Question

The inverse demand for a homogeneous-product Stackelberg duopoly is P $\mathbf{16}\mathbf{,}\mathbf{000}$- 4Q. The cost structures for the leader and the follower, respectively, are CL (QL)localid="1657080008156" $\mathbf{4},\mathbf{000}$QL and CF (QF)$\mathbf{6}\mathbf{,}\mathbf{000}$QF.

a. What is the follower’s reaction function?

b. Determine the equilibrium output level for both the leader and the follower.

c. Determine the equilibrium market price.

d. Determine the profits of the leader and the follower.

Short answer

a.${\text{Q}}_{\text{2}}=r_2\left(Q_1\right)=\frac{a-c_2}{2b}-\frac{1}{2}{Q}_1$

b. The equilibrium output level for the leader is 1750 unitsand for the follower 375 units.

c. The equilibrium market price is$\text{\$7,500}$

d. The profit of the leader in the equilibrium is$\text{\$1,312,500.}$

Step-by-step solution

Given by

The goal is to determine all of the factors in the given inverse demand for a homogeneous-product in Stackelberg duopoly.

The general demand function for the Stackelberg Duopoly is given by:

$\text{P}=a-b\left(Q_1+Q_2\right)$Compare it to the given demand function:

$\text{P}=16,000-4\left(Q_1-Q_2\right)$to obtain the coefficients:

$\text{a=16,000andb=4}$

What is more, the general cost functions for leader ($1$) and the follower($2$) in the Stackelberg Duopoly are given by:

$\begin{array}{l}{\text{C}}_{\text{1}}\left({\text{Q}}_{\text{1}}\right){\text{=c}}_{\text{1}}{\text{Q}}_{\text{1}}\\ {\text{C}}_{\text{2}}\left({\text{Q}}_{\text{2}}\right){\text{=c}}_{\text{2}}{\text{Q}}_{\text{2}}\end{array}$

Comparing it to the cost demand functions:

$\begin{array}{l}{\text{C}}_{\text{1}}\left({\text{Q}}_{\text{1}}\right){\text{=4,000Q}}_{\text{1}}\\ {\text{C}}_{\text{2}}\left({\text{Q}}_{\text{2}}\right){\text{=6,000Q}}_{\text{2}}\end{array}$

To obtain the coefficients:${\text{c}}_{\text{1}}{\text{=4,000andc}}_{\text{2}}\text{=6,000}$

Finding the followers reaction function

a.

The follower's reaction function is given by the formula:

${\text{Q}}_{\text{2}}=r_2\left(Q_1\right)=\frac{a-c_2}{2b}-\frac{1}{2}{Q}_1$

Where a, b, c are the coefficients in the demand and cost functions of Stackelberg Duopoly.

Substituting ${\text{a=16,000,b=4,andc}}_{\text{2}}\text{=6,000}$into the formula:

$\begin{array}{c}{Q}_2=\frac{16,000-6,000}{2\times 4}-\frac{1}{2}{Q}_1\\ Q_2=\frac{10,000}{8}-\frac{1}{2}{Q}_1\\ =1250-\frac{1}{2}{Q}_1\end{array}$

 Step 3: Determining the equilibrium output level

b.

In order to find the equilibrium output level of both, the leader and the follower, first, find the reaction function for the leader. Then, we should substitute that value into the equation obtained in the previous part.

The leader's reaction function is given by the formula:

${\text{Q}}_{\text{1}}{\text{=r}}_{\text{1}}\left({\text{Q}}_{\text{2}}\right)=\frac{a+c_2-2c_1}{2b}$8 of 16

where ${\text{a,b,c}}_{\text{1}}{\text{,c}}_{\text{2}}$are the coefficients in the demand and cost functions of Stackelberg Duopoly.

Substituting ${\text{a=16,000,b=4,c}}_{\text{1}}{\text{=4,000,andc}}_{\text{2}}\text{=6,000}$into the formula:

$\begin{array}{c}{Q}_1=\frac{16,000+6,000-2\times 4,000}{2\times 4}\\ Q_1=\frac{22,000-8,000}{8}\\ =\frac{14,000}{8}\\ =1750\end{array}$

Substitute ${\text{Q}}_{\text{1}}\text{=1750}$into the follower's reaction function:

$\begin{array}{c}{Q}_2=1250-\frac{1}{2}\times 1750\\ =1250-875\\ =375\end{array}$

Therefore, the equilibrium output level for the leader is 1750 and for the follower 375

Determining the equilibrium market price

c.

The equilibrium market price is determined by the equation given in the task:

$\text{P}=16,000-4\text{Q}$

where $\text{Q=Q\_}\left\{\text{1}\right\}\text{+Q\_}\left\{\text{1}\right\}$therefore:

$\text{P}=16,000-4\left(Q_1+Q_2\right)$

Substituting${\text{Q}}_{\text{1}}{\text{=1750andQ}}_{\text{2}}\text{=375}$ into the equation:

$\text{P}=16,000-4\left(1750+375\right)$

To calculate the equilibrium market price:

$\begin{array}{c}P=16,000-4\times 2125\\ =16,000-8,500\\ =7,500\end{array}$

Therefore, the equilibrium market price is $\text{\$7,500}$

 To Determine the profits of the leader and the follower

d.

The profit for the leader is given by the formula:

${\pi }_1=\left(a-b\left(Q_1+\left(\frac{a-c_2}{2b}-\frac{1}{2}{Q}_1\right)\right)\right)Q_1-c_1{Q}_1$Step 15

Substituting$\text{a=16,000,b=4}$ into the equation:

${\pi }_1=\left(16,000-4\left(Q_1+\left(\frac{16,000-c_2}{2\times 4}-\frac{1}{2}{Q}_1\right)\right)\right)Q_1-c_1{Q}_1$

Substituting${\text{Q}}_{\text{1}}{\text{=1750,Q}}_{\text{2}}\text{=375}$ into the equation:

${\pi }_1=\left(16,000-4\left(1750+\left(\frac{16,000-c_2}{2\times 4}-\frac{1}{2}\times 375\right)\right)\right)1750-c_1\times 1750$

Substituting ${\text{c}}_{\text{1}}{\text{=4,000,c}}_{\text{2}}\text{=6,000}$into the equation:

${\pi }_1=\left(16,000-4\left(1750+\left(\frac{16,000-6,000}{2\times 4}-\frac{1}{2}\times 375\right)\right)\right)1750-4,000\times 1750$

Calculating the profit:

$\begin{array}{l}{\pi }_1=\left(16,000-4\left(1750+\left(\frac{16,000-6,000}{2\times 4}-\frac{1}{2}\times 375\right)\right)\right)1750-4,000\times 1750\\ =\left(16,000-4\left(1750+\left(\frac{10,000}{8}-187.5\right)\right)\right)1750-7,000,000\\ =(16,000-4(1750+(1250-187.5)\left)\right)1750-7,000,000\\ =(16,000-4(1750+1062.5\left)\right)1750-7,000,000\\ =(16,000-4\times 2812.5)1750-7,000,000\\ =(16,000-11,250)1750-7,000,000\\ =\left(4750\times 1750\right)-7,000,000\\ =8,312,500-7,000,000\\ =1,312,500\end{array}$

Therefore, the profit of the leader in the equilibrium is $\text{\$1,312,500.}$