Monopoly and Oligopoly

Monopoly and Oligopoly

1. Third-degree price discrimination

Since the market is separated into two groups, submarket, then we face two demand functions, )(11y p and )(22y p . Thus, profit maximization problem can be described as

)()()(:21222111,2

1y y c y y p y y p Max y y +-⋅+⋅=π

Hence, FOC is

)()(2111y y MC y MR +=, and )()(2122y y MC y MR +=

Other situation: combination of these two separated markets

)()()(:2121212

1y y c y y y y p Max y y +-+⋅+=+π

Horizontal summation: )()()(1

121121p p p p p p y y y ---=+=+= Optimal solution: )()(y MC y MR =

2. Oligopoly behavior

A. Cournot model —— Simultaneously quantity setting

Conditions: market demand )(21y y b a p +-=, products are identical, marginal cost of these two firms are equal and constant,c MC MC ==21, they decide their output decision simultaneously without knowing other ’s decision advanced. Maximization problem:

)(:111y c y p Max -⋅=π s.t. )(21y y b a p +-= )(:222y c y p Max -⋅=π s.t. )(12y y b a p +-=

Solution:

(1) Since 11211)]([y c y y y b a ⋅-⋅+-=π, then, its FOC is shown as

02121

1

=---=∂∂c by y b a y π, after rearranging, this yields the reaction function of firm 1, b

y b c a y 22

1--=

(2) The same, we have the reaction function of firm 2

b

y b c a y 21

2--=

(3) Combine these two reaction functions, we have

⎪⎩

⎪⎨⎧-=

-=b c a y b

c a y 33*

2*1

B. Bertrand model —— Simultaneously price setting

C. Stackelberg model (quantity leadership model)

Conditions: market demand )(21y y b a p +-=, products are identical, marginal cost of these two firms are equal and constant,c MC MC ==21, however, firm 1 is quantity leader, who makes his output decision first, firm 2 is follower, who makes his decision with knowing firm 1’s decision. Maximization problem:

)(:111y c y p Max -⋅=π

s.t. )(21y y b a p +-= & firm 2’s reaction function )(122y f y =

)(:222y c y p Max -⋅=π s.t. )(12y y b a p +-=

Solution: backward induction

(1) Since 22122)]([y c y y y b a ⋅-⋅+-=π, thus,

022122=---=∂∂c by y b a y π, hence, firm 2’s reaction function is b

y b c a y 21

2--= (2) And 11211)]([y c y y y b a ⋅-⋅+-=π, substitute firm 2’s reaction function, then Iso-profit curves can be shown as

)(2

1

21111by cy ay --=π

thus, optimal choice appears when

0)2(2

1

111=--=by c a dy d π

So that ⎪⎩

⎪⎨⎧-=

-=b c a y b

c a y 42*

2*1

D. Price leadership model

Conditions: market demand )(21y y b a p +-=, products are identical, marginal cost of these two firms are 2212 ,y MC c MC ⋅==α, however, firm 1 is price leader, who makes his pricing decision first, firm 2 is follower, who makes his decision with knowing firm 1’s decision.

Traditional Method:

Taking use of backward induction: Maximization problems of follower

)(:2222y c y p Max -⋅=π

FOC: 222)(y y MC p α==

Thus, the reaction function of follow is α

2)(22p

p f y =

=, in other words, it ’s supply function of the follower.

Then we consider the maximization problem of price leader

)(:1111y c y p Max -⋅=π

s.t. )()(2121y y b a y y D p +-=+= Substituting α

2)(22p

p f y =

= into the above constraint, then, p b b a y )211(1α

+-=

, more generally, )(11p f y =, which is residual demand for firm 1. Then the maximization problem becomes

))(()(:1111p f c p f p Max p

-⋅=π

Thus, FOC is 0)(1

1111=-+dp

df df dc dp df p p f Hence,22*

c b a p ++=αα, then 2)211(2*1c b b a y α+-=, an

d then α

α424*

2c b a y ++=