9. Perfect competitionEcon 494Spring 2013

Most of these notes are taken directly from Silb §4.4

Agenda

• Quick recap from last class• Perfect competition

2

3

Recap…

• Desirable properties for a production function:• Positive marginal product • Diminishing marginal product • Isoquants should have

• Negative rate of technical substitution • Diminishing RTS

Where are we going with this?Why do we care?• Production function defines the transformation of inputs into

outputs• Postulates of firm behavior

• Profit maximization• Cost minimization

• Results: shape of production fctn is key to FONC and SOSC• Especially in evaluating comparative statics.

4

Application: Profit maximization and perfect competition• Profit maximizing firm sells its output, , at a constant per-unit

price .• Perfectly competitive output market

• Firm purchases two inputs, and , at a constant per-unit factor price and • Perfectly competitive factor market

• Strictly concave production fctn

5

Strict Concavity

• The function is strictly concave if its Hessian matrix is negative definite (ND).

• A negative definite matrix has leading principal minors with determinants that alternate in sign, starting with negative:• etc.

• Alternatively:

• Diagonal elements of H are all • This is a sufficient condition for ND

6

7

Perfect competition

1 21 2 1 2 1 2 1 1 2 2

,( , ; , , ) ( , )

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

1 1 2 1 1 2 1 1 1 2 1

2 1 2 2 1 2 2 2 1 2 2

( , ) ( , ) 0 ( , )

( , ) ( , ) 0 ,

NC

)

F

(

O

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

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

1 11 1 2 11 1 2 11

22 1 2 22 1 2 22

2 211 22 12 11 22 12

12 1 2 12 1 2

( , ) ( , ) 0 0

( , ) ( , ) 0 0

0 0

where ( , ) ( ,

SOSC ( is negative definit

)

e)

x x p f x x f

x x p f x x f

f f f

x x p f x x

H

H

H

Interpret FONC

• The profit maximizing firm will employ resources up to the point where the marginal contribution of each factor to revenues, , is equal to the cost of acquiring additional units of that factor, .

• Factors are paid () the value of their marginal products (, also referred to as )

• Note the FONC imply (positive MP)

8

1 1 2 1 1 2 1 1 1 2 1

2 1 2 2 1 2 2 2 1 2 2

( , ) ( , ) 0 ( , )

( , ) ( , ) 0 ,

NC

)

F

(

O

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

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

1 2

1 2 1 2 1 2 1 1 2 2,

( , ; , , ) ( , )x x

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

Interpret SOSC

• Diminishing MP• Production function strictly concave• Read Silb p. 76 for more discussion on SOSC

9

1 11 1 2 11 1 2 11

22 1 2 22 1 2 22

2 211 22 12 11 22 12

12 1 2 12 1 2

( , ) ( , ) 0 0

( , ) ( , ) 0 0

0 0

where ( , ) ( ,

SOSC ( is negative definit

)

e)

x x p f x x f

x x p f x x f

f f f

x x p f x x

H

H

H

Solve FONC implicitly

• By the IFT, because the SOSC are non-zero, we can solve the two FONC simultaneously for the explicit choice functions:

• These two functions represent factor demand functions• The quantity of each factor that will be hired as a function of factor prices

and output price.

10

1 1 2 1 1 2 1

2 1 2 2 1 2 2

( , ) ( , )

FONC

0

( , ) ( , ) 0

x x p f x x w

x x p f x x w

Comparative statics:How do prices affect factor demand?

• Substitute into FONC to get identities:

11

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

1 2 1 2

, , , , ,x x x x x x

w w p w w p

* *1 1 1 2 2 1 2 1

* *2 1 1 2 2 1 2 2

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

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

p f x w w p x w w p w

p f x w w p x w w p w

Effect of a change in input price

• Differentiate each identity wrt :

• Express in matrix form…

12

* *1 1 2 2 21 1 1

1* *

2 2 211 2 2

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

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

p f x w p x w p

p f x w p x w p w

w w w

w w

* ** * * *1 2

21 1 2 22 1 21 1

( ( ), ( )) ( ( ), ( )) 0x x

p f x x p f x xw w

* ** * * *1 2

11 1 2 12 1 21 1

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

p f x x p f x xw w

Express in matrix form

• Matrix form:

