Question

5.* MBI and Pear are the only two producers of computers. MBI started producing earlier than Pear. MBI's costs of production are given by C1(/1) = y?. Pear's cost function is C2(y2) = The national demand for computers is y 106 - 105p (a) Calculate the Stackelberg equilibrium in which MBI is the leader in this market. Indicate output levels, market price, and the profits of each firm 5y2 (b) Suppose that both firms enter this market at the same time. Calculate the Cournot equilibrium and compare it to the situation in part (a)

Answer 1

Solution:

(a) The cost function of MBI is C1(y1) = y1²

Cost function of Pears is C2(y2) = 5y2

aggregate demand = y1+y2 = y =10⁶ - 10⁵p

Therefore inverse demand function is:

p = 10 - y/10⁵

MBI is the leader and Pears is the follower firm

Profit function of Pears firm is

Profit2 = P*y2 - C2

= (10 - (y1+y2)/10⁵)*y2 - 5y2

By derivating profit function w.r.t to y2, we get

= 10- (y1+2y2)/10⁵ - 5 = 0

y2 = (5*10⁵ -y1)/2

Profit function of MBI

Profit 1 = P*y1 - C1

(10 - (y1+y2)/10⁵)*y1 - y1²

Substituting value of y2 in y1

(10-((5*10⁵ -y1)/2 )+y1) /10⁵)*y1 - y1²

⁼(10- (5*10⁵ +y1) /2*10⁵)*y1 - y1²

Derivating above equation w.r.t to y1

(10 - (5*10⁵ + 2y1) /2*10⁵) - 2y1=0

2y1 * 2*10⁵ +2y1 = 10-5*10⁵

y1 = (10-5*10⁵ )/(4*10⁵ + 2)

substituting value of y1 in y2

y2 = (5*10⁵ -y1)/2

y2 =( 20*10¹⁰ + 10^{6 _-} (10-5*10⁵ ) ) / (4*10⁵ + 2)

Substituing value of y1 and y2 in p

p = 10 - (y1+y2)/10⁵

p = 10 - (((10-5*10⁵ )/(4*10⁵ + 2))+(( 20*10¹⁰ + 10^{6 _-} (10-5*10⁵ ) ) / (4*10⁵ + 2)))/10⁵

= 10- (2*10¹¹ + 10⁶)/10⁵ (4*10⁵ + 2)

(B) Cournot equilibrium

Profit function of Pears firm is

Profit2 = P*y2 - C2

= (10 - (y1+y2)/10⁵)*y2 - 5y2

By derivating profit function w.r.t to y2, we get

= 10- (y1+2y2)/10⁵ - 5 = 0

y2 = (5*10⁵ -y1)/2 (BR eqution of firm 2)------------------------(1)

Profit function of MBI

Profit 1 = P*y1 - C1

= (10 - (y1+y2)/10⁵)*y1 - y1²

Derivating it w.r.t to y1, we get

10 - (2y1+y2)/10⁵ - 2y1 =0

substituting value of y2 in the above equation we get

= 10 - 3y1/2*10⁵ - 5/2 -2y1 =0

15/2 = (3+4*10⁵) *y1 / 2*10⁵

15*10⁵ = (3+4*10⁵) *y1

y1 = 15*10⁵ / (3+4*10⁵)

Substituting value of y1 in y2

y2 = (5*10⁵ -y1)/2

= (5*10⁵ -(15*10⁵ / (3+4*10⁵)))/2

=(15*10⁵+20*10¹⁰ - 15*10⁵)/ 2*(3+4*10⁵)

y2 = 10¹¹ / (3+4*10⁵)

Therefore p = 10 - (y1+y2)/ 10⁵

p = 10 - ((15*10⁵ / (3+4*10⁵))+(10¹¹ / (3+4*10⁵))/10⁵

p= 10 - (15*10⁵ + 10¹¹) / 10⁵*(3+4*10⁵)

Price in cournot model is more than stakalberg model