown on the vertical axis are the same as the units on

the horizontal axis.

Panel (a) Panel (b)

a. In panel (a), what is the slope of the line? Show that the slope is constant along

the line.

b. In panel (b), what is the slope of the line? Show that the slope is constant

3. a. In panel (a), the slope is –2. From any point on the line, moving one unit to the

right along the horizontal axis requires moving down two units along the ver-

b. In panel (b), the slope is 1

3. From any point on the line, moving three units

4. Answer each of the following questions by drawing a schematic diagram.

a. Taking measurements of the slope of a curve at three points farther and

farther to the right along the horizontal axis, the slope of the curve changes

from –0.3, to –0.8, to –2.5, measured by the point method. Draw a schematic

diagram of this curve. How would you describe the relationship illustrated in

your diagram?

Solution

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to 0.5, to 0, to –0.5, to –1.5, measured by the point method. Draw a schematic

diagram of this curve. Does it have a maximum or a minimum?

4. a. The accompanying diagram schematically shows this curve. The slope is nega–

tive throughout. That means that the curve is downward sloping. Because the

absolute value of the slope is increasing, the curve becomes steeper. The slope

is negative increasing.

Slope = –0.3

b. The accompanying diagram schematically shows this curve. The slope is posi-

Solution

2 APPENDIX Graphs in Economics S-25

5. For each of the accompanying diagrams, calculate the area of the shaded right

triangle.

100

Panel (a) Panel (b)

7. The accompanying table shows the relationship between workers’ hours of work

per week and their hourly wage rate. Apart from the fact that they receive a dif-

ferent hourly wage rate and work different hours, these five workers are other-

wise identical.

Name

Quantity of labor

(hours per week)

Wage rate

(per hour)

Athena 30 $15

Diego 36 60

Emily 32 75

a. Which variable is the independent variable? Which is the dependent variable?

b. Draw a scatter diagram illustrating this relationship. Draw a (nonlinear) curve

that connects the points. Put the hourly wage rate on the vertical axis.

c. As the wage rate increases from $15 to $30, how does the number of hours

worked respond according to the relationship depicted here? What is the average

slope of the curve between Athena’s and Boris’s data points using the arc method?

7. a . If the wage rate is greater than your opportunity cost of time, you will choose

to work. So the wage rate is the independent variable and the number of hours

worked is the dependent variable.

b. The accompanying diagram illustrates the relationship between the hourly

wage rate and the number of hours worked. Since the hourly wage rate is the

price paid for labor, economists place wages on the vertical axis—just as in

the case of other types of prices.

$75

Wage rate

(per hour)

Quantity of labor (hours per week)

28 29 30 31 32 33 34 35 36 37

c. As the wage rate increases from $15 to $30, the number of hours worked

increases by 5. The average slope of the curve between the two points is

therefore 15

Solution

2 APPENDIX Graphs in Economics S-27

8. An insurance company has found that the severity of property damage in a fire

is positively related to the number of firefighters arriving at the scene.

a. Draw a diagram that depicts this finding with number of firefighters on the

horizontal axis and amount of property damage on the vertical axis. What is

the argument made by this diagram? Suppose you reverse what is measured

on the two axes. What is the argument made then?

8. a. By drawing the diagram with number of firefighters on the horizontal axis

and amount of property damage on the vertical axis, you are assuming that

the number of firefighters is the independent variable and amount of property

damage is the dependent variable. That graph is shown here. It makes the argu–

ment that as the number of firefighters on the scene increases, the amount of

damage increases. You could also have drawn the graph with amount of prop–

Amount of

property

damage

Number of firefighters

b. The statement implies that there is a causal link between the number of fire–

fighters and the amount of property damage, and this is likely not the case. It

is instead likely that there is a third, omitted, variable that is related to both

9. This table illustrates annual salaries and income tax owed by five individu-

als. Despite receiving different annual salaries and owing different amounts of

income tax, these five individuals are otherwise identical.

Name Annual salary Annual income tax owed

Susan $22,000 $3,304

Peter 37,000 7,020

a. If you were to plot these points on a graph, what would be the average slope

of the curve between the points for Eduardo’s and Camila’s salaries and taxes

using the arc method? How would you interpret this value for slope?

Solution

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c. What happens to the slope as salary increases? What does this relationship

9. a. Annual salary is the independent variable and so is measured on the horizon-

c. The slope is positive increasing. This implies that the tax scheme is “progres-

sive”: the higher the annual salary, the greater the amount of income tax owed

per dollar of income. Therefore, the incentive to earn more and more income

becomes weaker and weaker, since more of the additional income earned is

owed as income taxes.

WORK IT OUT Interactive step-by-step help with solving this

problem can be found online.

10. Studies have found a relationship between a country’s yearly rate of eco–

nomic growth and the yearly rate of increase in airborne pollutants. It

is believed that a higher rate of economic growth allows a country’s resi-

dents to have more cars and travel more, thereby releasing more airborne

pollutants.

a. Which variable is the independent variable? Which is the dependent

variable?

b. Suppose that in the country of Sudland, when the yearly rate of economic

growth fell from 3.0% to 1.5%, the yearly rate of increase in airborne pol-

lutants fell from 6% to 5%. What is the average slope of a nonlinear curve

between these points using the arc method?

c. Assume that when the yearly rate of economic growth rose from 3.5% to

4.5%, the yearly rate of increase in airborne pollutants rose from 5.5% to

7.5%. What is the average slope of a nonlinear curve between these two

points using the arc method?

d. How would you describe the relationship between the two variables here?

–1.5 = 2

c. The change in the growth rate is +1. The change in the rate of increase in

Solution