Question

The flat screen TV market is dominated by Sony (firm 1) and Samsung (firm 2) in a
city. The demand during a typical month for Sony’s product is Q1 = 5000-10P1 +5P2 and
for Samsung’s product is Q2 = 5000-10P2 +5P1, where P1 is Sony’s price, P2 is Samsung’s
price. Both firms’ marginal cost is constant at $100 per TV. Both compete by choosing a
price for the whole month simultaneously.

Task 1: Find the Bertrand Competition Equilibrium. Illustrate it in a graph of best response curves.

Task 2: Now suppose the month before Christmas is a special month where the demand
is higher. The new demand functions are: Q1 = 8000-10P1+5P2 and Q2 = 8000-10P2+5P1.
Find the new equilibriumand show how the best response curves move in a figure with
Sony’s price choice on the horizontal axis and Samsung’s price choice on the vertical axis

Answer 1

Task 1:

Sony

Profit = Total Revenue - Total Cost

Profit = P1*Q1 - 100*Q1

Profit = P1(5000-10P1 +5P2) - 100(5000-10P1 +5P2)

Taking the first derivative of the profit function with respect to P1 and set it equal to zero we get,

5000 - 20P1 + 5P2 +1000 = 0

20P1 = 6000 + 5P2

P1 = 300 + 0.25P2................. This is the best response function of Sony.

Similarly, we get the best response function of Samsung

P2 = 300 + 0.25P1

Put the value of P1 from above we get

P2 = 300 + 0.25(300 + 0.25P2)

P2 = 300 + 75 + .0625P2

0.9375P2 = 375

P2 = 400

P1 = 300 + 0.25P2 = 400

Task 2:

Now Q1 = 8000 - 10P1 + 5P2 and Q2 = 8000 - 10P2 + 5P1

Sony

Profit = Total Revenue - Total Cost

Profit = P1*Q1 - 100*Q1

Profit = P1(8000-10P1 +5P2) - 100(8000-10P1 +5P2)

Taking the first derivative of the profit function with respect to P1 and set it equal to zero we get,

8000 - 20P1 + 5P2 +1000 = 0

20P1 = 9000 + 5P2

P1 = 450 + 0.25P2................. This is the best response function of Sony.

We can see that the slope of the response is same as before however the intercept has increased. Thus the response function will shift up.

Similarly, we get the best response function of Samsung

P2 = 450 + 0.25P1

Put the value of P1 from above we get

P2 = 450 + 0.25(450 + 0.25P2)

P2 = 450 + 112.5 + .0625P2

0.9375P2 = 562.5

P2 = 600

P1 = 450 + 0.25P2 = 600