Solutions1.8Solutions to Chapter 1 Solved Exercises
1.1Constructing demand
Checkpoint1.1.2.
Answer.
Anna’s pen demand schedule is shown in the table below:
Table1.1.3.Anna’s demand schedule for pens
\(P\)
\(Q_D\)
$15
0
$12
0
$9
1
$6
2
$3
2
Demand schedules reflect quantity values for a range of prices, even if consumer WTP has not changed, such as between \(P=\$6\) and \(P=\$3\text{.}\)
Checkpoint1.1.6.
Answer.
The consumer’s maximum willingness to pay tells us the most she will pay for an item. The demand schedule is a table that checks against this value for a given price. So if there’s a price given in the table, we get the relevant quantity by answering, "Is this price less than or equal to the maximum willingness to pay for another unit?"
Similarly, we can directly find the quantity demanded for a price listed in the table by looking at the corresponding quantity in the row. Or, considered differently, to fill in the Quantity column of a demand schedule, we can say, "What is the highest number of units for which this price is less than or equal to the consumer’s willingness to pay?"
1.2Individual demand to market demand
Checkpoint1.2.5.
Answer.
The slope of the curve is \(\frac{1}{5}\) or 0.2.
Solution.
We get to \(\frac{1}{5}\) by solving the demand function for price \(P\text{.}\)
The reason this is the slope of the demand curve is that when graphing demand, we have \(P\) (dependent variable) on the vertical axis and \(Q_{D}\) (independent variable) on the horizontal. Slope of a graph is defined as the change in the vertical axis variable divided by the change in the horizontal variable. Since the demand function gives quantity, \(Q_{D}\text{,}\) as a function of price, \(P\text{,}\) we need to find inverse demand by solving for price to get the slope of demand curve.
We get this by subtracting \(5Q_{D}\) from both sides, doing the same for \(P\text{,}\) and dividing through by 5.
Since the consumer surplus here is geometrically a triangle, we use the area of a triangle formula to solve. \(Area = \frac{1}{2}bh\) Where b is base and h is height.
The height is the difference between the vertical intercept, here 50, and the price we are interested in, here 30.
So at a price of $30, this market generates $400 in consumer surplus for consumers.
1.5Price elasticity of demand
Checkpoint1.5.1.
Answer.
The quantity demanded at \(P=10\) is \(Q_{D}=150 - 10(10)=50\text{.}\) The \(\frac{\Delta Q_D}{\Delta P}=-10\) from the demand function. (Each increase in P decreases \(Q_D\) by 10.)
So our \(\epsilon_{D}=\frac{\Delta Q_D}{\Delta P}\frac{P}{Q_D}=-10(\frac{10}{50})=-2\text{.}\)
Checkpoint1.5.2.
Answer.
Price elasticity of demand is always negative because it represents the relationship between quantity demanded and price. From the Law of Demand, we know that this is an inverse relation. If price rises, quantity demanded falls, and vice versa. So since these variables always move in opposite directions along a demand curve, the percent change in quantity is always negative for a positive percent change in price.
Checkpoint1.5.5.
Answer.
1. Because income elasticity is positive, this means that income and quantity demanded change in the same direction, so this is a normal good.
We interpret the value to mean that for each 1% change in income, quantity demanded for this good will increase by 0.5%, ceteris paribus.
2. Because cross-price elasticity is negative, this means that the quantity demanded for bagels (\(x\)) moves in the opposite direction when the price for coffee (\(y\)) changes, so these goods are complements. Higher prices for coffee means lower quantity demanded of coffee (Law of Demand). Since people consume bagels and coffee together, Less coffee demanded means fewer bagels demanded.
We interpret the value to mean that for each 1% change in the price of coffee, quantity demanded for bagels will fall by 0.75%, ceteris paribus. This is an indicator of the strength of the complementarity.
1.6Application: Price elasticity of demand and total revenue
Checkpoint1.6.7.
Answer.
We can use the definition of price elasticity to answer the first question: \(\epsilon_D = \frac{\%\Delta Q_D}{\%\Delta P}\) so if \(Q_{D}\) increased 20% when \(P\) decreased 6% we can say that \(\epsilon_D = \frac{20}{-6}=-3.33\text{.}\) Since this is less than -1, we know that Model S’s are relatively elastic.
We now can also answer the second part of the question. Total revenue would rise because the percentage increase in quantity demanded was greater than the percentage decrease in price. Hence \(TR = PQ_{D}\) must rise.
