Micro Econ1

8/6/2019 Micro Econ1

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Chapter 1TECHNOLOGY-PRODUCTIONFUNCTION


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INTRODUCTION


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Basic concepts Production: Production may be regarded as a

transformation from one state of the world to another.

Acts of production: There are four ways in which thestate of the world may be so change.:

(i) The quantity of a good may be changed: (We canproduce more motor cars )

(i) The quality of a good may be changed: ( We canproduce better motor cars )

(i) The geographical location of a good can bechanged: ( We can deliver a car to a customer )

(i) The time location of a good can be changed: (We canhold a car in stock until the consumer wishes to takedelivery )


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Measurement of inputs and outputs

Inputs: Economic resources such as labor,capital, landare used in producing goodsand services.

Outputs: Goods that a firm produces in theproduction process.

Inputs and outputs are measured in terms offlow (a certain amount of inputs per time

period are used to produced a certain amountof outputs per unit time period).

Example:


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II. Specification of technology

2.1. Definition 1.1: Net output

Suppose the firm has m possible goods toserve as inputs or outputs.

If a firm uses aj units of a goods j as an inputand produces b

j

units of the good j as anoutput , then the net output of good j is givenby : yj =bj -aj .

yj > 0, then the firm is producing more of

good j than it uses as an input; yj< 0, then the firm is using more of good j

than produces it


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Definition ofProduction plan

A production plan is a list of net outputs of

various goods.

We can represent a production plan by avector y in Rm, where yj0 if the jth good

serves as a net output.

Example


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Definition ofProduction possibilities set

The set of all technologically feasible

production plan is called the firms

production possibilities set and will be

denoted by Y, or

Production possibility set of a firm is a sub-

set Y of the space Rm. A firm may select any

vector y Y as its production plan

{ | }m

Y y R y is a feasible production plan


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Figure 1. Production possibilities set

.

Y

0

y

x


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ASSUMPTION .1.Axioms on the Production

Possibility Set, Y

(iv) Y is a closed and bounded set. .

0 ; (possibility of inaction)

( ) 0 ; (no free production).( ) ; (free disposal)

( )m

m

Y

ii Yiii Y

i


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The first of these axioms is called the

possibility of inaction. It says that no inputs

and produce no output.

Since costs are the expense of acquiring

inputs, and revenue the proceeds from the

sale of output

one of implications of this axiom is that

firm profits in the long run need never be

negative .


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Axiom 2 is called the axiom of no freeproduction at least some resources must be

used up in the production of any output). Axiom 3 is called the axiom of free disposal.

It says that the firm can always use unlimitedamounts of inputs to produce no output.

Axiom 4 ensures that the productionpossibility set contains its boundary so thatthere will be an efficient frontier, giving a

well - defined maximum amount of outputthan can be obtained from a given level ofinput.


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Figure 2. A production possibility set

not closed and closed

.

0

y2

y1

0

y1

y2


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Short-run production possibilities set

Suppose a firm produces some outputs from

labor and capital.

Production plans then look like (y,-l,-k).

Suppose that labor can varies immediately

but that capital is fixed as level of in the

short- run . Then

k

( ) ( , , ) :Y k y l k Y k k


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Input requirement set

The production possibility set allows for

multiple inputs and multiple outputs.

However, we will want to consider firms

producing only a single product from many

inputs. It is more convenient to describe the firm's

technology in terms of the inputs necessary to

produce different amounts of the firm's output.

The concept of input requirement set V(y) is

related to require positive amounts of n inputs to

produce a scalar output


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Definition of input requirement set The input requirement set is defined as all

combinations of inputs which produce at least yunits of output. Figure 3. input requirement set

( ) , : ( , )nV y x y y x Y

20 40 60 80 100

0

20

40

60

80

100

V(y)

x1

x2


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Assumptions on the Input Requirement Set

1. (Input regularity)

V(y) is non - empty, closed, and if y>0,then, 0 V(y)

2. (Monotonicity)

If x V(y) and x x, then xV(y);

3. (Convexity)

If x1, x2V(y) and with t 0,1 then tx1+(1-t)x2V(y).


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Input regularity is both a continuity

requirement and an implication of the "no

free production" axiom. Monotonicity says that adding more of any

input can never reduce the amounts of

output produced and it is implied by theaxiom of free disposal.

