Question

PC Connection and CDW are two online retailers that compete in an Internet market for digital cameras. While the products they sell are similar, the firms attempt to differentiate themselves through their service policies. Over the last couple of months, PC Connection has matched CDW’s price cuts, but has not matched its price increases. Suppose that when PC Connection matches CDW’s price changes, the inverse demand curve for CDW’s cameras is given by$\text{P=1,500-3Q}$ . When it does not match price changes, CDW’s inverse demand curve is$\text{P=900-0.50Q}$. Based on this information, determine CDW’s inverse demand and marginal revenue functions over the last couple of months. Over what range will changes in marginal cost have no effect on CDW’s profit-maximizing level of output?

Short answer

The range of marginal costs that will have no effect on CDW's profit-maximizing level of output will be 660 units.

Step-by-step solution

Given

PC Connection and CDW compete in a sweezy oligopoly, in which each firm thinks that the rival will reduce its prices if one of them reduces them first, but it will not happen in the opposite direction in which one of the firms increases its prices. the following inverse demand functions correspond for each scenario.

(1) Inverse demand function when PC Connection and CDW match in the price drop:

$\text{P}=1,500-3\text{Q}$

(2) Inverse demand function when PC Connection and CDW dont match in the price:

$\text{P}=900-0.5\text{Q}$

To obtain the marginal income

To obtain the marginal income of each situation we must multiply by Q both functions to obtain the total income:

(1) Total Revenue = Price x Quantity

$\begin{array}{l}=(1,500-3Q)Q\\ =1,500Q-3Q^2\end{array}$

Deriving to obtain the marginal revenue:

${\text{MR}}_{\text{-}}1=\frac{d{R}_1}{dQ}=1,500-6Q$

(2)Total Revenue=Price x Quantity

$=(900-0.5Q)=900Q-0.5Q^2$

Deriving to obtain the marginal revenue:

${\text{MR}}_{\text{-}}2=\frac{d{R}_2}{dQ}=900-Q$

The above marginal revenue correspond in the scenario in which PC connection macth and dont with the price of CDW.

To equalize the marginal costs

Given that in the sweezy oligopoly model the marginal revenues of both scenarios are associated with both inverse demand functions, the condition that:

${\text{P=MR}}_{\text{-}}{\text{1=MR}}_{\text{-}}\text{2}$

We can equalize the marginal costs of both inverse demand functions to obtain the equilibrium output:

$\begin{array}{c}1,500-6Q=900-Q5Q\\ Q=1500-900\\ =600\end{array}$

Thus, each firm will have an equilibrium output of $600/5$$120$ units and the total output will be $240(120\times 2)$

To substitute the equilibrium output level

We can substitute the equilibrium output level in each marginal revenue function to obtain the range that will changes marginal cost and have no effect on CDW's profit maximizing level of output.

$\begin{array}{c}M{R}_{-1}=1,500-6\left(240\right)\\ =60\{MR\_\_\}^2\\ =900-240\\ =660\end{array}$

Therefore, the range of marginal costs that will have no effect on CDW's profit-maximizing level of output will be 660 units