Question

In 2010, Americans smoked 315 billion cigarettes, or 15.75 billion packs of cigarettes. The average retail price (including taxes) was about \(5.00 per pack. Statistical studies have shown that the price elasticity of demand is -0.4, and the price elasticity of supply is 0.5.

1. Using this information, derive linear demand and supply curves for the cigarette market.

2. In 1998, Americans smoked 23.5 billion packs of cigarettes, and the retail price was about \)2.00 per pack. The decline in cigarette consumption from 1998 to 2010 was due in part to greater public awareness of the health hazards from smoking, but was also due in part to the increase in price. Suppose that the entire decline was due to the increase in price. What could you deduce from that about the price elasticity of demand?

Short answer

1. The demand curve will be Q = 22.05 – 1.26P, and the supply curve will be Q = 23.65 – 1.58P.

2. The price elasticity of demand will be0.22.

Step-by-step solution

Explanation for part (a)

Let the demand curve be Q = a + bP, where b is the slope, i.e., the change in quantity by changing the price; a is the intercept.

The price elasticity of demand will be:

$$\begin{gathered} E_D=\frac{△Q}{△P}\times \frac{P}{Q} \\ P=\$5 \\ Q=15.75 \\ {E}_D=-0.4 \\ -0.4=\frac{△Q}{△P}\times \frac{5}{15.75} \\ \frac{△Q}{△P}=\frac{-0.4\times 15.75}{5} \\ =-1.26 \\ b=-1.26 \end{gathered}$$

Substituting Q, P, and b value to get a, therefore:

15.75 = a - 1.26 (5)

a = 15.75 + 6.3

a = 22.05

The demand curve will be Q = 22.05 – 1.26P.

Let the supply curve be Q = c + dP, where c is the slope, i.e., the change in quantity by changing the price; d is the intercept.

The price elasticity of supply will be:

$$\begin{gathered} E_S=\frac{△Q}{△P}\times \frac{P}{Q} \\ P=\$5 \\ Q=15.75 \\ {E}_S=0.5 \\ 0.5=\frac{△Q}{△P}\times \frac{5}{15.75} \\ \frac{△Q}{△P}=\frac{0.5\times 15.75}{5} \\ =1.58 \\ c=1.58 \end{gathered}$$

Substitute Q, P, and c values to get a, therefore:

15.75 = c - 1.58(5)

c = 15.75 + 7.9

c = 23.65

The supply curve will be Q = 23.65 – 1.58P.

Explanation for part (b)

The price elasticity of demand is calculated below:

$$\begin{gathered} E_D=\frac{△Q}{△P}\times \frac{P}{Q} \\ =\frac{Q_2-Q_1}{P_2-P_1}\times \frac{P_1}{Q_1} \\ {Q}_1=23.5 \\ {Q}_2=15.75 \\ {P}_1=\$2 \\ {P}_2=\$5 \\ {E}_D=\frac{15.75-23.5}{5-2}\times \frac{2}{23.5} \\ =\frac{-7.75}{3}\times \frac{2}{23.5} \\ =-0.22 \\ \left|E_D\right|=0.22 \end{gathered}$$

The price elasticity of demand will be 0.22; the demand is inelastic.