The axiom of convexity say that any convex

combination of two processes which eachproduce at least y units of output as separate,

third process can produce at least y units.


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Example on properties of v(y)

For an input requirement set:

Assume that the parameters a, b and the output

level is strictly positive. Show that v(y) ismonotonic and convex, nonempty but not closed.

Proof: It is easy to show that v(y) is nonempty andnot closed (since x1>0)

21 2 1 2 1( ) , : , 0V y x x y ax bx x


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Understading v(y) and Y

V(y) is convex but Y may not be convex, for

example:

Consider the technology generated by aproduction f(x)=x2:

The production possibility set is

Y={(y,-x): y x2} which is not convex,

V(y)={x: x y1/2} which is convex set.


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Definition1.6: Isoquant

An isoquant shows the different combinations ofinputs ( labor (L) and capital(K) )with which a firm

can produce a specific quantity of output. A higherisoquant refers to a greater quantity of output and alower one, to smaller quantity of output.

The isoquant is the efficient frontier of the input

requirement set and is where we expect a firmproducing y units of output to choose to operatewhenever inputs are costly:

( ) ( ) & ( );0 1nQ y x x V y x V y


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Figure 4. Isoquant(ng lng)

.

0 20 40 60 80 100 x1

V(y)

20

40

60

80

100

x2

x

Q(y)x V(y)


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Figure 5. Slope of isoquant

The slope of the isoquant at x is the tangent of

the isoquant at x

0 20 40 60 80 100

Q(y) = {x | y = f(x)}

20

40

60

80

100

120

x1

x2

x

Production Function


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Production Function A real valued function f(x) is called a

production function if: f(x) max {y > 0 | x

V(y)}. Since by definition. V(y) = {x | f(x) y}

Q(y) = {x | f(x) = y}.

When V(y) is input regular, the productionfunction is continuous and f(0)=0. If y=f(x) andy>0, then xi>0 for at least one input i. If V (y) is

monotonic, the production function is non -decreasing. When V(y) is convex, f(x) isquasiconcave.


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Figure 6. Production function y=f(x) and

production possibility set

.

x

y

y=f(x)

Y

0


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Figure 7. Isoquant lines for Cobb-DouglasandLeontief technology

.

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100

Q(y2)

Q(y1)

x1

x2

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100

Q(y2)

Q(y1)

x1

x2


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Examples

Draw an isoquant map in the case of production

function: y=x11/2x2

1/2

1.. Assume that we hold output constant y=1,2,10

1(2,3,,10)=x11/2x2

1/2 , where only x1 and x2 are

allowed to vary. The range of combinations allowed

by that equation is the isoquant for the productionlevel: x2=1/x1.


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Figure 8. Production function in the short-

run

.

0 20 40 60 80 100 120

20

40

60

80

100

x1

y =f(x1,)


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The Cobb-Douglas technology is defined in the

following manner

3 1

1 2 1 2

2 1

1 2 1 2

2 1

1 2 1 2

3 1

1 2 1 2 2

1

1 2 1 2

1

1 2 1 2

, , :

( ) , :

( ) , :

( ) , , : ,

( , , )

( , )

a a

a a

a a

a a

a a

a a

Y y x x y x x

V y x x y x x

Q y x x y x x

Y z y x x y x x x z

T y x x y x x

f x x x x


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Homogeneous production functions

A production function f (x) is called:

1. Homogeneous of degree k iff : f(tx)=tkf(x)

Two special cases are worthy of note. f(x) is:

2. Homogeneous degree 1(or linear homogeneous) iff

f(tx)=tf(x) for all t>0

Homogeneous of degree zero iff f(tx)=f(x) for all t>0

Homogeneity is a global characteristic. When a function

homogeneous of degree zero, changes in all variables

leave the value of the function unchanged.


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Example : Consider the production function:

f(x)= Ax1ax2

b , A > 0, a > 0, b > 0

We check whether this function is homogeneous bymultiplying all variables by the same factor t and seeingwhat we get.

We find that

According to the definition, the CobbDouglas form ishomogeneous of degree a+b>0 .

1 2 1 2 1 2

1 2

( , ) ( ) ( ) .

