micro econ1
TRANSCRIPT
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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 capital- labor 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.
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21
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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
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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
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;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