Question

The market for a standard-sized cardboard container consists of two firms: Composite Box and fiber board. As the manager of Composite Box, you enjoy a patented technology that permits your company to produce boxes faster and at a lower cost than fiber board. You use this advantage to be the first to choose its profit-maximizing output level in the market. The inverse demand function for boxes is P$\mathbf{1}\mathbf{,}\mathbf{2006}\mathit{Q}$, Composite Box’s costs are CC(QC)$\mathbf{60}$QC, and fiber board’s costs are CF (QF)$\mathbf{120}$QF. Ignoring antitrust considerations, would it be profitable for your firm to merge with fiber board? If not, explain why not; if so, put together an offer that would permit you to profitably complete the merger

Short answer

The merger would be more profitable for our firm that keep as a leader

Step-by-step solution

Given by

As there are two companies in an oligopoly where Composite Box enjoys its position as leader and Fiberboard as a follower, we are in the presence of a Stackelberg model in which the leading firm will obtain higher profits than the following firm. Therefore, we must check if this condition will benefit us as a leading firm or merge with Flipboard and forming a monopoly will generate greater profit.

Under the Stackelberg model, the following inverse demand function and cost functions are presented for the leading and following firm.

Inverse Demand Function for both firm:$\text{P=1,200-6Q}$

Leader Firm Cost Function:${\text{C}}_{\text{c}}{\text{Q}}_{\text{c}}{\text{=60Q}}_{\text{c}}$

Follower Firm Cost Function:${\text{C}}_{\text{F}}{\text{Q}}_{\text{F}}{\text{=120Q}}_{\text{F}}$

To find the reaction function

We must find the reaction function of the follower firm so that they can make the first decision regarding its output level. For this, the marginal income must be equated to the marginal costs of the follower company, so first we need to calculate the total revenue. It is important to keep in mind that .$\text{Q=}\left({\text{Q}}_{\text{C}}{\text{+Q}}_{\text{F}}\right)$

Total Revenue = Price x Quantity

$\begin{array}{l}\text{=1,200-6}\left({\text{Q}}_{\text{c}}{\text{+Q}}_{\text{F}}\right){\text{Q}}_{\text{F}}\\ {\text{=1,200Q}}_{\text{F}}{\text{-6Q}}_{\text{c}}{\text{Q}}_{\text{F}}{\text{-6Q}}_{\text{F}}^{\text{2}}\end{array}$

Applying the first derivative to obtain the marginal revenue:

$\text{MR=}\frac{\text{dR}}{{\text{dQ}}_{\text{F}}}{\text{=1,200-6Q}}_{\text{c}}{\text{-12Q}}_{\text{F}}$

$\text{MC=}\frac{\text{dC}}{{\text{dQ}}_{\text{F}}}\text{=120}$

To equate the marginal revenue and costs

Equating marginal revenue and costs to obtain the follower firm's reaction function, we can solve for${\text{Q}}_{\text{F}}$:

$\begin{array}{l}{\text{1,200\&-6Q}}_{\text{c}}{\text{-12Q}}_{\text{F}}\\ {\text{=120Q}}_{\text{F}}\text{=}\frac{{\text{1,200-120-6Q}}_{\text{c}}}{\text{12}}{\text{Q}}_{\text{F}}\\ \text{=90-}\frac{\text{1}}{\text{2}}{\text{Q}}_{\text{c}}\text{nendaligned}\end{array}$

To determine the equilibrium output level

To determine the equilibrium output level for both the leader and the follower we can use the profit formula for the leading firm and substitute the reaction function of the follower firm${\text{Q}}_{\text{F}}$.

$\begin{array}{l}{\text{II=TotalRevenue}}_{\text{c}}{\text{-TotalCost}}_{\text{c}}\\ {\text{=P}}_{\text{-}}{\text{c×Q}}_{\text{-}}{\text{c-C}}_{\text{-}}\text{c}\end{array}$