( , ).

a b a b a b

a b

f tx tx A tx tx t t Ax x

t f x x


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Theorem:(Shephard) Linear Homogeneous

Production Functions Are Concave

Let f(x) be a production function and suppose

that it is homogeneous of degree 1. Then f (x)

is a concave function of x (Exercise)


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Homothetic production function

A function is homothetic if it can be writtenas a monotonic transformation of a

homogeneous function. More formally

z = f(x): f: is homothetic if there

exist two functions h and g , where h: is

homogeneous at degree r and g

With g > 0 such that f(x) = g[h(x)].

n

n


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For examples

1. Prove that production functions in (i) and

(ii) are homothetic production functions and(iii) is not homothetic production function

(i) y= f(x)=x11/2x2

1/2,.

3 21 2 3

1 2 3

2 2

1 2 1

( ) ( , , )

( ). ( , ) ( 2)

x x xii f x x x e

iii f x x x


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Figure 9. Homogeneous and homothetic

production functions

.

20 40 60 80 100

0

20

40

60

80

100

20 40 60 80 100

0

20

40

60

80

100

x2

x1

x2

x1

2x

x

2x'

x' y2 = 2y1

y1

y2 2y1

y1

2x

2x'

x

x'

(b)(a)


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Figure (a). Presents an homogeneous function

at degree 1. If x and x' can produce y , then

2x and 2x' produce 2y. Figure (b) shows an homothetic function . If

x and x' produce y then 2x and 2x' can

produce the same level of output , but notnecessarily 2y.


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Cobb-Douglas production function Cobb-Douglas production function with two inputs

satisfied assumption constant return to scale:

: y=f(K,L)=AKL1-, where K is capital, L is labor. y isoutput. A , 1> > 0 are parameters.

The many input- Cobb-Douglas production function

(1) It is easy to show that this function exhibits constantreturn to scale if 1+2++ n=1.

(2) Since i [0,1] for all i, it exhibits diminishingmarginal productive for each input.

(3) Any degree of increasing return to scale can beincorporated into this function depending on the value of1+2++ n

n

i

iixAxfy

1

)(


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Isoquant of Cobb-Douglas Production

function

Cobb-Douglas production function with two

variable in the form:y=f(K,L)= AKa Lb, where

K is capital, L is labor. y is output. A , a,b> 0

are parameters.

The equation for an isoquant is obtained if we

fix the value of output y=y0. We then obtain:

y0= AKa Lb,

Rearranging, we find L=(y0/ A)1/b K-a/b


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Figure 10 Isoquant map of Cobb-Douglas

Production function with two variables

.

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100

Q(y2)

Q(y1)

L

K


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CES production function The two input constant elasticity of substitution

production function is given by

1. This function exhibits constant return to scale

2. It becomes the linear production function (if=1)

3. It becomes the Cobb-Douglas production function (=0).When =0 the CES is not defined.

However, one can show that as approaches zero, theisoquants of the CES function look like the isoquants of theCobb-Douglas production function.

4.The Leontief production function (-)

10,0;)1(),( /12121 AxxAxxfy


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A general CES production function

The CES production function can be

generalized to any degree of homogeneity.

Consider the production function

x1 ,x2 > 0; B , and k are positive. This

function is homogeneous of degree k .

If k


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The many input constant elasticity of

substitution

The many input constant elasticity of substitution isgiven by:

is a CES form with for all . It can

be shown that as giving Leontief form

1/

1 1

, 1n n

i i i

i i

y x where

1/(1 )

ij i j

, 0ij

1min{ ,..., }

n y x x


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Figure 11. Isoquant map of Leontief

production function

.

0 10 20 30 40 50 60 70 80 90 100

0

10

20

30

40

50

60

70

80

90

100

Q(y2)

Q(y1)

x1

x2


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Linear production function

( perfect substitute)

Linear production function can be as

y=f(x1,x2,,x2) = a0+a1x1+.+anxn

where ai is the quantity of the ith inputrequired to produce one unit of output.

In the case of two variables:

y=f(x1,x2) = a0+a1x1+a2x2


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Translog production function

N-input case of the translog production function can

be as

1. Note that the Cobb-Douglas function is a special

case of this function j

=ji

=0 for all i and j.

2. The condition k=ki is required to assure equality

of the cross-partial derivatives.

3. This function can assume any degree of returns to

scale . If for all i, this function

exhibits constant return to scale .

