Eco311, First Midterm Exam, Spring 2018

Prof. Bill Even

1

Your Name (Please print) __ANSWER KEY TO FORM 1______________________________

Your row (Please circle) 1 2 3 4 5

Directions

Place your answers to all questions in the space provided. Organize your answer so it is easy for me to

see the logic behind your answer. Clearly circle your answer to each question so it can be easily

distinguished from your work. If the logic of your work looks correct but there is a math error, you may

receive partial credit for your answer. It’s important that you present your work in a way that is easy to

follow.

Round all numerical answers to the nearest 100th (e.g. 1.23) unless specifically told otherwise. Unless

indicated otherwise, each question is worth 4 points.

The formula sheet and a table with the standard normal CDF are attached to the last page of the exam.

Eco311, First Midterm Exam, Spring 2018

Prof. Bill Even

2

To answer the next 3 questions, consider the following relationship between annual household

spending on water, the household head’s age, and age-squared that was estimated using data from the

2016 American Community Survey.

=258 +10.9 −.10 2

1. If the household head is 30 years old, use a derivative to calculate the marginal effect of one more

year of age on water cost.

=.−. =. =

In the ACS data, heads of household range between 16 and 97 years of age.

2. What age maximizes water cost? (Your answer must be somewhere in the range of 16 to 97 years of

age).

Since the second derivative implies that this is a concave function, set the first derivative to zero and

solve for age to obtain the age that maximizes water cost.

=10.9 −.2 = 0 ==> =. .

3. What age minimizes annual water cost? (Your answer must be somewhere in the range of 16 to 97

years of age).

Since the function is concave, the minimum is at either age 16 or age 97. Comparing water cost at

the two ages reveals water cost is minimized at age 97.

4. As an alternative to the linear model above, consider the following log-linear model of annual water

cost:

ln(cost of water)= 5.4 + .023age −.0002age2

For a household head who is 30 years old, use a log-approximation to estimate the percent change in

water cost associated with an additional year of age. Be sure to give your answer as a percent and

include the percent sign in your answer (e.g. 20.32%, not .2032).

()

=. −. =. = ==> by 1.1 percent if age

increases from 30 to 31.

Eco311, First Midterm Exam, Spring 2018

Prof. Bill Even

3

Consider the following alternative to the earlier equation for the cost of water that allows cost to

depend on age, the number of children living in the household (nchild.), and the interaction between

age and nchild.

=5218 −24.8 +16 ∗ℎ −11.9 ∗(ℎ ∗)

5. Based on this equation for water cost, use a first order approximation to estimate the marginal

effect of an additional child on water cost for a person who is 40 years old with 2 children. Round

your answer to the nearest cent.

ℎ =16 −11.9 ∗ =−460 =40.

6. Based on the above equation, which of the following is true?

a. The marginal effect of age on water cost is more negative for families with more children.

b. The marginal effect of children on water cost is negative for all households since household

heads range in age from 16 to 97.

c. Both of the above are true

d. None of the above are true.

To answer the next 3 questions, consider the following data on the number of children living with the

household head. Those with 3 or more children are counted as having 3. You can ignore this issue in

answering the questions. I also rounded the probabilities to the nearest .05 to make calculations easier.

Show the basic steps behind your work for each answer.

Number of

children

0

1

2

3

Probability

.60

.20

.15

.05

5. What is the expected number of children living with a household head?

0.65

6. What is the variance of number of children living with a household head?

0.83

7. What is [( − )3] where X is number of children living with a household head and ()=?

0.86