13

* ** * * *1 2

11 1 2 12 1 21 1

* ** * * *1 2

21 1 2 22 1 21 1

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

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

x xp f x x p f x x

w w

x xp f x x p f x x

w w

*11 12 1 1

*21 22 2 1

1

0

pf pf x w

pf pf x w

1

1

*111 12 1 1

*21 22 22 1

w

w

x w

x w

Cramer’s rule (for )

14

1

1

*111 12 1 1

*221 22 2 1

1

0w

w

pf pf x w

pf pf x w

12

*221 22

1

22

1

00

SOSC: 0 and 0

pf

pfx pf

w

pf

H H

H

11

*212 12

1

12

1

00

SOSC: 0 0

pf

pfx pf

w

pf

H H

H

¤

¤

Comparative statics for

• We already know where this will wind up:• Solve FONC for using IFT. • Substitute into FONC to get two identities• Differentiate identities with respect to • Express simultaneous linear equations in matrix form to get the

following:

15

2

2

*111 12 1

*21

2

222 22

0

1

w

w

w

w

pf pf x

pf pf x

1 1 2 1 1 2 1

2 1 2 2 1 2 2

( , ) ( , )

FONC

0

( , ) ( , ) 0

x x p f x x w

x x p f x x w

16

Use Cramer’s rule (for )

*11 12 1 2

*21 22 2 2

0

1

pf pf x w

pf pf x w

*2

12

*221 12

12

0

1reciprocity relationship

pf

pfx pf x

ww

H H

11

*212 11

2

0

10

pf

pfx pf

w

H H

Comparative statics for

• We already know where this will wind up:

• Apply Cramer’s rule…

17

*111 12 11

*221 22 22

p

p

pf pf fx p

pf pf fx p

1 1 2 1 1 2 1

2 1 2 2 1 2 2

( , ) ( , )

FONC

0

( , ) ( , ) 0

x x p f x x w

x x p f x x w

Cramer’s rule (for )

18

*11 12 11

*21 22 22

pf pf fx p

pf pf fx p

1 12

*2 221 1 22 2 12

f pf

f pfx f pf f pf

p

H H

11 1

*21 22 2 11 1 21

pf f

pf fx f pf f pf

p

H H

No refutable implications emerge.

PS6#2, show both cannot be negative.

The supply function

• It is also possible to ask how output changes with prices.• Since , at the optimal solution:

• where is the profit maximizing level of output

• The factor demand curves are functions of prices: •

• Substituting yields the supply function:

• Use this to derive comp. statics for

19

Homogeneity

• Definition: The function is homogeneous of degree if:

• A production fctn is when increasing all inputs by results in a increase in output.

• Example: is :

•

20

Chiang §12.6SH, §12.6Hoy, §11.5

Production functions:What makes sense?

• Does not make economic sense • e.g., doubling all inputs () leads to less output

• Decreasing returns to scale• Double all inputs, get less than double output

• Constant returns to scale• Double all inputs double output

• Increasing returns to scale• Double all inputs, get more than double output

21

Euler’s theorem

22

Silb p. 56Chiang, p. 385-389

1

1 11 1

If ( ,..., ) is HOD(k), then:

( ,..., )

n

n

i n ni i n

g x x

g g gx x x k g x x

x x x

1 1By homogeneity: ( ,..., ) ( ,...,

P OO

)

R Fk

n ng tx tx t g x x

111

1

Differentiate wrt : ( ,..., )( ) ( )

knn

n

txg tx gt k t g x x

tx t tx t

11 1

1

Note : ( ,..., )( ) ( )

kii n n

n

tx g gx x x k t g x x

t tx tx

1 11

Since this holds for all and , it must hold for 1:

( ,..., )

i

n nn

t x t

g gx x k g x x

x x

Corollary to Euler’s Theorem

• Application• If production function has constant returns to scale, , then each marginal

product is .

