Question

The table below shows the retail price and sales for instant coffee and roasted coffee for two years.

1. Using these data alone, estimate the short-run price elasticity of demand for roasted coffee. Derive a linear demand curve for roasted coffee.

2. Now estimate the short-run price elasticity of demand for instant coffee. Derive a linear demand curve for instant coffee.

3. Which coffee has the higher short-run price elasticity of demand? Why do you think this is the case?

Short answer

a. The price elasticity of demand in year 1 will be -0.43, in year 2 will be -0.38, and arc elasticity of demand will be -0.40. The demand curve will be Q = 1172.2 – 85.7P.

b. The price elasticity of demand in year 1 will be -5.31, in year 2 will be -5.76, and arc elasticity of demand will be -5.53. The demand curve will be Q = 473.5 – 38.5P.

c. Instant coffee has greater elasticity of demand than roasted coffee because it is a cheap substitute for roasted coffee.

Step-by-step solution

Explanation for part (a)

Let the demand for roasted coffee be Q = a – bP.

The change in quantity by change in price is the slope of the demand curve; thus, it is calculated as:

$$\begin{gathered} \frac{△Q}{△P}=\frac{850-820}{3.76-4.11} \\ =\frac{30}{-0.35} \\ b=-85.7 \end{gathered}$$

The price elasticity of demand for year 1, year 2, and the arc elasticity, i.e., at the average point, are calculated below:

$$\begin{gathered} E_D^1=\frac{△Q}{△P}\times \frac{P}{Q} \\ =-85.7\times \frac{4.11}{820} \\ =-0.43 \\ \left|E_D^1\right|=0.43 \\ {E}_D^2=-85.7\times \frac{3.76}{850} \\ =-0.38 \\ \left|E_D^2\right|=0.38 \\ {E}_D^{ARC}=-85.7\times \frac{{\displaystyle \frac{4.11+3.76}{2}}}{{\displaystyle \frac{820+850}{2}}} \\ =-85.7\times \frac{3.935}{835} \\ =-0.40 \\ \left|E_D^{ARC}\right|=0.40 \end{gathered}$$

The price elasticity of demand in year 1 will be -0.43, in year 2 will be -0.38, and arc elasticity of demand will be -0.40.

The intercept value of the demand curve is calculated below:

when Q = 820, P = $4.11

820 = a - 85.7(4.11)

820 = a - 352.227

a = 820 + 352.227

a = 1172.2

when Q = 850, P = $3.76

850 = a - 85.7(3.76)

850 = a - 322.232

a = 850 + 322.232

a = 1172.2

At both prices, the intercept value is the same; thus, the demand curve will be Q = 1172.2 – 85.7P.

Explanation for part (b)

Let the demand for instant coffee be Q = a – bP.

The change in quantity by change in price is the slope of the demand curve; thus, it is calculated as:

$$\begin{gathered} \frac{△Q}{△P}=\frac{70-75}{10.48-10.35} \\ =\frac{-5}{0.13} \\ b=-38.5 \end{gathered}$$

The price elasticity of demand for year 1, year 2, and the arc elasticity, i.e., at the average point, are calculated below:

localid="1654042632401" $$\begin{gathered} E_D^1=\frac{△Q}{△P}\times \frac{P}{Q} \\ =-38.5\times \frac{10.35}{75} \\ =-5.31 \\ \left|E_D^1\right|=5.31 \\ {E}_D^2=-38.5\times \frac{10.48}{70} \\ =-5.76 \\ \left|E_D^2\right|=5.76 \\ {E}_D^{ARC}=-38.5\times \frac{{\displaystyle \frac{10.35+10.48}{2}}}{{\displaystyle \frac{75+70}{2}}} \\ =-38.5\times \frac{10.415}{72.5} \\ =-5.53 \\ \left|E_D^{ARC}\right|=5.53 \end{gathered}$$

The price elasticity of demand in year 1 will be -5.31, in year 2 will be -5.76, and arc elasticity of demand will be -5.53.

The intercept value of the demand curve is calculated below:

when Q = 75, P = $10.35

75 = a - 38.5(10.35)

75 = a - 398.475

a = 75 + 398.475

a = 473.5

when Q = 70, P = $10.48

70 = a - 38.5(10.48)

70 = a - 403.48

a = 70 + 403.48

a = 473.5

At both prices, the intercept value is the same; thus, the demand curve will be Q = 473.5 – 38.5P.

Explanation for part (c)

Instant coffee has higher elasticity than roasted coffee, i.e., instant coffee has elastic demand, and roasted coffee has inelastic demand. Roasted coffee has inelastic demand as people think that roasted coffee is a necessity; thus, the demand does not change much with the price increase. Instant coffee might be inferior, but it is a substitute for roasted coffee; therefore, the rise in price for instant coffee will lead to a large fall in demand. Consumers may substitute roasted coffee as they won’t pay more for an inferior good.