= CB q ( ) p x dx ( ) i i 0 Hence the gain to type i if she - - PDF document

Optimal Pricing - - solving analytically A plan ( , ) q R is a fee R for q units ( ) ( )

q i i

CB q p x dx = ∫ Hence the gain to type i if she chooses the plan ( , ) q R is ( , ) ( ) ( )

q i i i

U q R CB q R p x dx R = − = −

∫

. An indifference curve is depicted below. ( )

i i i

p q a b q = −

i

a

i

q ( )

i i

CB q ( , )

i

U q R k = q R

i

q slope is ( )

i i

p q

Around an indifference curve ( , )

i

U q R k = we have ( )

q i

p x dx R k − =

∫

and hence ( )

q i

R p x dx k = −

∫

. Differentiating, the slope of the indifference curve is ( )

i

dR p q dq = Special case: Linear demands

2 1 2

( ) ( )

q i i i i

CB q p x dx a q b q = = −

∫

.

Hence the gain to type i if she chooses the plan ( , ) q R is

2 1 2

( , )

i i i

U q R a q b q R = − − . Example:

1 2

( ) 14 2 , ( ) 24 2 , 4 p q q p q q c = − = − = Choose any

1

q and then the total payment which leaves this type indifferent between choosing this plan and taking the smaller plan (buying nothing.)

2 1 1 1 1 1 1

( , ) 14 U q R q q R = − − = . Hence

2 1 1 1

14 R q q = − . Let

2 2

( , ) q R be the plan chosen by type 2. From the figure below, ( , )

i

U q R k = q R

i

q cq

* i

q

the firm maximizes profit by moving around the indifference curve for type 2 until the slope of the indifference curve equals the slope of the cost line. Therefore

* * 2 2 2

( ) 24 2 4 p q q = − = and so

* 2

10 q = . If type 2 selects plan 1 his utility is

2 2 2 2 1 1 1 1 1 1 1 1 1 1

( , ) 24 24 (14 ) 10 U q R q q R q q q q q = − − = − − − = . If he selects plan 2 his utility is

* * * 2 2 2 2 2 2 2 2

( , ) 24 ( ) 140 U q R q q R R = − − = − . The firm then chooses the payment so that he is (almost) indifferent between this plan and plan 1. Therefore

* 2 2 2 2 1 1

( , ) ( , ) U q R U q R = and so

2 1

140 10 R q − = Collecting results,

2 1 1 1

14 R q q = − and

2 1

140 10 R q = − . Let n be the number of type 1 buyers and let an be the number of type 2

• buyers. Total profit is

1 1 2 2

( 4 ) ( 4 ) n R q an R q Π = − + −

2 1 1 1

[10 (100 10 )] n q q a q = − + − It is now an easy matter to solve for the optimal quantity for plan 1.

1 1 1

[10 2 10 ] [10(1 ) 2 ] n q a n a q q ∂Π = − − = − − ∂ .

If 1 a ≥ this is always negative hence

* 1

q = . If 1 a < ,

* 1

5(1 ) q a = − . Suppose 0.4 a = . Then

* 1

3 q = Since

2 1 1 1

14 R q q = − and

2 1

140 10 R q = − . We can solve for the optimal fees for each plan. Exercise 1: Solve analytically for the profit maximizing plan if

1( )

20 2 p q q = − ,

2( )

24 2 p q q = − , 2 c = and there are equal numbers of each type. Check your answer using Solver. Exercise 2: Solve again if instead

2( )

20 p q q = − . Hint: Carefully check the constraint for ype 2.