Chapter 2, Problem 2.17P

To determine

(a)

To calculate the equilibrium price and quantity of golf balls, when R = 2 and T = 10.

Explanation of Solution

The following equations for demand and supply are given:

  $\begin{array}{l}Demand:Q^d=90-2P-2T\\ Supply:Q^s=-9+5P-2.5R\end{array}$

Substitute the values of R and T to solve for demand and supply equations in terms of price only as follows:

  $\begin{array}{l}Demand:\\ Q^d=90-2P-2T\\ Q^d=90-2P-2\left(10\right)\\ Q^d=90-2P-20\\ Q^d=70-2P\\ Supply:\\ Q^s=-9+5P-2.5R\\ Q^s=-9+5P-2.5\left(2\right)\\ Q^s=-9+5P-5\\ Q^s=-14+5P\end{array}$

At equilibrium,

  $\begin{array}{l}{Q}^d=Q^s\\ 70-2P=-14+5P\\ 7P=84\\ P=\$12\end{array}$

Substitute the value of price in any of the two equations to solve for equilibrium quantity as follows:

  $\begin{array}{l}{Q}^d=70-2P\\ Q^d=70-2\left(\$12\right)\\ Q^d=70-24\\ Q=46\end{array}$

To determine

(b)

To calculate the price elasticity of demand and price elasticity of supply at equilibrium values.

Explanation of Solution

The price elasticity of demand measures the degree of responsiveness of a change in the quantity demanded for a given change in the price level. It is calculated as follows:

  $\begin{array}{c}{e}_d=\frac{d\left(Q^d\right)}{dP}\times \frac{P}{Q^d}\\ =\frac{d\left(70-2P\right)}{dP}\times \frac{P}{\left(70-2P\right)}\\ =-2\times \frac{\$12}{46}\\ =\boxed{-0.52}\end{array}$

The price elasticity of demand at equilibrium values is '-0.52'.

The price elasticity of supply measures the degree of responsiveness of a change in the quantity supplied for a given change in the price level. It is calculated as follows:

  $\begin{array}{c}{e}_s=\frac{d\left(Q^s\right)}{dP}\times \frac{P}{Q^s}\\ =\frac{d\left(-14+5P\right)}{dP}\times \frac{P}{\left(-14+5P\right)}\\ =+5\times \frac{\$12}{46}\\ =\boxed{+1.30}\end{array}$

The price elasticity of supply at equilibrium values is '+1.30'.

To determine

(c)

Cross-price elasticity of demand for golf balls with respect to the price of titanium.

Explanation of Solution

The cross-price elasticity of demand measures the degree of responsiveness of a change in the quantity demanded of a good for a given change in the price of some other good.

It is calculated as follows:

  $\begin{array}{c}{\epsilon }_{golf,ti\tan ium}=\frac{d\left(Q^d\right)}{d{P}_T}\times \frac{P_T}{Q^d}\\ =\frac{d\left(90-2P-2T\right)}{dT}\times \frac{T}{\left(90-2P-2T\right)}\\ =-2\times \frac{10}{\left[90-2\left(12\right)-2\left(10\right)\right]}\\ =-2\times \frac{10}{46}\\ =\boxed{-0.43}\end{array}$

The negative sign indicates that titanium and golf balls are complements, i.e., when the price of titanium goes up, the demand for golf balls decreases.