23

1

1

If ( ,..., ) is HOD( ), then:

The first partial derivatives ( ,..., ) are HOD( 1)n

i n

g x x k

g x x k

Paying for the inputs

• The FONC for the perfectly competitive firm tell us: • Factors are paid () the value of their marginal products ()

• This analysis was developed in a “partial equilibrium” framework• Each factor analyzed independently

• How is it possible to be sure that the firm is capable of making payments to both factors?• Will enough output be produced to pay for inputs?

24

Paying for the inputs

• If the production function exhibits constant returns to scale, , the sum of factor payments will identically equal total revenue from output:• If each factor, , is paid , then total payment to each is • This means total payment to all factors is:

• And by CRS of production function & using Euler’s Theorem:

25

Paying for inputs

• All this together implies:

• Or…total costs identically equal total revenues, and the product of the firm is exactly “exhausted” in making payments to all factors.

26

1 1 2 2 1 1 2 2 1 1 2 2( )w x w x pf x pf x p f x f x py

Homogeneity of factor demands

• The profit-maximizing factor demand functions are in prices. For any , that is:

• To prove this, we will compare the input choice with prices to the input choice when prices are instead.

• To do this, let’s set up and solve two separate profit-max problems with different sets of prices and compare solutions

27

See Silb §4.5

Homogeneity of factor demands

• When prices are , the firm’s objective function is:

• We’ve already shown the two FONC imply:

• Assuming the SOSC hold, by the IFT, the choice of inputs that maximizes profits is the simultaneous solution to the FONC:

28

1 2

1 2 1 2 1 2 1 1 2 2,

( , ; , , ) ( , )x x

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

1 2( , ) 1,2i ip f x x w i

*1 2( , , ) 1,2ix p w w i

Homogeneity of factor demands

• Now consider the case in which prices are . The firm’s objective function is:

• The two FONC imply:

• Assuming the SOSC hold, by the IFT, the choice of inputs that maximizes profits is the simultaneous solution to the FONC:

29

1 2

1 2 1 2 1 2 1 1 2 2,

( , ; , , ) ( , )x x

Max x x p w w p f x x w xt t t t tw xt

*1 2( , , ) 1, 2ix p w w it t t

1 2( , ) 1,2i it tp f x x w i

Compare results• With prices , the FONC are:

• And with prices , the FONC are:

• Note that the in [2a] cancels out:

• Note that the FONC in [1] and [2c] are identical

30

*1 2 1 2[1] ( , ) ( , , ) 1, 2i i ip f x x w x p w w i

*1 2 1 2[2 ] ( , ) ( , , ) 1, 2i i ia p f x x w x p wt t twt t i

*1 2 1 2

*1 2 1 2

[2 ] ( , ) ( , , ) 1, 2

[2 ] ( , ) ( , , ) 1, 2

i i i

i i i

b p f x x w x p w w i

c p f x x w x p w

t t t t t

t t tw i

Compare results

• Remember that is always the solution to the FONC. • So…

• If the FONC with prices in [1] are identical to the FONC with prices in [2c]

• Then the solutions to [1] and [2c] must also be identical.

• Therefore

is .

31

Le Châtelier PrincipleShort-run vs long-run• It is common to assert that certain factors are “fixed”, at least

in the short-run• Or that certain inputs are more easily varied (less costly)

• Let’s assert that one factor is fixed

• How would a profit maximizing firm react to changes in the wage of one factor, say , if the firm found that it could not vary ?

32

See Silb §4.6

Le Châtelier Principle

• Consider the following constrained maximization problem:

33

1 2

1 2 1 2 1 1 2 2,

02 2

( , ) ( , )

subject to

s

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

x x

Substitute into the objective function, to get an unconstrained problem:

1

1 1 2 1 1 1 20 0 02 2 2( , ; , , ) ( , )s

xMax x p w w px w xf x w xx

What are the choice variables? Parameters?

Optimality conditions

• Note that does not enter this factor demand function. • is fixed cost & irrelevant for choice of in the short-run.

34

1

0 0 01 2 1 2 1 2 1 1 2 2( , ; , , ) ( , )s

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

01 1 1 2 1 0

1 1 1 2

011 11 1 2

FONC

SO

( , ) 0( , , )

( , )

SC

0

ss

s

p f x x wx x p w x

p f x x

Comparative statics ()

Substitute into FONC:

35

0 01 1 1 2 2 1( ( , , ), ) 0sp f x p w x x w

Differentiate wrt

1 111

1 1 11

11 0 0 by SOSC

s sx xp f

w w pf

How does this compare with ?

