Answer to Question #264848 in Microeconomics for manuze

Question

Suppose two firms are the sole producers of widget in West Africa, and they are faced with a market demand function given as P = 40 - 20Q While Dally Limited is located in Nigeria, Joy Manufacturing operates from Ghana. The firms' total cost function is given as TC = 12 + Q

a.) Determine the output and profit for each firm under Cournot's assumptions. [3 marks] b.) To aid Joy Manufacturing increase its output to the Stackelberg leader's output level, the Ghanaian government plans to support the firm with subsidies. In monetary terms what should be the value of the subsidy that will make Joy Manufacturing the leading firm in the market? [3 marks]

c.) Assume the fims now operates under Stackelberg' s assumptions, with Joy Manufacturing as the leader, determine output and profit for each firm. [4 marks]

Answer 1

a)

Under the Cournot model,

Firm 1,

Profit1 = Total Revenue1 - Total cost1

$π1= TR1−TC1\\π1=P×Q1−TC1\\π1= [40−20(Q1+Q2)]×Q1− (12+Q1)\\π1= [40−20Q1− 20Q2]×Q1− 12−Q1\\π1= 40Q1−20Q12− 20Q2Q1− 12−Q1$

Differentiate the profit function with respect to Q1 to get the Best response (BR1) function of firm 1. 

$\frac{∂π1}{∂Q1}= 40−40Q1− 20Q2− 0−1\\\frac{∂π1}{∂Q1}=0\\ 40−40Q1− 20Q2− 0−1= 0\\40Q1= 39−20Q2\\Q1=\frac{39−20Q2}{40}$

This equation is known as BR1.

Firm2,

Profit2 = Total Revenue2 - Total cost2

$π2= TR2−TC2\\π2=P×Q2−TC2\\π2= [40−20(Q1+Q2)]×Q2− (12+Q2)\\π2= [40−20Q1− 20Q2]×Q2− 12−Q2\\π2= 40Q2−20Q1Q2− 20Q22− 12−Q2$

Differentiate the profit function with respect to Q2 to get the Best response (BR2) function of firm 2. 

$\frac{∂π2}{∂Q1}= 40−20Q1− 40Q2− 0−1\\\frac{∂π2}{∂Q1}=0\\ 40−20Q1− 40Q2− 0−1= 0\\40Q2= 39−20Q1\\Q2=\frac{39−20Q1}{40}$

This equation is known as BR2.

Now, solve BR1 and BR2 to get Q1 and Q2.

$Q1=\frac{39−20(\frac{39−20Q1}{40})}{40}\\Q1=\frac{39−(\frac{39−20Q1}{2})}{40}\\Q1=\frac{78−39+20Q1}{80}\\80Q1=78−39+20Q1\\60Q1=39\\Q1= 0.65$

Put the value of Q1 in the BR2 function,

$Q2=\frac{39−20(0.65)}{40}\\Q2=0.65$

So,

Q = Q1 + Q2 

Q = 0.65 +0.65 

Q= 1.3

The price is,

P = 40-20Q

P= 40- 20 (1.3)

P = 14

The profit of firm 1,

$π1= P*Q1−(TC1)\\π1= 14*0.65−(12+0.65)\\π1= 9.1−12− 0.65\\π1=−3.55$

The profit of firm 2,

$π2= P*Q2−(TC2)\\π2= 14*0.65−(12+0.65)\\π2= 9.1−12− 0.65π2=−3.55$

So, the output is 0.65 and the profit is -3.55 of the Daily limited firm. The output is 0.65 and the profit is -3.55 of Joy firm. 

 

 b.

From c part, we know,

The Stackelberg leader firm (Joy) has an output of Q2= 0.975 and the follower firm (Daily limited) has an output of Q1= 0.4875

We have a demand function,

P= 40-20Q1 - 20Q2

When the subsidy is provided to firm 2, the subsidy is deducted from the price.

Since Q1 = 0.4875, the demand function of firm 2 will be,

P= 40-20(0.4875) - 20Q2

P = 30.25- 20Q2

If subsidy given by the government is S per unit to the Joy firm, 

P- S= 30.25- 20Q2

Since Q2 should equal to the Stackelberg leader firm output, but the price remains the same.  

14- S= 30.25- 20 (0.975) 

14-S= 10.75

S=14-10.75

S= 3.25

The subsidy should be 3.25 per unit given to joy firm to increase its output to Stackelberg leaders' output. 

c.

The leader firm is Joy limited, that is, firm 2, and the follower firm is daily limited, that is, firm 1.

We have the profit function of firm 2 and BR2 function of firm 1 from a part. 

Profit function of firm 1,

$π2= 40Q_2−20Q_1Q_2− 20Q_2^2− 12−Q_2$

BR2 function of firm 2,

$Q_1=\frac{39−20Q_2}{40}$

In the Stackelberg model, we put the BR function of the follower firm in the profit function of the leader firm.

$π2= 40Q2−20(\frac{39−20Q2}{40})Q2− 20Q_2^2− 12−Q2\\π2= 40Q2−(\frac{39−20Q2}{2})Q2− 20Q_2^2− 12−Q2\\π2= 40Q2−\frac{39Q_2}{2}+\frac{20Q_2^2}{2}− 20Q_2^2− 12−Q2\\π2= 40Q2−19.5Q2+10Q_2^2− 20Q_2^2− 12−Q2\\π2= 19.5Q2−10Q_2^2− 12$

Now, differentiate the profit function to find Q2.

$\frac{∂π2}{∂Q1}= 19.5−20Q2= 0\\20Q2= 19.5\\Q2= 0.975$

Put Q2 in the BR1 function of firm 1 to get Q1.

$Q1=\frac{39−20(0.975)}{40}\\Q1=0.4875$

So,

Q = Q1 + Q2 

Q = 0.975+ 0.4875

Q=1.4625

The price is,

P = 40-20Q

P= 40- 20 (1.4625)

P = 10.75

The profit of firm 1,

$π1= P*Q1−(TC1)\\π1= 10.75*0.4875−(12+0.4875)\\π1= 5.24−12− 0.4875π1=−7.25$

The profit of firm 2,

$π2= P*Q2−(TC2)\\π2= 10.75*0.975−(12+0.975)\\π2= 10.48−12− 0.975\\π2=−2.5$

So, the output is 0.4875 and the profit is -7.25 of the Daily limited firm. The output is 0.975 and the profit is -2.5 of Joy firm