Let the price elasticity of demand for a soft drink be - 2. In the year 2005 , the per capita consumption of soft drinks was about 500 cans per person, and the average price was $\$ 1.00$ per can. If we suppose that demand for the soft drink is linear, $Q ^ { d } = a - b P$, where $a$ and $b$ are constants, $Q ^ { d }$ is quantity demanded and $P$ is price, an estimate of the demand equation could be:
A) $Q ^ { d } = 100 - 2 P$
B) $Q ^ { d } = 1500 - 2 P$
C) $Q ^ { d } = 1500 - 1000 P$
D) $Q ^ { d } = 1000 - 1500 P$

Question

Quizplus · Accepted Answer

We can use the midpoint method formula to estimate the demand equation:

$$
\begin{aligned} 	ext{Price Elasticity of Demand} &= \frac{	ext{Percent Change in Quantity Demanded}}{	ext{Percent Change in Price}} \ &= \frac{\frac{\Delta Q/Q}{\Delta P/P}}{-2} \ &= \frac{-\Delta Q}{\Delta P} \cdot \frac{P}{Q} \ &= \frac{	ext{Change in Quantity Demanded}}{	ext{Change in Price}} \cdot \frac{	ext{Average Price}}{	ext{Average Quantity}} \end{aligned}
$$

We are given that the price elasticity of demand is -2, which means that a 1% increase in price will cause a 2% decrease in quantity demanded. Using the given data for 2005, we can plug in the values to solve for the constant terms:

$$
\begin{aligned} -2 &= \frac{	ext{Change in Quantity Demanded}}{	ext{Change in Price}} \cdot \frac{	ext{Average Price}}{	ext{Average Quantity}} \ &= \frac{(Q_2 - Q_1)/((Q_1 + Q_2)/2)}{(P_2 - P_1)/((P_1 + P_2)/2)} \cdot \frac{(P_1 + P_2)/2}{(Q_1 + Q_2)/2} \ &= \frac{(Q_2 - Q_1) \cdot (P_1 + P_2)}{(Q_1 + Q_2) \cdot (P_2 - P_1)} \end{aligned}
$$

Plugging in the values, we get:

$$
\begin{aligned} -2 &= \frac{(Q_2 - Q_1) \cdot 1}{(Q_1 + Q_2) \cdot 0.5} \ -4(Q_1 + Q_2) &= Q_2 - Q_1 \ -3Q_1 &= 3Q_2 \ Q_2 &= 2Q_1 \end{aligned}
$$

So, in 2005, we have:

$$
\begin{aligned} Q_1 &= 500 \ P_1 &= 1.00 \end{aligned}
$$

We can then use the demand equation to solve for the constant terms:

$$
\begin{aligned} Q ^ d &= a - bP \ Q_1 &= a - bP_1 \ 500 &= a - b(1.00) \ a &= 500 + b \end{aligned}
$$

Let's now find the value for $Q_2$ (quantity demanded in 2006):

$$
\begin{aligned} -2 &= \frac{(Q_2 - Q_1) \cdot 1}{(Q_1 + Q_2) \cdot 0.5} \ -4(Q_1 + Q_2) &= Q_2 - Q_1 \ 3Q_2 &= -3Q_1 \ Q_2 &= -500 \end{aligned}
$$

Since we cannot have a negative value for quantity demanded, it is clear that our assumption of linear demand is not reasonable for our purposes.

However, if we still want to proceed with the given answer choices, we can plug in the values for $Q_1$ and solve for $b$:

$$
\begin{aligned} Q_1 &= a - bP_1 \ 500 &= a - b(1.00) \ a &= 500 + b \end{aligned}
$$

Substitute $a = 500 + b$ into the demand equation and use the price and quantity information from 2005 to solve for $b$:

$$
\begin{aligned} Q ^ d &= a - bP \ 500 &= (500 + b) - b(1.00) \ b &= -1000 \end{aligned}
$$

Therefore, the demand equation is:

$$
\boxed{Q ^ d = 1500 - 1000P}
$$

Let the Price Elasticity of Demand for a Soft Drink $1.00\$ 1.00$1.00

Let the Price Elasticity of Demand for a Soft Drink $\$ 1.00$