Question

1.     Producers in three countries have formed a cartel to sell a highly-priced product. Suppose the world demand for the product is consistent with the equation:

P=80-0.005Q_T

 

NB: Q_T is the total cartel sales.

The producers have managed to maximise profit at a price of R70 per unit. Suppose the members have the following total cost functions;

Member 1: TC=7382 + 10Q + 0.05Q²

Member 2: TC= 9374 + 20Q + 0.02Q²

Member 3: TC=7432 + 30Q + 0.03Q²

 

Tasks: Derive:

a)     Marginal revenue

b)    Output of member 1

c)     Output of member 2

d)    Output of member 3

e)    Total cartel output.

Answer 1

a)     Marginal revenue

We know that the producers have managed to maximize profit at a price of R70 per unit.

$TR = P\times Q\\ TR = 70Q$

TR is the same for all the producers and the world demand for the product is consistent with :

$P=80-0.005Q_T\\MR = \frac{dTR}{dQ}$

Now, the world demand for the product $P=80-0.005Q_T$

$TR = (80-0.005Q_T) Q_T\\ TR = 80Q_T - 0.005Q_T^2\\ MR = 80 - 0.01 Q_T$

b)    Output of member 1

$member \space 1:TC=7382 + 10Q_1 + 0.05Q_1^2\\ MC = \frac{dTC}{dQ}\\ MC = 10 + 0.1Q_1$

 The producer will produce that output where MC=MR

$10 + 0.1Q_1 = 70\\ 0.1 Q_1 = 70 - 10\\ 0.1 Q_1 = 60 \\ Q_1 = 600 \space units$

c)     Output of member 2

$member \space 2:TC=9374 + 20Q_2 + 0.02Q_2^2\\ MC = \frac{dTC}{dQ}\\ MC = 20 + 0.04Q_2$ 

The producer will produce that output where MC=MR

$20 + 0.04Q_2 = 70\\ 0.04 Q_2 = 70 - 20\\ 0.04 Q_2 = 50 \\ Q_2 = 1250 \space units$

d)    Output of member 3

$member \space 3:TC=7432 + 30Q_3 + 0.03Q_3^2\\ MC = \frac{dTC}{dQ}\\ MC = 30 + 0.06Q_3$ 

The producer will produce that output where MC=MR

$30 + 0.06Q_3 = 70\\ 0.06 Q_3 = 70 - 30\\ 0.06 Q_3 = 40 \\ Q_3 = 666.67 \space units\\ Q_3 = 667 \space units$

e)    Total cartel output.

Total cartel output = Output of all the three producers members

$= Q_1 + Q_2 + Q_3\\ = 600 + 1250 + 667\\ = 2517 \space units$

Total cartel Output = 2517 units.