$\begin{array}{l}\text{=1,200-6}\left({\text{Q}}_{\text{c}}{\text{+Q}}_{\text{F}}\right){\text{Q}}_{\text{c}}{\text{-60Q}}_{\text{c}}\\ {\text{=1,200-6Q}}_{\text{c}}^{\text{2}}{\text{-6Q}}_{\text{F}}{\text{Q}}_{\text{c}}{\text{-60Q}}_{\text{c}}\\ {\text{=1,200-6Q}}_{\text{c}}^{\text{2}}\text{-6}\left(\text{90-}\frac{\text{1}}{\text{2}}{\text{Q}}_{\text{c}}\right){\text{Q}}_{\text{c}}{\text{-60Q}}_{\text{c}}\\ {\text{=1,200Q-6Q}}_{\text{c}}^{\text{2}}{\text{-540Q-3Q}}_{\text{c}}^{\text{2}}{\text{-60Q}}_{\text{c}}\\ {\text{=660Q-3Q}}_{\text{c}}^{\text{2}}{\text{-60Q}}_{\text{c}}\\ {\text{=600Q-3Q}}_{\text{c}}^{\text{2}}\end{array}$

To derive and solve

Deriving and solving for${\text{Q}}_{\text{C}}$:

$\begin{array}{l}\frac{{\text{dΠ}}_{\text{c}}}{{\text{Q}}_{\text{c}}}{\text{=600-6Q}}_{\text{c}}\\ {\text{Q}}_{\text{c}}\text{=}\frac{\text{600}}{\text{6}}\\ \text{=100}\end{array}$

Already having the equilibrium output of the leading firm Composite Box, we can substitute that value in the reaction function of the following firm to obtain the corresponding output.

${\text{Q}}_{\text{F}}\text{=90-}\frac{\text{1}}{\text{2}}\text{100}$

${\text{Q}}_{\text{F}}\text{=40}$

We can obtain the equilibrium price for both firms by substituting the output of each in the inverted demand function.

$\begin{array}{l}\text{P=1,200-6(100+40)}\\ \text{=1,200-840}\\ \text{=360}\end{array}$

To calculate the profits of both firms

The profits of both firms can be calculated as follows:

${\text{II=TotalRevenue}}_{\text{c}}{\text{-TotalCost}}_{\text{c}}{\text{=P}}_{\text{-}}{\text{c×Q}}_{\text{-}}{\text{c-C}}_{\text{-}}\text{c=360×40-120×40=14,400-4,800=9,600}$

Under the Stackelberg oligopoly model, the leading company Composite Box will make a profit of$\$30,000$ while the follower company Fiber board will have a profit of$9,600\$.$

To obtain marginal revenue

Assuming now that Composite Box absorbs Fiber board and a monopoly is established, the condition that MR = MC will be maintained, however the inverse demand function will be unique for a level $Q$ and the cost function will be that reflected by the Composite Box company. Therefore, we can find the total revenue for be marginal revenue and equate it to the marginal costs.

Total Revenue =Price x Quantity

$\begin{array}{l}\text{=(1,200-6Q)Q}\\ {\text{=1,200Q-6Q}}^{\text{2}}\end{array}$

Deriving to obtain marginal revenue:

$\text{MR=}\frac{\text{dR}}{\text{dQ}}\text{=1,200-12Q}$

And deriving to obtain the marginal cost:

$\text{MC=}\frac{\text{dC}}{\text{dQ}}\text{=60}$

To Equate the marginal revenues and costs

Equating marginal revenues and costs to obtain the equilibrium output of the monopoly:

$\begin{array}{l}\text{1,200-12Q=60Q}\\ \text{=}\frac{\text{1,200-60}}{\text{12}}\text{Q}\\ \text{=95}\end{array}$

And the price of equilibrium:

$\begin{array}{l}\text{P=1,200-6(95)}\\ \text{P=630}\end{array}$

calculate the profit

Finally, we can calculate the profit that the monopoly company will obtain in the following way:

$\begin{array}{l}\text{Π\&=TotalRevenue-TotalCost}\\ \text{=630×95-60×95}\\ \text{=59,850-5,700}\\ \text{=54,150}\end{array}$

Taking the benefits obtained in the two scenarios, we can conclude that Composite Box in the Stackelberg model will obtain benefits of $\$ 30,000$ while the follower firm Flip board will have income of $\$ 9,600$.

If Composite Box decides to merge with Flip board it will make profits of$54,150\$,$which are higher for both companies. Flip board will accept the merger as long as it is paid more than$9,600\$.$

Therefore, the merger would be more profitable for our firm that keep as a leader.