6. Optimization with 2 variables: Price discrimination

Econ 494

Spring 2013

See Handout 5 for general case with

two variables

Agenda

• Moving to more than one variable• Application: Price discriminating monopolist• Until now, there has only been one choice variable: .• Many economic problems involve more than one choice variable.• For now, we will deal with only 2 variables: and .

• Will later extend to variables.

• Problem set 4 due Wed 2/20• Midterm 1 Wed 2/27 – up to and including Linear Algebra.

2

3

Calculus review: Optimization with 2 variables

Maximum Minimum

FONC(both must hold)

SOSC(all 3 must hold)

( , )z g x y

( , ) 0; ( , ) 0x yg x y g x y

2 0xx yy xyg g g

0; 0xx yyg g 0; 0xx yyg g

These are the optimality conditions when there are only 2 variables. Later, you will see a more general version of this using matrices.

Calculus review: Young’s theorem

• The order of differentiation does not matter• Result extends to any 2nd partial derivative of a function of

many variables• This symmetry result will come in handy…

4

2 2

For any function ( , ) with continuous 2nd derivatives,

( , ) ( , )or ( , ) ( , )xy yx

g x y

g x y g x yg x y g x y

x y y x

5

Application: price discrimination

• A monopolist produces a single good that is sold in 2 separate markets.• Assume the monopolist is able to segment the 2 markets. Example:

• Student and adult prices at movies• Sells product in two different states or countries

• Inverse demand in market is • Total output sold is • Total costs are

See Silb p. 71, example 3

Aside: Total output

• Recall that , which we can also write as

6

1 2 1 2

1

1 2

2

If ( , )

Then 1

1

and 1

y y y y y

y

y y y

y y y

y

Aside: The Cost Function

• Using , there are a number of identical ways to express the cost function :

7

1

1 2 1 2

1 1 1

Differentiate wrt :

( ) ( , ) ()

)(1

y

dC y y C y y C y

dC

y yy

y

y y

2

1 2 1 2

2 2 2

Differentiate wrt :

( ) ( , ) ()

)(1

y

dC y y C y y C y

dC

y yy

y

y y

When , the marginal cost of an additional , i.e., , is the same as the marginal cost of an additional unit of , i.e., .

Intuitively, the good produced is exactly the same regardless of where it is sold, so it should make sense that the marginal costs are identical.

1. Set up objective function

8

Cost:Demand:

Output:

1 2

1 2 1 1 1 2 2 2,

, ) ( ) ( ) ( )y y

Max y y p y y p y y C y

Total revenue:1 1 1 2 2 2( ) ( ) ( )R y p y y p y y

Objective function:

2. Find FONC

9

Interpret FONC:

1 2

1 2 1 1 1 2 2 2,

1. Obj. fctn.: , ) ( ) ( ) ( )y y

Max y y p y y p y y C y

1 1 2 1 1 1 1 1

2 1 2 2 2 2 2 2

( )

(

2. FONC , ) ( ) ( ) 0

, ) 0)( ) ( )

C yy y p y y p

C y

y

y y p y y p y

1 1 2 2( ) ( ) ( )MR y MR y C y

The profit-maximizing monopolist will choose a level of production such that the marginal revenue in each market is the same, and equals the marginal cost of producing the good.

1 2 1 2Recall, if ( , ), the FONC for maximum are: 0 and g 0z g x x g

MC is exactly the same in both equations

3. Find SOSC

10

11 1 2 1 1

22 1 2 2 2

211 22 12 12

3. SOSC , ) ( ) ( ) 0

, ) ( ) ( ) 0

0, where ( )

y y R y C y

y y R y C y

C y

1 2

1 2 1 1 2 2,

1. Obj. fctn.: , ) ( ) ( ) ( )y y

Max y y R y R y C y

1 1 2 1 1

2 1 2 2 2

2. FONC , ) ( ) ( ) 0

, ) ( ) ( ) 0

y y R y C y

y y R y C y

1 2

211 22 11 22 12

Recall, if ( , ), the SOSC for maximum are:

0; 0; 0

z g x x

g g g g g

As long as the SOSC hold, we know profits are maximized.

By the IFT, as long as the SOSC hold, we know that we can, in principle, solve the FONC simultaneously for the explicit choice functions.

Extend this example

• Using the previous example of the price discriminating monopolist…

• We said there were 2 markets. • Assume these 2 markets are:

• 1. China• 2. Japan

• Further, let’s assume that each country imposes a tariff on imports, and that these tariffs differ by country .

