Question

Two firms produce closely-related products and have marginal costs MC1=10 and MC2=20. The market supplied by firm 1 has demand Q1=100-2p1+p2, while 2's market has demand Q2=100+p1-2p2. The two firms are engaged in Bertrand price competition.

Two firms produce closely-related products, and have marginal costs MC1-10 and MC2-20. The market supplied by firm 1 has demand Q1 = 100-2p1+P2, while 2's market has demand Q2=100+p1- 2p2. The two firms are engaged in Bertrand price competition. 3(a)What is the intercept of firm 1's price reaction curve? (One digit after the decimal point only) Question 8 1 pts 3(b) What is the slope of firm 1's price reaction curve? (One digit after the decimal point only) Question 9 1 pts 3(c) What is the intercept of firm 2's price reaction curve?

Answer 1

In the Bertrand model, firms compete in prices rather than quantities. Thus, we focus on choosing price that maximises profits.

3.a. The profit for a firm is given by

$\prod$ = Total revenue - Total cost

Where Total Revenue = Price*Quantity and Total price = Average Cost*Quantity.

Since we are given that marginal costs are constant thus, average cost= Marginal cost.

The profit function for firm 1 will thus be given by-

$\prod$1 = p1*Q1- MC1*Q1

=> p1*(100-2p1+p2) - 10*(100-2p1+p2) = 100p1-2p1²+p1p2-1000+20p1-10p2

Using partial differentiation,

$\partial \prod1/\partial p1$ = 100-4p1+p2+20 = 120-4p1+p2

For maximizing, set this equal to zero,

120-4p1+p2 = 0=> 4p1=120+p2 => p1= 30+0.25p2

The last equation is the price reaction of firm 1.

The intercept of this price reaction function is 30.

Question 8. And the slope is the coefficient of p2, i.e, 0.25.

Question 9. Similarly, for firm two we can derive the price reaction curve.

$\prod2$ = p2*Q2 - MC2*Q2 = p2*(100+p1-2p2)- 20*(100+p1-2p2)=100p2+p1p2-2p2²-2000-20p1+40p2

Using partial differentiation,

$\partial \prod2/ \partial p2$= 100+p1-4p2+40 = 140+p1-4p2

Setting this equal to zero for maximizing,

140+p1-4p2=0 => 4p2=140+p1 => p2= 35+0.25p1

Thus,p2= 35+0.25p1 is the price reaction of firm 2.

The intercept is 35 and slope is 0.25.