0;111

n

j

ij

n

i

i

jiijji

n

i

n

j

iji

n

i

i xxxy

;lnln21ln

1 11

0


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Average product of an input

Total product divided by the number of units

of the input used

ix

xxf ),( 21


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Marginal physical product

Maginal physical product: The change in

total product per unit change in the quantity

used of one input

The marginal physical product of an input is

the additional output that can be produced by

employing one more unit of that input while

holding all other inputs constant.

If y= f(K,L)( , )

K K

f K L MP f

K

( , )L L

f K L MP f

L


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Law of diminishing returns

As more units of an input are used per unit of

time with fixed amounts of another input, the

marginal physical product declines after a

point

M h i ll i f


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Mathematically, an assumption of

diminishing marginal physical productivity is

an assumption about the second-orderderivatives of the production function

2

2

( , )

0

K

KK

MP f K L

fK K

2

2

( , )0L

LL

MP f K Lf

L L


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Definition of Technical rate of

substitution (TRS) (or marginal rate of

technical substitution)

The amount of an input that a firm can give

up by increasing the amount of other input by

one unit and still remain on the same

isoquant


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Explanation

Suppose that we have some technology

summarized by a smooth production function

y=f(x1,x2).

Suppose that we want to increase the amount

of input 1 and decrease the amount of input 2

so as to maintain a constant level of output.

How can we determine this technical rate ofsubstitution (TRS) between these two factors?

Consider the particular change in which only


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Consider the particular change in which onlyfactor 1 and factor 2 change, and the change issuch output remain constant.

f(x1,x2(x1))=y. Differentiating the identity yields

This gives an explicit expression for the TRS.

2 1

1 2 1

2 1

1 1 2

( *) ( *) ( *)0

( *) ( *) ( *)/

f x f x x x

x x x

x x f x f xor

x x x

Here is another way to derive the technical


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Here is another way to derive the technical

rate of substitution (TRS).

This expression is known as the total

differential of the function f(x) .

21 2

1 2 1 1 2

( ) ( ) ( ) ( )0 / f x f x dx f x f xdx dx x x dx x x

1 2

1 2

( ) ( ) f x f xdy dx dx

x x


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In general, the TRS of factor i for factor j is

defined as

( )

( ) /( 1)

( ) /j iij

i jalongQ y

dx f x xTRS dx f x x

Fi 12 Ill d TRS


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Figure 12. Illustrated TRS

.

0 20 40 60 80 100

Q(y) = {x | y = f(x)}

20

40

60

80

100

120

x1

x2

x

1

2

( ) /( ) /

f x x f x x


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Example

Compute the TRS for a Cobb-Douglas

technology : f(x1,x2) =x1ax2

1-a

El ti iti


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Elasticities

Definition: The elasticity of f(x) with respect to

x is (approximately) the percentage change inf(x) corresponding to a one per cent increase inx.

Partial elasticity of f(x1,x2,..., ) with respect to

xi can be defined as

.

( ( ))

'( ) ( ) '( )( ) ( ) ( )x

x x d Lnf x

f x or f x f x f x f x d Lnx

( )

( )

ii

i

x f x

f x x


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General rules for calculating elasticities

2. ( ) / ( ) ( ) ( ) x x xf x g x f x g x

3. ( ) ( ) ; ( ) x u xf g x f u u u g x

( ) ( ) ( ) ( )

4. ( ) ( )

( ) ( )

x x

x

f x f x g x g xf x g x

f x g x

( ) ( ) ( ) ( )

5. ( ) ( )( ) ( )

x x

x

f x f x g x g xf x g x

f x g x

1. ( ) ( ) ( ) ( ) x x xf x g x f x g x


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Special rules for calculating elasticities

6. 0; tanx A A cons t

7.a

x

x a

18.

xLnx

Lnx

19. logx a

xLnx


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Definition of Elasticity of Substitution

The elasticity of substitution () is defined to

be the ratio of two factors (capitallabor ratio

(K/L)) and the TRS changes. It is measure of

how curved the isoquant is.

In moving from A to B on the Q = Q0 isoquant,

both the capitallabor ratio (K/L) and the TRS

will change.

Figure 13 Description of the elasticity of


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Figure.13.Description of the elasticity of

substitution

.