11

Price discriminating monopolist

12

1 2

1 2 1 2 1 1 1 2 2 2,

, ; , ) (1 ) ( ) (1 ) ( ) ( )y y

Max y y R y R y C y

1 1 2 2

1 2 1 2

4 comparative static results: , , ,y y y y

Gross revenue in each market

( ) 1, 2( ) ii i i ip y yR iy

To keep notation simple, we will keep the general form of the revenue function.

Net revenue in each market

(( ) 1,) 21 ii iR y i

Step 1: Objective Function:

Optimality conditions

13

1 2

1 2 1 2 1 1 1 2 2 2,

1. Obj. fctn.: , ; , ) (1 ) ( ) (1 ) ( ) ( )y y

Max y y R y R y C y

1 1 2 1 1 1 1

2 1 2 2 2 2 2

2. FONC , ; ) (1 ) ( ) ( ) 0

, ; ) (1 ) ( ) ( ) 0

where ( ) ( ) 1,2i i i i i i

y y R y C

y y R y C

R p y p

y

i

y

y y

11 1 2 1 1 1

22 1 2 2 2 2

211 22 12 12 21

3. SOSC , ) (1 ) ( ) ( ) 0

, ) (1 ) ( ) ( ) 0

0, where ( )

and ( ) 2 ( ) ( )i i i i i i i

y y R y C y

y y R y C y

C y

R y p y y p y

Don’t forget

4. Find explicit choice functions

14

1 1 2 1 1 1

2 1 2 2 2 2

2. FONC , ) (1 ) ( ) ( ) 0

, ) (1 ) ( ) ( ) 0

y y R y C y

y y R y C y

Choice variables:Parameters in FONC:

By the IFT, as long as the SOSC are satisfied, we can solve the FONC simultaneously for the explicit choice functions y1*(t1, t2) and y2*(t1, t2).

* *1 1 2 2 1 2

* *1 1 2 2 1 2

* *1 1 2 1 1 1 1 1 2 1 2( , ), ( , )

* *2 1 2 2 2 2 2 1 2 1 2( , ), ( , )

( , ; ) (1 ) ( ( , )) ( ( , )) 0

( , ; ) (1 ) ( ( , )) ( ( , )) 0

y y

y y

y y R y C y

y y R y C y

5a. Substitute explicit choice function into FONC to get identities:

Understanding Pi

• The 2 first derivatives of the objective fctn. are:

15

1 1 2

2

1

1 2

1 1

2 2 2

(1 ) ( ), ) (

,

)

(1 ) ( )) ( )

R y C y

R y C y

y y

y y

• Note that we do not use the superscript *. This is because these are just first derivatives, and hold for any (y1, y2).

• The maximizing the objective function requires that we set these equal to zero, but this relationship is still an equality. We have not yet substituted the optimal solution back into the FONC.

• Note that Pi are a function of both variables y1 and y2.

Understanding Pi

• When we substitute back into the FONC, we have an identity.• Moreover, these functions are evaluated at a particular

the set of that guarantee that the objective function will be maximized for any given set of parameters, i.e. .

16

Understanding Pi

• In summary…• denotes the first derivative of the objective function and can be

evaluated at any . • If we happen to evaluate at , then • At any other ,

17

*1 2

1 2 ( , ), ) 0

ii y

y y

5. Comparative statics with respect to

• How would an increase in the Chinese import tariff () affect sales in each country (i.e. find and ).

• Remember that the FONC are a system of simultaneous equations

• How to approach this simultaneous equation problem:1. Differentiate both identities wrt

2. This will give us a system of linear equations which we can express with matrices.

3. We will then use Cramer’s rule to find the comparative static results.

• Begin with the 1st identity,

18

Aside: Total Output ()

19

1 2 1 2

Recall that we can express total output as:

( , )y y y y y

* *1 1 2 2 1 2

* * * * *1 1 2 2 1 2 1 2 1 1 2 2 1 2

Substitute the explicit choice functions ( , ) and ( , ) :

*( ( , ), ( , )) ( , ) ( , ) ( , )

y y

y y y y y y

* *1 1 2 2 1 2

* *1 1 2 2 11 2 2

Take the derivative wrt

*( ( , ), ( , )) *( ( , ), ( )) ,

ii i

i

i

y y y y yy

Aside: More on the cost function

20

* *1 1 1 1 2 1 2

* *2 2 2 1 2 1 2

Identity from before (by substituting explicit choice fctns into FONC

(1 ) ( ( , )) ( ( , )) 0

(1 ) ( ( , )) ( ( , )) 0

):

R y C y

R y C y

1 2 1 2 1 2

* *1 1 2 2 1 2

* * * *1 1 2 2 1 2 1 1 2 2

Recall the different ways to express the cost function:

( ( , )) ( , ) ( ) ( )

Substitute the explicit choice functions ( , ) and ( , ) :

( ( ( , ), ( , ))) ( ( , ),

C y y y C y y C y y C y

y y

C y y y C y y

1 2

* *1 1 2 2 1 2

*1 2

( , ))

( ( , ) ( , ))

( ( , ))

C y y

C y

These are all identical ways of expressing the same relationship.