*1 22

1

Recall: 0x pf

w

H

Compare with (cont.)

36

*1 1 22

1 1 11

1sx x pf

w w pf

H

2 2 2 211 22 11 22 12

11

p f f p f f p f

pf

H

11 222 2

12

11

11 22( )f f f fp f

pf

H

212

11

0 by SOSCpf

f

H

1

1 11

10

sx

w pf

*1 22

1

0x pf

w

H

211 22

11

p f f

pf

H

H

Interpret result

• Both and are negative.• The change in due to a change in its own price is larger in

absolute value when is freely variable than when is fixed.• Demand curves with fewer fixed factors are more elastic• “Second law of demand”

• This comparison only makes sense when the level of employed is the same in both cases.• Local result holds only where the demand curves intersect

37

* 2 *1 1 12 1 1

1 1 11 1 1

0s sx x pf x x

w w f w w

H

Result is local

38

* 21 1 12

1 1 11

0sx x pf

w w f

H

w1

x1

x1*(p,w1 ,w2)

x1s(p,w1 , x2

º)

Note that the comparative static is dx/dw, but the graph is in the opposite order (w-x, not x-w).

Le Châtelier PrincipleAlternative derivation• Consider the identity:

39

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

• “Conditional demand”• Fixed factor set equal to level of that would be chosen by the

firm if all factors were freely variable• i.e., when

• Note the distinction between the identity above and the equality:

*2 1 2

* 01 1 2 1 1 2 ( , , )( , , ) ( , , )s

x p w wx p w w x p w x

Example

• Construction firm has to dig a ditch.• Without constraints, the firm chooses:

• 10 skilled workers with machines ()• 10 unskilled workers with shovels ()

• Suppose labor union steps in.• Contract requires 10 unskilled workers

• Firm still chooses 10 skilled workers ()• Matches unconstrained choice

• What if the cost of skilled workers () increases?

40

Differentiate with respect to

• Note that is a function of (not )• Can we say anything about the sign of this?

41

* *1 2 1 21 1 21( , , ) ( , , ( , , ))sx p w x p wx p ww w

* *

01 1 1 2

1 1 12

s sx x x x

w w wx

* *

01 1 1 2

1 1 12

s sx x x x

w w wx

42

Find sign

• Recall the reciprocity relation:

* *1 1 1 2

01 1 2 1

s sx x x x

w w x w

* *2 1

1 2

x x

w w

• To find , differentiate identity wrt :

* *1 1 1 1 2 12 2( , , ) ( , , ( , , ))sx p w x p x p w www

* *1 1 2

02 2 2

sx x x

w x w

*2

1

x

w

Reciprocity

43

Find sign (cont)*1 1 1

01 1 2

*2

1

s s x

w

x x x

w w x

*11

022

Using reciprocity relationships xx

x w

* * *1 1 2 2

02 2 2 1

Using:sx x x x

w x w w

*1 2102 2

02

ss x x

w

x

xx

2 *1 202 2

0sx x

x w

Long- and short-run output supply

• Show that the long-run output supply function is more elastic than the short-run supply.

• Fundamental identity:

44

*1 2 1 2 1 2*( , , ) ( , , ( , , ))sy w w p y w p x w w p

What does this mean?

45

Differentiate wrt *

1 2 1 2 1 2*( , , ) ( , , ( , , ))sy w w p y w p x w w p

*2

02

* s sy y y x

p p x p

*2

02

* s sy y y

p px

x

p

Reciprocity relation from PS6:* *

i

ix

p

y

w

02 2

* *s sy y y

p p x

y

w

Need to find this

46

Differentiate wrt *

1 2 1 2 1 2*( , , ) ( , , ( , , ))sy w w p y w p x w w p

2

*2

02 2

* sy x

x

y

w w

02 2

*From before:

*s sy y y

p p

y

x w

2 *2

02 2

*0

s sy y y x

p p x w

*L-run supply * is more elastic

sy yy

p p