A TRSA

10 20 30 40 50 60 70 80 90

L

0

10

20

30

40

50

60

70

80

90

K

Q=Q0

B TRSB

(K/L)A

(K/L)B

Mathematical definition of The Elasticity


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Mathematical definition of The Elasticity

of Substitution

For a production function f(x), the elasticityof substitution between factors i and j at the

point x is defined as

ln( / ) ( / ) ( ) / ( )

ln( ( ) / ( )) / ( ( ) / ( )) '

ln( / )ln

j i j i i j

ij

i j j i i j

j i

d x x d x x f x f x

d f x f x x x d f x f x

d x xd TRS

An alternative formula for the elasticity of substitution


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y

If f(x1,x2 ) is homogeneous of degree 1, then

ij

ij

i

ixx

xxffx

xxffcxxf

f

f

ff

f

f

ffxfx

),(;),(;),(,

)(2

)(

11

21221

21

2

2

22

21

12

2

1

11

2211

12

21

ff

ff

Th l i t th t i tl


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The closer is to zero, the more strictly

convex the isoquants and the more "difficult"

substitution between factors. The larger is, the flatter the isoquants and

the "easier" substitution between factors.

Isoquants for two extreme and oneintermediate value of are illustrated in

Figures below.

Figure 14. Isoquant map for two extreme and


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g q p

one intermediate values.

.

0

Q(y)

Q(y)

x2

x1

0

Q(y)

Q(y)

x2

x1

0

Q(y)

Q(y)

x2

x1

0

fi ( ) i l d l b f


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In figure (a) capital and labor are perfect

substitutes. In this case the TRS will not

change as the capitallabor ratio changes. In figure (b) the fixedproportion case, no

substitution is possible. The capital-labor

ratio is fixed. A case of limited substitutability is illustrated

in figure (c)


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Example

Calculate for the Cobb - Douglas

production function y= Ax1ax2

b, where A>0,a>,

and b>0

Example : In the case of homogeneous of


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Example : In the case of homogeneous of

degree 1

The elasticity of substitution can be phrased in terms of

the production function and its derivatives in the constant

return to scale

For example: given f(K,L)= AKaL1-a

2/ /

( , ) /

f L f K

f K L f L K

22

/ / 1 / ( / ) 11 /( , ) /

f L f K a f L a f K f a a KLf K L f L K

Returns to Scale and Varying Proportions


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Returns to Scale and Varying Proportions

Returns to scale refers to how output responds

when all factors are varied in the same proportion,i, e,..

Constant returns to scale: When all inputs are

increased in a given proportion and the outputproduced increases exactly in the same proportion.

Decreasing returns to scale: The case when outputgrows proportionately less than inputs.

Increasing returns to scale: The case when outputgrows proportionately more than inputs.

Returns to scale and production


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Returns to scale and production The scale properties of the technology may

be defined either locally or globally.

A production function is classified as havingglobally constant, increasing, or decreasingreturns to scale according to the followingdefinitions.

(Global) Returns to ScaleA production function f(x) has the property of(globally):

1. Constant returns to scale if, and only iff(tx) =tf(x), for all t > 0 and all x.


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2. Increasing returns to scale if, and only if,

f(tx)>tf(x) for all t > 1 and all x.

3. Decreasing returns to scale if, and only if,

f(tx) 1 and all x.

Notice from these global definition of returnsto scale that a production function has constant

returns if and only if it is a (positive) linear

homogeneous function.

Many technologies exhibit increasing


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Many technologies exhibit increasing,constant, and decreasing returns over onlycertain ranges of output. It is therefore oftenuseful to have a local measure of returns toscale.

One such measure, defined at a point, tells us

the instantaneous percentage change inoutput that occurs with a 1 percent increasein all inputs.

It is known as the elasticity of scale or the(overall) elasticity of output, and defined asfollows.

(Local) Returns to Scale


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(Local) Returns to Scale

The elasticity of scale at the point x is

defined as

Returns to scale are locally constant,

increasing, or decreasing as is equal togreater than, or less than 1. The elasticity of

scale and the output elasticities of the factors

are related as follows:

1

1

( )log ( )( ) limlog ( )

n

i ii

t

f x xd f txxd t f x

1

( ) ( )n

i

i

x x

Example


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Example

What is the elasticity of scale of CES technology:

Since

Implies that the CES production function

exhibits constant returns to scale and hence

Has elasticity of scale of 1.

babxaxy ;10')(/1

21

tfxaxttxtxatxtxf /1

21

/1

2121 )('))()((),(


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Example

Let's examine a production function with

variable returns to scale:

(E.1)

Where >0 ,>0 , and k is an upper bound

on the level of output, so that 0 y


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Technical progress

Suppose that we let

y(t)=f(K(t),L(t),t) Where t - time.