5. Comparative statics ()

21

* *1 1 2 21 1 1

1st identity:

(1 ) ( , ) ( , ) 0R y C y

*1 1

** * *

*1

11 1 1

2

1

Simplify:

(1 ) ( ( ) ((

) ( ( )) ( ( ))( )

( ))

)R y C yy

C y Ry

y

1

* ***2

1

*

1 1

*

1 11

11 1

1

Differentiate wrt :

(1 ) ( (( )

)) ( (( ) (

()

( ) 0)) )R yy

R y yy

Cy

*

1

( )y

1 2To simplify notation, let ( ) ( , )

5. Comparative statics ()

22

* *2 2 2 2121

2nd identity:

(1 ) ( ( , )) ( ( , )) 0R y C y

*2

*1

1

* * *2

12 2

( )

Simplify:

( ( )) (1 ) ( ( )) ( 0( )

( ))y

C y R y C yy

Now combine the 2 results into matrices…

* *2

1

* * 2

1

*1

1 12 2 2

(

Differentiate wrt :

(1 ) ( ( ))) ( )

) 0( ()

)(

Ry

y C yyy

*

1

( )y

Express in matrix form:

23

** * **1

11 1 1

*

11

21

from 1st identity:

(1 ) ( ( )) ( ( )) ( ( )) )(

())

( )(y

R yR y C y C yy

** * *

2 2 21 1

*1 2

from 2nd identity:

( ( )) (1 ) ( ( )( )

) ( (( )

)) 0C y R y C yy y

Notice that the terms above in black are all the 2nd partial derivatives, Pij, evaluated at yi*(). Using this, rewrite the equations above:

*1 1

*2

111 12

*1

1

( ( ))))

)(

(1(

R yyy

*2

1

*

21 221

1

( ) ( )0(2)

y y

The 2 comparative static results are the simultaneous solution to this pair of equations.

In matrix form

24

*1 1

*2

111 12

*1

1

( ( ))))

)(

(1(

R yyy

*2

1

*

12 221

1

( ) ( )0(2)

y y

This pair of simultaneous equations can be expressed in matrix form Ax = b:

11 12

12

*1 1

*

1

2

1

22

*

1

(

( )

(3)( ( ))

0)

R

y

y

y

Use Young’s theorem P12=P21

“Hessian” matrix (H): contains all 2nd partial derivatives

Determinant of the Hessian: 211 22 12 0 (by SOSC)H

Review: Cramer’s rule (2 eqns)

25

The terms of the matrix equation Ax = b are:

11 12 1

21 22 2

1

2

a a x

a a x

b

b

Let A(i) be the matrix obtained by replacing column iof the matrix A with the column vector b:

11 2 11

22 21

1

2 2

(1) and (2)a a

A Aa

b b

bab

Then the solution for xi is:

1 2

(1) (2)and

A Ax x

A A

Apply Cramer’s rule

26

*

11

*

*1 112

2 2

1

1

1

12 2 (Eq

( ( ))

0n (3

)

(

)

)

R

y

y

y

12

22 22

11 12

1

0 0

2

0

22

*

1

1

*1

*1

1 1

Using Cramer's rul

( ( ))

0 ( ( ))( )0

e:

R y

R y

H

y

*1

1

( )0

y

because marginal revenue is always positive (slope of MR is <0).

*22 < 0 by SOSC

|H| > 0 by SOSC

Interpret result

• We get the usual result. • If China () increases the import tariff on goods sold in China (), then the

firm will reduce the quantity sold in China () because the firm’s costs in China, including taxes, have increased.

27

*1 1 2

1

( , )0

y

On your own…

• We just showed that decreases when increases. • What happens to sales in Japan () when the Chinese import

tariff () increases?• Using the same steps, show that if marginal costs are continually

increasing, that is , then:

• Explain the result in economic terms.

28

*2 1 2

1

( , )0

y

Apply Cramer’s rule

29

*

11

*

*1 112

2 2

1

1

1

12 2 (Eq

( ( ))

0n (3

)

(

)

)

R

y

y

y

11

12 12

11 1

0

2

12 22

*1 1 0

*2 1

0

*

1

1

( ( )

Using Cramer's r

(

)

0 ( ( )

u

)

l :

)0

e

y

y

R y

H

R

*2

1

( )0

y

because marginal revenue is always positive.

by assumption.

|H| > 0 by SOSC