10. Envelope TheoremEcon 494

Spring 2013

2

Agenda

• Indirect profit function• Envelope theorem for general unconstrained optimization

problems• Theorem• Proof• Application to profit maximization• Graphical illustration

• Reading• Silb §7.1-7.3

Recall Profit maximization

• How would you answer the question:• “What are the firm’s profits at the optimal solution?”

• Substitute the optimal solution, xi*, back into the objective function

• This will give us the indirect objective function or value function.• Note: Until now, we have been substituting xi

* into the FONC for comparative statics

3

1 2

1 2 1 2 1 1 2 2,

* *1 1 2 2 1 2

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

1 1 2 2

( , ) ( , )

solution to FONC:

( , , ) and ( , , )

comparative statics:

, , , , ,

x xMax x x pf x x w x w x

x p w w x p w w

x x x x x x

w w w w p p

Example: Profit maximization

1 2

* *1 1 2 2 1 2 1 2

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

1 2 1 2 1 1 2 2,

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

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

( , ) ( ,

,

)

)

x x

x p w w x p w w p w w

pf x p w w x p w w w x p w w w x p w w

Max x x pf x

p w

w x

w

x x w

4

1 2 1 2 1 2 1 1 2 2Obj Fctn: ( , ; , , ) ( , )x x p w w pf x x w x w x

Substitute solution, xi*(p,w1,w2), into obj. fctn.

P*(p,w1,w2) is the indirect objective function, or indirect profit functionNote difference betw. P*(p,w1,w2) & P(x1,x2)

The indirect objective function is also referred to as the value function.

A simple example

• Consider a monopolist with:• Total Cost:• Inverse demand: • Per-unit tax on output:

• Set up objective function:

5

2

2

( ) ( ) ( )

( ) ( )

(200 2 )

200 3

y

y R y C y t y

p y y C y t yMax

y y y t y

y y t y

6

Simple example (cont.)

• FONC:

• SOSC:

• Solve FONC:

( ) ( ) ( ) 0

( ) ( ) ( ) 0

200 6 0

y R y C y t

p y y p y C y t

y t

( ) ( ) ( ) 0

2 ( ) ( ) ( ) 0

6 0

y R y C y

p y p y y C y

200( ) 200 6 0 *( )

6

ty y t y t

7

Simple example (cont.)

• Find :• Substitute into inverse demand function:

• Find profits at the optimal level of output:• Substitute into the profit function:

200*( )

6

ty t

*( ) ( ) 20000

*( ) 22

6

tp t p y t

2

*( ) ( *( )) 200 3200 200 200

6 6 6

t tt t

ty t

8

Two different functions

• The profit function gives you the level of profits for any given level of output, . • The level of output () may, or may not, be optimal.

• In indirect profit function gives you the optimal level of profits for any given tax, .

2200 200 200

*( ) ( *( )) 200 36 6 6

t t tt y t t

2

( ) ( ) ( )

200 3

y R y C y t y

y y t y

Compare

9

2

2

( ) ( ) ( )

200 3

200 200 200*( ) ( *( )) 200 3

6 6 6

y R y C y t y

y y t y

t t tt y t t

t y*(t) *(t) y (y)

2 33 3267 32 32642 33 3267 33 32672 33 3267 34 32643 32.8 3234.1 32.0 3232.03 32.8 3234.1 32.8 3234.13 32.8 3234.1 34.0 3230.08 32 3072 31 30698 32 3072 32 30728 32 3072 33 3069

10

Indirect profit function is HOD(1)1 2

1 2 1 2

If *( , , ) is HOD(1), then for any :

*( ,

0

, ) *( , , )

t

t t t

p w w

p w w p wt w

1 2

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

*( , , )

( ( , , ), ( , , ))

Proo

( , , ) ( , , )

f

t t t

t t t t t t t t t t t t

p w w

pf x p w w x p w w w x p w w w x t wtw tp

* * *

1 2 1 2

1 2

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

Since are HOD(0): ( , , ) ( , , )

*( , , )

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

i i ix x p w wt t x p w w

p w w

pf x p w w x p w w w x p w w w

t

t t

x p

t

t wt t w

1 2

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

1 2

*( , , )

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

*( (, ) 1),

p w w

pf x p w w x

t t

HOD

p w w w x p w w w x p w

t w

t

t w

p w

Envelope Theorem

• The envelope theorem is one of the most important theorems in economic theory.

