Question

In Example 2.8 we examined the effect of a 20-percent decline in copper demand on the price of copper, using the linear supply and demand curves developed in Section 2.6. Suppose the long-run price elasticity of copper demand were -0.75 instead of -0.5.

1. Assuming, as before, that the equilibrium price and quantity are P_ = $3 per pound and Q_ = 18 million metric tons per year, derive the linear demand curve consistent with the smaller elasticity.

2. Using this demand curve, recalculate the effect of a 55-percent decline in copper demand on the price of copper.

Short answer

1. The demand curve will be Q = 31.5 – 4.5P.

2. The demand falls by 55% then price fall by 30%.

Step-by-step solution

Explanation for part (a)

Let the demand curve be Q = a + bP. The coefficient of P is the change in quantity demand by a change in price; the value of a is the intercept of the demand curve.

The elasticity of demand will be:

$$\begin{gathered} E_D=\frac{△Q}{△P}\times \frac{P}{Q} \\ {E}_D=-0.75 \\ -0.75=\frac{△Q}{△P}\times \frac{3}{18} \\ \frac{△Q}{△P}=\frac{-0.75\times 18}{3} \\ b=-4.5 \end{gathered}$$

The intercept value is calculated below:

18 = a - 4.5(3)

a = 18 + 13.5

a = 30.5

The demand curve will be Q = 31.5 – 4.5P.

Explanation for part (b)

The demand curve falls by 55%; thus, the new demand curve will be:

$$\begin{gathered} Q_D^{\text{'}}=0.45Q_D \\ =0.45(31.5-4.5P) \\ =14.18-2.03P \\ \mathrm{The}\mathrm{supply}\mathrm{curve}\mathrm{is}\mathrm{Q}=-9+9\mathrm{P} \\ \mathrm{At}\mathrm{equilibrium}, \\ \mathrm{D}=\mathrm{S} \\ 14.18-2.03\mathrm{P}=-9+9\mathrm{P} \\ 2.03\mathrm{P}+9\mathrm{P}=14.18+9 \\ 11.03\mathrm{P}=23.18 \\ \mathrm{P}=2.10 \end{gathered}$$

The demand falls by 55%, then the price fall by 30%$\left(\left|\frac{2.1-3}{3}\right|\times 100=30\%\right)$