# 1 Appendix. Working With Numbers and Graphs - Sample Problems

Contents

1. Working With Numbers

1. Percentage Change. Last year you were paid \$10 per hour as a teaching assistant. This year you received a \$0.50 per hour raise. By what percentage did your wage rate increase? If the average inflation rate over the last year was 8 percent, are you better or worse off?

Answer: Your wage rate increased by 5 percent = 100 • (\$10.50 - \$10.00)/ \$10.00. The second part of the question relates to the Chapter 1 discussion on Interest Rates. If the average inflation rate was 8 percent you are likely worse off. The average level of prices increased more than your wage rate increased.

2. Percentage Change. Nominal gross domestic product (GDP) is the product of the average level of prices (P) times total output in an economy (Q), or GDP = P • Q. If nominal GDP increased by 5 percent while the average level of prices increased by 6 percent, what was the change in total output?

Answer: An approximation for decomposing growth rates is that the percent change in GDP equals the percent change in prices plus the percent change in output. Even though gross domestic product was reported to have grown by 5 percent, when we correct for price inflation we find that total output declined by 1 percent.

3. Marginal Cost. You want to buy a hamburger and a small soda. If you buy a hamburger for \$1.60, the soda will cost \$0.80. If you buy the Super Value Deal, a hamburger and 20 oz. soda will cost you \$2.50. What is the marginal cost of the additional 8 ounces of soda? Should you buy the Super Value Deal?

Answer: The total cost of the burger and 12 oz. coke is \$2.40. For \$2.50 you get the burger and 20 oz. soda. The marginal cost of the additional 8 oz. of soda is \$0.10. You should buy the Super Value Deal only if the marginal benefit you get from consuming the additional 8 ouces of soda exceeds the marginal opportunity cost. In other words, is there something better that you can spend that 10 cents on?

2. Working With Graphs

1. Draw the table below as a graph with X on the horizontal axis and Y on the vertical axis.

 A B C D E X 0 1 2 3 4 Y 1 2 5 10 17

Does this graph show a direct or inverse relationship between X and Y?

Answer: It is a direct or positive relationship because Y increases as X increases at all points.

What is the slope of the line between the points A and B? What is the slope between C and D? Is this a straight line?

Answer: The slope of a line is the rise (change in Y) divided by the run (change in X). Between A and B, the slope is 1 (change in Y or +1 divided by change in X of +1). Between C and D the slope is 5 (change in Y of +5 divided by change in X of +1). It is not a straight line. The curve is positively sloped with increasing slope.

2. Curves can be represented in algebraic form by an equation:

Y = 20 - 2 • X

Fill in the blanks in the following table by solving the equation for the values of Y for each value of X given:

 X 0 1 2 3 4 5 Y 20 18

Draw the table as a graph with X on the horizontal axis and Y on the vertical axis. Does the curve illustrate a positive (direct) or negative (inverse) relation between X and Y? What is the slope of this line between the first and second points (i.e., between X=0 and X=1)? Between the second and third? Is this a straight line?

 X 0 1 2 3 4 5 Y 20 18 16 14 12 10

There is an inverse (negative) relationship between X and Y: as X rises Y falls.
The slope is -2 between all points.
It is a straight line since it has constant slope.

3. If you measured income on the horizontal axis and consumption expenditures on the vertical axis, what would be the slope of a curve that showed the relation between these two variables?

Answer: If increased income leads to an increase in consumption expenditures, then the slope of the line should be positive. We could test this theory by plotting income and consumption statistics published by the Bureau of Economic Analysis over a period of years in which real income is increasing.