Question

Consider a market for energy drinks consisting of only one firm. The firm has a linear cost function: C(q)=4q, where q represents quantity produced by the firm. The market inverse demand function is given byr P(Q)=24-2Q, where Q represents total industry output. Based on the given information answer the following.

1. What price will the firm charge? What quantity of energy drinks will the firm sell?

2. Now suppose a second firm enters the market. The second firm has an identical cost function. What will be the Cournot equilibrium output for each firm?

3. Whay is the Stackelberg equilibrium output for each firm of firm 2 enters second?

4. How much profit will each firm make in yhe Cournot game? How much in Stackelberg?

5. Which type of market do consumers prefer: monopoly, Cournot duopoly or Stackelberg duopoly?

Answer 1

$1.\\C(q)=4q\\MC=\frac{\Delta TC}{\Delta q}\\=4$

$TR=P\times q\\=(24-2q)q\\=24q-2q^2\\MR=\frac{\Delta TR}{\Delta q}\\=24-4q$

$MR=MC\\24-4q=4\\20=4q\\q=5$

$P=24-2q\\=24-2(5)\\=14$

Price charged by the firm is 14 and quantity of energy drinks sold is 5

2:

A second firm with the same identical cost enters the market

The Cournot Equilibrium is calculated as follows :

Inverse Demand curve is$P(Q)=24−Q$

$C(q)=4q$

It can be written as $P(Q)=24−2(q_1+q_2)$

where q₁ is the output of firm 1 and q₂ is the output of firm 2 

The firms have identical cost functions with the cost of 4q 

Therefore,$P=24−2q_1−2q_2$

Now, For Firm 1 :

$TR=Price×q_1\\TR_1=(24−2q_1−2q_2)q_1\\=24q_1−2q_1^2−2q_1q_2\\MR_1=\frac{dTR}{dq_1}=24−4q_1−q_2$

Now, we equate MR=MC

$24−4q_1−q_ 2=4\\4q_1=20−q_2\\q_1=5−\frac{1}{4}q_2$

For Firm 2 :

$TR=Price×q_2\\TR_2=(24−2q_1−2q_2)q_2\\=24q_2−2q_1q_2−2q_2^2\\MR_2=\frac{dTR}{dq_2}=24−4q_2−q_1$

Now, we equate MR=MC

$24−q_1-4q_2 2=4\\q_2=5−\frac{1}{4}q_1$

Cournot equilibrium is 

$q_1=q_2\\ q_1=5−0.5q_2\\ q_2=5−0.5q_1\\ q_1=5−0.5(5−0.5q_1)\\ q_1=3.33$

Now putting 3.33 in $q_2=5−0.5q_1$

We get

$q_2=3.33$

Therefore, under Cournot model, both firms produce the same output.

3:

Now for Stackleberg Model, 

We have the same Inverse function and cost function

We find the TR of Firm 1

$TR_2=24-4q_1-2(5-0.5q_1)q_1\\=(14-3q_1)q_1\\=14q_1-3q_1^2 \\MR_2=\frac{dTR_2}{dq_1}\\=14-3q_1$

Now we equate 

$MR=MC\\14-3q_1=4\\3q_1=10\\q_1=3.33$

Now putting q₁=3.33 in the reaction curve of firm $2, i.e 5-0.5(q_1)$

We get

$5-0.5(3.33)\\5-1.66\\=0.33$

Therefore, both firms produce the same output. 

4:

First we calculate the price

Since quantity produced under Cournot and Stackleberg models are the same, i.e. q=3.33

We put this in the inverse demand function and get

$P=24−2(q_1+q_2)\\ P=10.68$

Now to calculate profit,

$Profit =Price−Cost\\ =10.68−4\\ =6.68$

Therefore profit under both the models is 6.68

5:

consumers prefer Cournot Model.

Cournot Model is defined as a model where the competing firms choose a quantity to produce independently

Stackleberg Model is defined as a model where the leader firm moves first and the follower firms move simultaneously.