Chapter 9, Problem 19PAA

To determine

To know:Whether $200 to be invested or not.

Explanation of Solution

There are two firms: TT and ST

Following is the cost function of TT:

  $\begin{array}{l}\frac{\partial T{R}_{TT}}{\partial Q_{TT}}=\frac{\partial C_{TT}}{\partial Q_{TT}}\\ \frac{\partial (160Q_{TT}-2Q_{TT}^2-2Q_{TT}{Q}_{ST})}{\partial Q_{TT}}=4\end{array}$

Following is the cost function of ST:

  $C_{ST}(Q_{ST})=4Q_{ST}$

Following is the inverse demand function:

  $\begin{array}{l}P=160-2Q\\ =160-2(Q_{TT}+Q_{ST})\end{array}$

There exists cournot oligopoly between two firms which sells homogeneous product. Profit maximizing condition is as follows:

  $M{C}_{TT}=M{R}_{TT}$

This implies:

  $\begin{array}{l}\frac{\partial T{R}_{TT}}{\partial Q_{TT}}=\frac{\partial C_{TT}}{\partial Q_{TT}}\\ \frac{\partial P{Q}_{TT}}{\partial Q_{TT}}=\frac{\partial 4Q_{TT}}{\partial Q_{TT}}\end{array}$

  $\frac{\partial (160Q_{TT}-2Q_{TT}^2-2Q_{TT}{Q}_{ST})}{\partial Q_{TT}}=4$

  $160-4Q_{TT}-2Q_{ST}=4$

  $Q_{TT}=\frac{1}{4}(160-4-2Q_{ST})$

  $Q_{TT}=39-\frac{1}{2}{Q}_{ST}$

Reaction function of TT firm is given as: $Q_{TT}=39-\frac{1}{2}{Q}_{ST}$ .

There exists cournot oligopoly between two firms which sells homogeneous product. Profit maximizing condition is as follows:

  $M{C}_{ST}=M{R}_{ST}$

This implies:

  $\frac{\partial T{R}_{ST}}{\partial Q_{ST}}=\frac{\partial C_{ST}}{\partial Q_{ST}}$

  $\frac{\partial P{Q}_{ST}}{\partial Q_{ST}}=\frac{\partial 4Q_{ST}}{\partial Q_{ST}}$

  $\frac{\partial (160Q_{ST}-2Q_{TT}{Q}_{ST}-2Q_{ST}^2)}{\partial Q_{ST0}}=4$

  $160-2Q_{TT}-4Q_{ST}=4$

  $Q_{ST}=\frac{1}{4}(160-4-2Q_{TT})$

In cournot oligopoly, equilibrium is attained when reaction function of both firms are solved simultaneously.

This implies:

  $Q_{TT}=39-\frac{1}{2}\left(39-\frac{1}{2}{Q}_{TT}\right)$

  $\begin{array}{l}{Q}_{TT}=39-19.5+\frac{1}{4}{Q}_{TT}\\ Q_{TT}=26\\ Q_{ST}=26\end{array}$

Solving both the equations,

Thus, equilibrium output of TT is 26 and of ST is26

The equilibrium market price is given by:

  $P=160-2(Q_{TT}+Q_{ST})$

  $\begin{array}{l}P=160-2(26+26)\\ P=56\end{array}$

Profit is calculated by subtracting total revenue and total cost. For TT firm,

  $\begin{array}{l}{\pi }_1=T{R}_{TT}-C_{TT}\\ =P{Q}_{ST}-C_{ST}\\ =56\times 26-4\times 26\\ =1352\end{array}$

Profit is calculated by subtracting total revenue and total cost. For ST firm,

  $\begin{array}{l}{\pi }_1=T{R}_{ST}-C_{ST}\\ =P{Q}_{ST}-C_{ST}\\ =56\times 26-4\times 26\\ =1352\end{array}$

Now, Stackelberg duopoly is followed in which cost functions are same but TT firm is leader and ST is follower.

Under stackelberg duopoly model, profit maximizing condition is as follows:

  $M{R}_{ST}=M{C}_{ST}$

This implies:

  $\frac{\partial T{R}_{ST}}{\partial Q_{ST}}=\frac{\partial C_{ST}}{\partial Q_{ST}}$

  $\frac{\partial P{Q}_{ST}}{\partial Q_{ST}}=\frac{\partial 4Q_{ST}}{\partial Q_{ST}}$

  $\frac{\partial (160Q_{ST}-2Q_{TT}{Q}_{ST}-2Q_{ST}^2)}{\partial Q_{ST}}=4$

  $160-2Q_{TT}-4Q_{ST}=4$

  $Q_{ST}=39-\frac{1}{2}{Q}_{TT}$

The leader firm TT in Stackelberg duopoly takes into account the reaction function of the follower firm when it selects Q_TT.

The profit of the leader firm is given by:

  $\begin{array}{l}{\pi }_{TT}=T{R}_{TT}-C_{TT}\\ =P{Q}_{TT}-4Q_{TT}\\ =82Q_{TT}-Q_{TT}^2-4Q_L\\ =78Q_{TT}{Q}_{TT}^2-\end{array}$

The leader firm chooses Q_TT that maximizes the given profit function.

Hence, Q_TT that maximizes the profit function of the leader firm is:

  $\begin{array}{l}\frac{\partial {\pi }_{TT}}{\partial Q_{TT}}=0\\ 2Q_{TT}=78\\ Q_{TT}=39\end{array}$

Hence, the equilibrium output level for the leader firm is 39.

Calculation of equilibrium output:

  $\begin{array}{l}{Q}_{ST}=39-\frac{1}{2}{Q}_{TT}\\ =39-\frac{1}{2}39\\ =19.5\end{array}$

Firm St has 19.5 units of output.

Calculation of equilibrium price:

  $\begin{array}{l}P=160-2Q\\ =160-2(Q_{TT}+2Q_{ST})\\ =160-2(39+19.5)\\ =43\end{array}$

Calculation of profit of TT firm is as follows:

  $\begin{array}{l}\pi =T{R}_{TT}-C_{TT}\\ =P{Q}_{TT}-C_{TT}\\ =43\times 39-4\times 39\\ =1521\end{array}$

Calculation of profit of ST firm is as follows:

  $\begin{array}{l}\pi =T{R}_{ST}-C_{ST}\\ =P{Q}_{ST}-C_{ST}\\ =43\times 19.5-4\times 19.5\\ =760.5\end{array}$

Thus, the equilibrium profit of the leader firm TT under the conditions of Stackelberg oligopoly is $1521, which is higher by $169, compared to the equilibrium profit of $1352 of the firm TT under the condition of Cournot duopoly.

The unrecoverable fixed investment is $200, greater than $169.

That means, the firm TT should not invest $200 as the cost of establishing the first mover advantage exceeds the benefits.

Economics Concept Introduction

Introduction:

Duopoly is a form of oligopoly market in which there are two firms selling goods in a competitive market where consumer’s choice does not affect them.