• It concerns the rate of change of the objective function (rather than the choice functions) when a parameter changes.

• It is useful for deriving comparative static results

11

Unconstrained optimization

12

Assuming the SOSC hold, the IFT implies that the FONC can be solved simultaneously for the choice functions xi = xi

*(a1,…,an)

1 1( , , ; , , ) 0 1, ,i n ng x x i n

(F a) is the indirect objective function

The FONC are:

1 1 1( , , ) ( , , ; , , )n n nx

Max g x x

Consider the general unconstrained maximization problem:

Unconstrained optimization

• Substituting into the objective function yields:

13

1 1( , , ; , , )n nx

Max g x x

(F a) is the indirect objective functionF represents the maximum value of for any .

This is because the that maximizes for any was substituted back into

*( ) ( ( ); )g α x α α

* *1 1 1 1( ( , , ), , ( , , ); , , )n n n ng x x 1( , , )n

14

Envelope Theorem:How does change when changes?

Define ( ) ( *( ); ) ( ; )

where *( ) is the solution to the

Th

FONC

(assuming SOSC hold).

eorem

xg Max g

α x α α x α

x x α

*( )

Then

( *( ); ) ( ; )

*( *( ); ) ( *( ); )

x x

x

g x g x

xg x g x

Remember that and are vectors

15

Envelope Theorem Proof

* *1

*

22

1

Differentiate ( *( ); ) with

( *( ); ) ( *( ); )

( *( );

respec

( *(

t to :

) ); )i

i

ii

n

i

n

i

g x g x

g x

g

x

xg x

x

x

*( )

By FONC: , therefor

( *( ); ) ( ;

( *( ); ) 0 e

)i i x

i

i xg x g x

g x

Remember that and are vectors

The last equality above follows from the definition of the partial derivative of wrt evaluated at .

* *1 1 1 1( , , ), , ( , , ); , ,n n n ng x x

16

Envelope Theorem Proof

*( )( *( ); ) ( ; )

i ii x xg x g x

**1

1( *( ); ) ( *( ); ) ( *( ); )

i j

nx xi j i i n

j j

xxg x g x g x

Rewriting the above yields:*

1

( *( ); ) ( *( ); )n

kxi j i j i k

k j

xg x g x

Now differentiate the envelope result wrt :

17

Envelope Theorem Recap

• The partial derivative of the indirect objective function wrt a parameter º

The partial derivative of the objective function wrt aparameter, when evaluated at .

• As changes, the rate of change of the max. value of , where and vary optimally as varies =

Rate of change of as changes holding and constant at optimal levels.

*( )

( *( ); ) ( ;

Enve

)

*( *( ); ) (

l

*( ); )

ope Theorem Results

x x

x

g x g x

xg x g x

Proving vs. using the envelope theorem

• If you are asked to prove the envelope theorem for a particular problem, you must follow all of the preceding steps exactly.

• If you are going to apply the envelope theorem, then you can simply use the ET results on the previous slide• To apply the ET:

• Differentiate objective function directly wrt parameter• That is, BEFORE substituting • Evaluate this derivative at the solution

18

19

ET proof for profit max problem

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

2 1 2 2 1 2 2

*1 2

( , ; , , ) ( , ) ( ,

Objective function FONC

Solution

) ( , ) 0

( , ) ( , ) 0

( , ,

to FONC

) 1,2i i

x x p w w pf x x w x w x x x pf x x w

x x pf x x w

x x p w w i

1* * * *

1 2 1 2 2 2 11 1 1 2 2 2 1 2

I

*( , , ) ( ( , , ),

ndirect objective functio

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

n

p w w pf x p w x pw w ww x p w w x p ww w

** * * *

1 1 2 1 2 1

*1 2

1 12 2

1

*1

S

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

0

implif

by FONC 0 by

y

FONC

:

( )* x x

pf x xx w pf x xw

ww w

* * * *

1

* * * * *1 2 1 21 1 2 2 1 2 1 1 2

1 1 1 1 1

*( ( ), ( )) ( (

Differentiate wrt

), ( )) )

:

(x x x x

pf x x pf x x w x ww w w w w

w

20

Factor demand functions

*1

1

*( )x

w

At the P-max input choice, the rate of change of profit when a factor price changes is the same whether we let the input choices vary optimally or they are held fixed at profit maximizing levels.

