# PROBLEM #6: WAGE DIFFERENTIALS

**Introduction**

When one looks at the labor market, one sees that not all jobs pay the same
wage, nor do

all individuals earn the same wage, even when they hold the same job. Wages vary
for

numerous reasons, including job characteristics, the amount of education and
training

required, the industry and region of the country in which they occur, and
whether the

work force is unionized of not. Wage differentials between jobs is a powerful
mechanism

in the allocation of labor, as higher paying jobs tend to be more attractive to
workers than

lower paying jobs. Where shortages of workers exsist, employers often raise
wages in

order to attract more workers. However, there also appears to be wage
differentials that

are associated with race or gender of the workers involved, and not with the

characteristics of the job or the qualifications of the worker.

In this section we will calculate some of these wage differentials.

**Mathematics: Comparing Two Functions **

If and
are two functions of x , then we may graph both functions

on the same set of axes. This enables us to compare the two functions . Some
questions

we might ask are:

1. When is ?

2. When is greater than
? When is
less than ?

3. When is the distance between the two functions the greatest? What

does this mean?

4. Which function is growing faster? Why?

**Example 1:**

The cost, in dollars, of manufacturing x items is given by the function

and the revenue obtained by selling x items is given by the function

Both and
are linear functions . We graph them both on the same set of

axes, using the window [0 , 200 ]× [ 0 , 5,000 ] .

**Number of Items**

We now use the TI-83 Calculator to find the intersection
point of the two lines.

PressCALC
to use the intersect feature of the calculator.

As the graph appears on the screen, press ENTER three times , and the
coordinates of the

intersection point will appear on the bottom of the screen.

Here, the intersection point of the graphs of
and
is the point (100,2925 ) .

This means that, if you manufacture and sell 100 items, your cost and revenue
will be

exactly the same $2,925.

This intersection point can also be found algebraically ,
as follows:

when

Subtracting from both sides yields

Dividing both sides by 4.25 yields

Now , so that the
intersection point of the two curves

is (100,2925) . Notice that also R (100) = 29.25 (100) = 2925.

What is the significant of this intersection point?

The point at which Cost equals Revenue is called the Break Even Point. Here, if
the

company manufactures 100 items, the Cost and Revenue are both $2925 .

We see from the graphs of the two functions, that for x less than 100, the graph
of

is above the graph of
so that Cost is greater than Revenue and the company

loses money. At the Break Even Point, the Cost equals the Revenue, so that there
is no

profit and no loss. For x greater 100, the Revenue is greater than the Cost and
the

company is making a profit.

**Economics Problem:** Compare the income of male and
female workers as a function of

education.

Use the TI-83 and the data below to fit lines through the data points.

**Mean Monthly Income by Highest Degree Earned
and by Gender 1993**

Gender |
Years of Education |
||||

12 | 14 | 16 | 18 | 20 | |

Male |
$1,812 | $2,561 | $3,430 | $4,298 | $4,421 |

Female |
$1,008 | $1,544 | $1,809 | $2,505 | $4,020 |

Source: Statistical Abstract of the United States, 1996.

Solution : The years of schooling are on the x-axis and the mean monthly earnings
are on

the y-axis. You can simultaneously plot the mean monthly earnings for males and

females. Thereby getting two different functions on the same set of axes. Thus
the points

to graph are:

for male earnings

(12, 1812)

(14, 2561)

(16, 3430)

(18, 4298)

(20, 4421)

for female earnings

(12, 1008)

(14, 1544)

(16, 1809)

(18, 2505)

(20, 4020)

Put these values into lists **L1, L2,** and **L3**

Steps to use the TI -83 to plot the data

Press [Y=] and clear any function that is there. Then,

[STAT] [EDIT]

Enter corresponding values into **L1, L2,** and **L3.**

[2nd] [STATPLOT] [PLOT1] [on] Xlist: **L1**, Ylist: **L2**

[2nd] [STATPLOT] [PLOT2] [on] Xlist: **L1**, Ylist: **L3**

choose the second mark.

[Graph] to view the scatter plot. Adjust the window as necessary.

The two scatter plots will look as below. The lower, triangle points, represent
female

income.

Differences in earnings between male and female workers
are large. Women are earning

less than men at every educational level. Does this suggest labor-market
discrimination?

Not necessarily. Several factors can cause earnings disparities. Women may have
less

work experience than men, women may be less likely than men to work overtime or
to

choose occupations that offer jobs with high pay but long hours. Women may
choose

jobs closer to home than men. In addition, many women are concentrated in
occupations

that tend to pay lower wages. Even after taking into account these objective
factors, there

still remains a gap between the wages of males and females, that some attibute
to labor

market discrimination. In recent years, however, there has been a trend for this
gap to

decrease, especially for highly educated workers.

An inspection of the two graphs also suggests that there are different types of
lines that

best fit each of the data points. For males, a straight line seems acceptable,
but for

females the income points go up at an ever increasing rate. This suggests an
exponential

function (of the form ) is a better fit than
a straight line.

To see this we will put the linear equation for the male incomes into equation
, and

the exponential regression equation for female incomes into equation

This puts the linear regression equation into the
equation.

This puts the exponential equation into the
equation.

Press [Y=] to see the equations of the best fitting functions.

Males:

Females:

Now press graph to see these equations graphed through the points.

What do these graphs suggest regarding the importance of more education for
females as

compared with males? Does advanced education result in a proportionately larger
gain

for women than for men? How can this pattern be explained? Does this provide an

additional incentive for women to persue advanced degrees?

**Additional Problems:**

1. The male income points show an increase and then a decrease, this indicates
that

perhaps the male income can best be fit by a quadratic equation of the form

X^{2} + bX + c . Steps to use the TI-83 to graph a quadratic function.

Clear the existing functions. Then enter:

a. What is the equation of the quadratic equation.

b. Which gives a better fit, the linear or the quadratic function?

c. If the quadratic fits better, what is its significance?

2. To be supplied.

3**. Research:** Use the Statistical Abstract of the United States to collect
data on

mean monthly income by highest degree earned according to race. Are any

differences in earnings between white and black workers? Graph the data and fit
a

straight line or exponential regression line to each data set. Can you give any

explanations for the differences in earnings between black and white workers

(other than discrimination)? There have been federal equal employment laws

against job discrimination based on race for over 30 years, as well as many

voluntary efforts to improve the employment of blacks and other minorities. If

these laws were effective, how would you expect them to affect the wage

differential between workers of different races?

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