Question

Q.3 Two firms (i 1, 2) produce differentiated products. The demand function for the product of firm i is given by: qiVi, pj) 4-pi + 2pj firm i and pj the price chosen by its competitor. Firm 1 chooses its price first and firm 2 chooses its price after observing the price of firm 1. The cost function of each firm is G(%) 21. Find the subgame-perfect Nash equilibrium. , where Pi is the price chosen by

Answer 1

For firm i,

Total revenue (TRi) = p x qi = 60qi - qi² - 0.5qiqj

Marginal revenue (MRi) =
For firm i, Total revenue (TRi) = p times qi = 60qi - qi^2 - 0.5qiqj Marginal revenue (MRi) = partial differential Tri/partial differential qi = 60 - 2qi - 0.5qj Equating MRi and MCi, 60 - 2qi - 0.5qj = 10 2qi + 0.5qj = 50 4qi + qj = 100........(1) [Best response, firm i] For firm j, Total revenue (TRj) = p times qj = 60qj - qiqj - 0.5qj^2 Marginal revenue (MRj) = partial differential Tri/partial differential qj = 60 - qi - qj Equating with MCj, 60 - qi - qj = 10 qi + qj = 50.........(2) [Best response, firm j] From equation (1), When qi = 0, qj = 100 (Vertical intercept) and when qj = 0, qi = 100/4 = 25 (Horizontal intercept) From equation (2), When qi = 0, qj = 50 (Vertical intercept) and when qj = 0, qi = 50 (Horizontal intercept) Best response functions are in following graph. Nash equilibrium is obtained by solving equations (1) and (2). Subtracting (2) from (1) we get, 3qi = 50
TRi/
For firm i, Total revenue (TRi) = p times qi = 60qi - qi^2 - 0.5qiqj Marginal revenue (MRi) = partial differential Tri/partial differential qi = 60 - 2qi - 0.5qj Equating MRi and MCi, 60 - 2qi - 0.5qj = 10 2qi + 0.5qj = 50 4qi + qj = 100........(1) [Best response, firm i] For firm j, Total revenue (TRj) = p times qj = 60qj - qiqj - 0.5qj^2 Marginal revenue (MRj) = partial differential Tri/partial differential qj = 60 - qi - qj Equating with MCj, 60 - qi - qj = 10 qi + qj = 50.........(2) [Best response, firm j] From equation (1), When qi = 0, qj = 100 (Vertical intercept) and when qj = 0, qi = 100/4 = 25 (Horizontal intercept) From equation (2), When qi = 0, qj = 50 (Vertical intercept) and when qj = 0, qi = 50 (Horizontal intercept) Best response functions are in following graph. Nash equilibrium is obtained by solving equations (1) and (2). Subtracting (2) from (1) we get, 3qi = 50
qi = 60 - 2qi - 0.5qj

Equating MRi and MCi,

60 - 2qi - 0.5qj = 10

2qi + 0.5qj = 50

4qi + qj = 100........(1) [Best response, firm i]

For firm j,

Total revenue (TRj) = p x qj = 60qj - qiqj - 0.5qj²

Marginal revenue (MRj) =
For firm i, Total revenue (TRi) = p times qi = 60qi - qi^2 - 0.5qiqj Marginal revenue (MRi) = partial differential Tri/partial differential qi = 60 - 2qi - 0.5qj Equating MRi and MCi, 60 - 2qi - 0.5qj = 10 2qi + 0.5qj = 50 4qi + qj = 100........(1) [Best response, firm i] For firm j, Total revenue (TRj) = p times qj = 60qj - qiqj - 0.5qj^2 Marginal revenue (MRj) = partial differential Tri/partial differential qj = 60 - qi - qj Equating with MCj, 60 - qi - qj = 10 qi + qj = 50.........(2) [Best response, firm j] From equation (1), When qi = 0, qj = 100 (Vertical intercept) and when qj = 0, qi = 100/4 = 25 (Horizontal intercept) From equation (2), When qi = 0, qj = 50 (Vertical intercept) and when qj = 0, qi = 50 (Horizontal intercept) Best response functions are in following graph. Nash equilibrium is obtained by solving equations (1) and (2). Subtracting (2) from (1) we get, 3qi = 50
TRj/
For firm i, Total revenue (TRi) = p times qi = 60qi - qi^2 - 0.5qiqj Marginal revenue (MRi) = partial differential Tri/partial differential qi = 60 - 2qi - 0.5qj Equating MRi and MCi, 60 - 2qi - 0.5qj = 10 2qi + 0.5qj = 50 4qi + qj = 100........(1) [Best response, firm i] For firm j, Total revenue (TRj) = p times qj = 60qj - qiqj - 0.5qj^2 Marginal revenue (MRj) = partial differential Tri/partial differential qj = 60 - qi - qj Equating with MCj, 60 - qi - qj = 10 qi + qj = 50.........(2) [Best response, firm j] From equation (1), When qi = 0, qj = 100 (Vertical intercept) and when qj = 0, qi = 100/4 = 25 (Horizontal intercept) From equation (2), When qi = 0, qj = 50 (Vertical intercept) and when qj = 0, qi = 50 (Horizontal intercept) Best response functions are in following graph. Nash equilibrium is obtained by solving equations (1) and (2). Subtracting (2) from (1) we get, 3qi = 50
qj = 60 - qi - qj

Equating with MCj,

60 - qi - qj = 10

qi + qj = 50.........(2) [Best response, firm j]

From equation (1),

When qi = 0, qj = 100 (Vertical intercept) and when qj = 0, qi = 100/4 = 25 (Horizontal intercept)

From equation (2),

When qi = 0, qj = 50 (Vertical intercept) and when qj = 0, qi = 50 (Horizontal intercept)

Best response functions are in following graph.

Nash equilibrium is obtained by solving equations (1) and (2).

Subtracting (2) from (1) we get, 3qi = 50

qi = 16.67

qj = 50 - qi [From (1)] = 50 - 16.67 = 33.33

p = 60 - 16.67 - (33.33/2) = 43.33 - 16.67 = 26.67