P*wi = – xi

* is what many economists refer to as Hotelling’s Lemma As you can see, this is just an application of the envelope theorem applied to

a profit max problem.

* * * *1 1

* *1 2

1

*2 1 2 2 11 2

1 1

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

0 by FONC 0 by FO

(

NC

*)pf x x w pf x x w x

x x

w w w

21

Proving vs using the ET revisited*1

1

*( )x

w

To prove that you must go through

all of the steps just shownSet up indirect objective functionDifferentiate wrt w1

Simplify and use FONC=0 to eliminate terms

To use the ETDifferentiate objective function wrt w1

Evaluate at optimal solution:*1 2 1 2

*1 1 1 21 2*1 1

1 2

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

( , , ) i ii i

p w w x xx x p w wx x p w ww w x x p w w

22

Supply function (prove with ET)

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

Indirect objective

( , , )) ( , ,

fun

) ( , , )

ction

p w w f x w w x w w w x w wp p p p pw x w w

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

1 1 2 2 1 2 1 2 1 2

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

Differentiate wrt

(

)) ( ( ), (

:

))x x x x

p f x x f x x f x x w wp p p p

p

p

0 by FONC 0 by

* ** * * * * *1 2

1 1 2 1 2 1 2 2 1FONC

2

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

Simplify:

x xpf x x w pf x x w f x x

p p p

* *1 2

*( ( ), ( )) *( )f x x y

p

* *1 2

* that ( ( ), ( )) *Prove ( )f x x y

p

Supply function (use ET)

1 2 1 2

*1 2

By the Envelope Theorem:

*( , , ) ( , )

( , , )i i

p w w x x

p p x x p w w

23

* *1 2

* that Use ET to sh ( ( ), ( )) *( )ow f x x y

p

1 2 *1 2

( , ) ( , , )i if x x x x p w w

* *1 1 2 2 1 2( ( , , ), ( , , ))f x p w w x p w w

1 2*( , , )y p w w

24

Reciprocity relationships

1

* *1 1 2

1

*( , , )w x p w w

w

1 2

2 ** 1

1 2 2

*w w

x

w w w

Differentiate with respect to w2:

By Young’s Theorem P*w1w2 = P*

w1w2

1 2 2 1

* ** * 1 2

2 1w w w w

x x

w w

Using symmetry

2 1

2 ** 2

2 1 1

*w w

x

w w w

25

Reciprocity relationships

1

2 ** 1

1

Differentiate wrt p

*

:

w p

x

w p p

1 1

* ** * 1

1

Young's Theorem

w p pw

x y

p w

*1 1 2

1

*( , , )x p w w

w

*

1 2

*( , , )y p w w

p

1

2 **

1

1

1

Differentiate wrt :

*pw

y

p w w

w

Note that these reciprocity results are identical to those we derived earlier using Cramer’s rule to solve for the comparative statics

Graphical Illustration of Envelope Theorem Applied to -max• Suppose we fix prices at • The -max input choices, given these prices:

• and

• Remember:• gives you the optimal choice of for any set of prices.• gives you the optimal choice of for a particular set of prices.

• What is ? Be careful !!!

26

See Silb §7.2

Graphical Illustration of Envelope Theorem Applied to -max• Now let’s see how profits vary as varies,

• holding the other prices constant (at ) • and holding inputs fixed at and

• This gives us the “constrained” profit fctn:

27

0 0 0 0 0 0 0 0 0 02 1 2 1 1 2 2121( , , , , ) ( , )w p x x p f x x x ww w x

• Note that every variable has a superscript 0, except w1.

• Remember that xi0 = xi

*(p0, w10, w2

0) is optimal given that prices are (p0, w1

0, w20).

• If prices are something different, say (p0, w11, w2

0), then xi

0 ¹ xi*(p0, w1

1, w20) is not optimal.

Constrained profit function

• Notice that this constrained profit function is linear in .• What is the slope?• What is the intercept?

• As varies, and are unchanged

• Let’s graph this fctn.

28

0 0 0 0 0 0 0 0 0 02 1 2 1 1 2 2121( , , , , ) ( , )w p x x p f x x x ww w x

w12

P(w12, w2

0 , p0 )

w11

P(w11, w2

0 , p0 )

w10

(w10, w2

0, p0)

w1

(w1, w20, x1

0, x20, p0)

Indirect profit function

• Now think about the indirect profit function• This is different from the constrained profit function.