Differentiating the equation with respect to time gives.

The first two terms on the right indicate the change in output

due to increased inputs of labor and capital , respectively.The

last term on the right indicates the change in output due totechnical change

dy f dL f dK f

dt L dt K dt t

Dividing both sides of the equation by output y


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Dividing both sides of the equation by output y ,

to convert to proportonate rates of change ,

yields:(1.26).

The first two terms on the right are theproportionate rates of change of the two inputs,

each weighted by the elasticity of output with

respect to the input. The third term is theproportionate rate of the technical change.

1 1 1 1dy L f dL K f dK f

y dt y L L dt y K K dt y t

Assume that the elasticity of output with


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Assume that the elasticity of output with

respect to labor and capital are constant

and given by and , respectively.Assume further that the proportionate rate

of technical change is constant at the rate

m , then the equation above implies that .

The rate of technical change , m, can be as

1 1 1dy dL dK m

y dt L dt K dt

1 1 1dy dL dK m

y dt L dt K dt

Classifying technical change


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Classifying technical change 1. Neutral technical progress

Y=A(t) f(K,L)-Here technical progress affects all theinputs equally-

2. Capital augmenting technical progress: Y=f[A(t)K,L].In this case , technical progress affects only capital. K

becomes more productive overtime in which newtechnology is applied.

3. Labor augmenting technical progress: Y=f[K,A(t)L].In this case , technical progress affects only the quality

of labor-hours that enter into the production function.The productive power of labor is augmented over time,perhaps because workers learn to do their jobs better.

i


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Exercise 1

For each input requirement set determine if it

is regular, monotonic, and/or convex. Assume

that the parameters a, b and the output levels

are strictly positive 1.1. v(y)={x1,x2: ax1logy, bx2 logy}

1.2. v(y)={x1,x2: ax1+ bx2 y, x1>0}


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Exercise 2

Prove that production functions in (i) and (ii)

are homothetic production functions and (iii)

is not homothetic production function

2 21 2

2 4

1 2 1 2

1 2

2 2

1 2 1

( ) ( , )

( ) ( , )

( ) ( , ) ( 1)

x xi f x x x xii f x x e

iii f x x x

Exercise 3: Marginal physical


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Exercise 3: Marginal physical

product of factor Given that :

1. Y=Axayb

2. Y=(x1r+x2

r)1/r

Compute the marginal physical product of

each input from two technologies.

Exercise 4: Relation between marginal

h i l d t d h i l


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physical product and average physical

product

Veryfy that at the optimal point of the

average physical product of the factor is

equal to the marginal physical product of

that factor.

E i 5


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Exercise 5

Prove Shephards theorem that:

Linear Homogeneous Production Functions

Are Concave

Exercise 6. Law of a diminishing


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TRS

Assume that y=f(K,L) and fk,fl>0, fkk


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Exercise 7: Compute TRS

What is the TRS for CES and Cobb-Douglas

production functions

2. y=f(x)= Ax1ax2

b

10,0;)1(),(.1

/1

2121

AxxAxxfy


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Exercise 8

Let's examine a production function with

variable returns to scale:

(E.1)

Where >0 ,>0 , and k is an upper bound

on the level of output, so that 0 y


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1.A generalization of the CES productionfunction is given by

for A > 0, 0>0, i>0 and 0 < 1.Calculate ij for this function and show

that the elasticity of scale is measured bythe parameter . Is this functionhomogeneous?

2. Calculate the elasticity of substitution

for the production in the form of y=k(1+x1

-ax2-b)

/

0

1( )

n

i i

i y A x

Exercise 10


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1. A Leonief production has the form:

for >0, >0. Sketch the isoquant map for

this technology and verify that the elasticityof substitution is equal to zero.

2. The CMS (constant marginal shares)production function is the form y= Ax1

ax2b-

mx2. Calculate for this function and showthe relationship with AP2

1 21 1min( , ) y x x

E i 11


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Exercise 11

1. To calculate the elasticity of substitution

for CES production function

2. Veryfy that If f(x1,x2 ) is homogeneous

of degree 1, then12

21

ffff

;10')(/1

21

xxy

Exercise 12


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Exercise 12

Prove that if f(x1,x2 ) is homogeneous of degree 1, then

12

21

ff

ff

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