Chapter 3, Problem 1E

The Potomac Range Corporation manufactures a line of microwave ovens costing $500 each. Its sales have averaged about 6,000 units per month during the past year. In August, Potomac’s closest competitor, Spring City Stove Works, cut its price for a closely competitive model from $600 to $450. Potomac noticed that its sales volume declined to 4,500 units per month after Spring City announced its price cut.

1. What is the arc cross elasticity of demand between Potomac’s oven and the competitive Spring City model?
2. Would you say that these two firms are very dose competitors? What other factors could have influenced the observed relationship?
3. If Potomac knows that the arc price elasticity of demand for its ovens is $-3.0$, what price would Potomac have to charge to sell the same number of units it did before the Spring City price cut?

To determine

To find: Arc cross elasticity of demand between P’s oven and S City model.

Explanation of Solution

Given information:

The unit price of the P’s oven: $\$500$

P’s average monthly sales: $6000$ units

The unit price of the S City model before price reduction: $\$600$

The unit price of the S City model after price reduction: $\$450$

P’s average monthly sales after price reduction of S City model: $4500$ units

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 the other good.

The arc cross-price elasticity between the two goods is calculated as follows:

  $\begin{array}{c}{E}_{cross}=\left[\frac{\raisebox{1ex}{$Q_2-Q_1$}\!\left/ \!\raisebox{-1ex}{$\raisebox{1ex}{$\left(Q_1+Q_2\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$}\right.}{\raisebox{1ex}{$P_2-P_1$}\!\left/ \!\raisebox{-1ex}{$\raisebox{1ex}{$\left(P_1+P_2\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$}\right.}\right]\\ =-\frac{\Delta Q}{\Delta P}\left(\frac{P_1+P_2}{Q_1+Q_2}\right)\\ =\frac{4500-6000}{450-600}\left(\frac{600+450}{6000+4500}\right)\\ =\frac{-1500}{-150}\left(\frac{1050}{10500}\right)\\ =+1\end{array}$

The above formula gives the cross-price elasticity of $1$ .

To determine

To find: Whether the two firms are close competitors and the factors that could have influenced the observed relationship.

Explanation of Solution

P’s Range Corporation and S City Stove Works are close competitors because the cross-price elasticity is positive.

As the cross-price elasticity is positive, these two products are substitutes for each other.

When S City model price reduced, the P’s average monthly sales also reduced. Here, the cross-price elasticity is $1$ . It means that the percentage change in the price of the S City model equals the percentage change in quantity demanded of P’s average monthly sales.

Following factors influence the observed relationship:

1. The availability and Closeness of Substitutes: If there are more possible substitutes, a given good or service, the greater will be the elasticity. When there are many close substitutes, then the consumers can conveniently switch from one commodity to another, even in the case of a small change in price.
2. Percentage of the Consumer’s budget: There is a great elasticity when products that consume a part of the purchaser’s budget that will have greater elasticity.
3. Positioning as Income Superior
4. Time Period of Adjustments: In the case of non-durable goods, the elasticity will be larger in the long-run than in the short-run.

To determine

To find: When arc price elasticity of demand is $-3.0$ for P ovens, the amount by which the P should change the price to maintain the same amount of sales before the S City price cut.

Explanation of Solution

Given information:

Arc price elasticity for P ovens = $-3.0$

After price reduction of S City sales of the P ovens = $4500$ units

Sales of the P ovens before the price reduction = $6000$ units

Potomac current selling price = $\$500$

Formula used:

  $\begin{array}{l}{E}_d=\%\Delta D/\%\Delta P\\ \frac{\frac{\frac{\left(Q2-Q1\right)}{\left(Q2+Q1\right)/2}}{(P2-P1)}}{(P2+P1)/2}\end{array}$

E_D − Percentage change in quantity demanded to percentage change in determinant

%?D − Percentage change in quantity demanded

%?P - Percentage change in price

By using the above-mentioned formula, P’s changes in price to maintain the same amount of sales before the S City price cut is calculated as follows:

  $\begin{array}{l}e=\frac{\left(\%\Delta Q\right)}{\left(\%\Delta P\right)}\\ \%\Delta Q=\left[\raisebox{1ex}{$Q_2+Q_1$}\!\left/ \!\raisebox{-1ex}{$\raisebox{1ex}{$\left(Q_2+Q_1\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$}\right.\right]\\ =\frac{\left[6000-4500\right]}{\left[\left(6000+4500\right)/2\right]}\\ =\frac{1500}{5250}\end{array}$

  $Withe=-3=\left[\%changeinquantity\right]/[\%changein\underset{\_}{}price]$

  $\%changein\underset{\_}{}price=-\left[\raisebox{1ex}{$1500$}\!\left/ \!\raisebox{-1ex}{$\raisebox{1ex}{$5250$}\!\left/ \!\raisebox{-1ex}{$3$}\right.$}\right.\right]=-\raisebox{1ex}{$300$}\!\left/ \!\raisebox{-1ex}{$5250$}\right.$

Current Price = 500

New Price = x

Hence, the % change in price will be:

  $\begin{array}{l}-\raisebox{1ex}{$300$}\!\left/ \!\raisebox{-1ex}{$5250$}\right.=\left[\frac{\left(P_2-P_1\right)}{\raisebox{1ex}{$\left(P_2+P_1\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\right]\\ -\raisebox{1ex}{$300$}\!\left/ \!\raisebox{-1ex}{$5250$}\right.=\left\{\raisebox{1ex}{$\left[x-500\right]$}\!\left/ \!\raisebox{-1ex}{$\left[\left(x+500\right)/2\right]$}\right.\right\}\\ -150\left(x+500\right)\text{= }5250(x-500)\\ -150x-75,000=5250x–2,625,000\\ 5400x=2,550,000\\ x=\$472.22\end{array}$

P should reduce the price from $\$500$ to $\$472.22$ , to maintain the same amount of sales before the S City price cut.