• Here, as varies, will adjust optimally.• Other prices still constant at .

• Thus, we consider the indirect profit function:

• As opposed to the constrained profit function:

29

xi* are functions that vary optimally with prices

and are not parameters of this function.

xi0 are fixed and cannot vary with

prices. They are parameters of this function.

30

Indirect vs. constrained profit function

0 02

0 * 0 0 * 0 0 * 0 0 0 * 0 01 2 2 2 1 2 2 2 2

0 0 0 02 1 2

0 * 0 0 * 0 0 *1 2

1

1 1 1 1 1

1

10 01 2 11 2

Indirect profit function:

*( , , )

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

Constrained profit function:

( , , , , )

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

p w

p f x w p x w p x w p w x w p

p w x x

p f x w p x ww

w

w w w w w

w

pw xw

0 0 0 * 0 02 2 2 2

0 01 1

0 0 0 0 02 21

01 1 2

, , ) ( , , )

( , )

w p w x w p

p f

w

x x x

w

w w x

Shape of indirect profit function

• Consider where the indirect profit function lies in relation to the constrained profit function.

• By definition is the maximum profits for any ( and are fixed at )

• are optimal given prices are

• Therefore, when prices are , the values of the indirect and constrained profit functions must be equal:

31

Shape of indirect profit function

• However, is not optimal when prices are different, say .

• Therefore, if , then does not yield maximum profits. • And, the indirect profit function must in general lie above the straight line

32

Graph: Indirect profit function is convex in

33

w10

(w10, w2

0, p0)

w1

*(w1, w20, p0)

(w1, w20, x1

0, x20, p0) = p0 f(x1

0, x20)-w1 x1

0-w20 x2

0

The indirect profit function lies above the straight line Except when prices are

Graphical illustration in more detail

34

w1

P

2 21 1

* 0 0 0 0 0 02 2 1 2( , , ) ( , , , , )w p w p xw w x

21w 0

1w 11w

0 0 0 02 1 21( , , , , )xw w p x

1* 0 0

2( , , )w pw

0 0 0 02

21 21( , , , , )w p x xw

0 0 0 02

01 21( , , , , )w p x xw

0 0 0 02

11 21( , , , , )w p x xw

0 01 1

* 0 0 0 0 0 02 2 1 2( , , ) ( , , , , )w p w p xw w x

1 11 1

* 0 0 0 0 0 02 2 1 2( , , ) ( , , , , )w p w p xw w x

A

B

C

Convexity in

• lies above on both sides of • must be more convex (or less concave) than • Since is linear, must be convex in

• Convexity in has important implications for comparative statics.

35

2

1 21

*1 1 2

1

2 *1

21 1

*Convexity in 0

*Hotelling's Lemma ( , , )

*Differentiate both sides 0

ww

x p w ww

x

w w

Consider another constrained profit fctn

• Now, consider a different price for input : (other prices unchanged, )

• At prices , the profit maximizing choices of inputs are • We can define another constrained profit function:

• This is another linear function, with slope • Where does this function lie in relation to the indirect profit function and

the previous constrained profit function ?

36

37

Another constrained profit function• We can show that the two constrained profit functions are

related to each other as follows:

w10 w1

P

P (w1, w20, p0, x1

1, x21)

w11

A: P*(w 10, w2

0, p0) > P (w10, w2

0, p0, x11, x2

1)

B: P*(w 11, w2

0, p0) = P (w11, w2

0, p0, x11, x2

1)

P (w1, w20, p0, x1

0, x20)

Show relationship in previous graph

• Since and are profit maximizing input choices at prices , the constrained and indirect profit functions are equal when

• Constrained profit function passes through point B and is equal to the indirect profit function at that point.

• Now decrease from to • Since input choices vary optimally along indirect profit function and are

fixed along a constrained profit function, it must be true that the constrained profit function lies below point A.

• Imagine doing this for all possible values of envelope

38

39

Indirect profit function as “envelope” of all constrained profit functions

w10 w1

*(w1, w20, p0)

(w1, w20, x1

0, x20, p0)

w11

A

B

(w1, w20, x1

1, x21, p0)