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```					Part 3 PRODUCTION AND SUPPLY

• PRODUCTION FUNCTIONS

• COST FUNCTION

• PROFIT MAXINIZATION
Chapter 7
PRODUCTION FUNCTIONS
Contents

   Marginal productivity
   Isoquant Maps and the rate of technical
substitution
   Returns to Scale
   The elasticity of substitution
   Four simple production function
   Technical progress

Lee, Junqing               Department of Economics , Nankai University
Production Function
Production Function

Lee, Junqing             Department of Economics , Nankai University
Production Function

Lee, Junqing             Department of Economics , Nankai University
Symmetry Between Consumer and
Firm Theory

Lee, Junqing     Department of Economics , Nankai University
Production Function

   The firm’s production function for a
particular good (q) shows the maximum
amount of the good that can be produced
using alternative combinations of capital (k)
and labor (l)

q = f(k,l)

Lee, Junqing                 Department of Economics , Nankai University
Production When Only One Input is
Variable

Lee, Junqing           Department of Economics , Nankai University
Total Product (TP)（

Lee, Junqing              Department of Economics , Nankai University
The Production Function: TP increases
with L

Lee, Junqing         Department of Economics , Nankai University
Average Physical Product

   Labor productivity is often measured by
average productivity
output    q f (k , l )
APl              
labor input l    l

   Note that APl also depends on the amount
of capital employed

Lee, Junqing                   Department of Economics , Nankai University
Average Physical Product

Lee, Junqing          Department of Economics , Nankai University
Average Physical Product

Lee, Junqing          Department of Economics , Nankai University
Marginal Physical Product

   To study variation in a single input, we define
marginal physical product as the additional
output that can be produced by employing
one more unit of that input while holding other
inputs constant

q
marginal physical product of capital  MPk      fk
k
q
marginal physical product of labor  MPl      fl
l
Lee, Junqing                   Department of Economics , Nankai University
Marginal Physical Product

Lee, Junqing          Department of Economics , Nankai University
Diminishing Marginal Productivity

   The marginal physical product of an input
depends on how much of that input is used
   In general, we assume diminishing marginal
productivity

MPk  2 f                  MPl  2 f
 2  f kk  f11  0         2  fll  f 22  0
k  k                      l  l
Lee, Junqing                Department of Economics , Nankai University
Diminishing Marginal Productivity

   Because of diminishing marginal productivity,
19th century economist Thomas Malthus
worried about the effect of population growth
on labor productivity
   But changes in the marginal productivity of
labor over time also depend on changes in
other inputs such as capital
   we need to consider flk which is often > 0

Lee, Junqing                     Department of Economics , Nankai University
The Relationship between TP and MP

Lee, Junqing        Department of Economics , Nankai University
The Reasons of Diminishing Returns

Lee, Junqing         Department of Economics , Nankai University
Relationship between MP and AP

Lee, Junqing          Department of Economics , Nankai University
Relationship between MP and AP

Lee, Junqing          Department of Economics , Nankai University
Three Stages of Production

Lee, Junqing                 Department of Economics , Nankai University
Isoquant Maps and the rate of
technical substitution
Isoquant Maps

   To illustrate the possible substitution of
one input for another, we use an isoquant
map
   An isoquant shows those combinations of
k and l that can produce a given level of
output (q0)

f(k,l) = q0

Lee, Junqing                Department of Economics , Nankai University
Isoquant Map

   Each isoquant represents a different level of
output
   output rises as we move northeast
k per period

q = 30
q = 20

l per period
Lee, Junqing                         Department of Economics , Nankai University
Production Function : Multiple Inputs

Lee, Junqing             Department of Economics , Nankai University
Production Function : Multiple Inputs

Lee, Junqing             Department of Economics , Nankai University
Marginal Rate of Technical
Substitution (RTS)

   The slope of an isoquant shows the rate at
which l can be substituted for k
k per period
- slope = marginal rate of technical
substitution (RTS)

RTS > 0 and is diminishing for
kA
A             increasing inputs of labor
B
kB
q = 20

l per period
lA      lB
Lee, Junqing                           Department of Economics , Nankai University
Marginal Rate of Technical
Substitution (RTS)
   The marginal rate of technical substitution
(RTS) shows the rate at which labor can be
substituted for capital while holding output
constant along an isoquant

 dk
RTS (l for k ) 
dl    q q0

Lee, Junqing                  Department of Economics , Nankai University
Marginal Rate of Technical
Substitution (RTS)

Lee, Junqing          Department of Economics , Nankai University
RTS different from diminishing
marginal product

Lee, Junqing           Department of Economics , Nankai University
RTS and Marginal Productivities

   Take the total differential of the production
function:
f      f
dq   dl      dk  MPl  dl  MPk  dk
l      k
   Along an isoquant dq = 0, so
MPl  dl  MPk  dk
 dk             MPl
RTS (l for k )                 
dl    q q0     MPk
Lee, Junqing                     Department of Economics , Nankai University
RTS and Marginal Productivities

   Because MPl and MPk will both be
nonnegative, RTS will be positive (or zero)

   However, it is generally not possible to derive
a diminishing RTS from the assumption of
diminishing marginal productivity alone

Lee, Junqing                Department of Economics , Nankai University
RTS and Marginal Productivities

   To show that isoquants are convex, we would
like to show that d(RTS)/dl < 0
   Since RTS = fl/fk

dRTS d (fl / fk )

dl      dl

dRTS [fk (fll  flk  dk / dl )  fl (fkl  fkk  dk / dl )]

dl                          (fk )2
Lee, Junqing                          Department of Economics , Nankai University
RTS and Marginal Productivities

   Using the fact that dk/dl = -fl/fk along an
isoquant and Young’s theorem (fkl = flk)
dRTS ( f k2 f ll (  )  2 f k f l f kl (  ?)  f l 2 f kk (  ))quasi-concave function
                                          3
dl                                   ( fk ) ()
   Because we have assumed fk > 0, the
denominator is positive
   Because fll and fkk are both assumed to be
negative, the ratio will be negative if fkl is
positive quasi-concave function
Lee, Junqing                                  Department of Economics , Nankai University
RTS and Marginal Productivities

   Intuitively, it seems reasonable that fkl = flk
should be positive
   if workers have more capital, they will be more
productive
   But some production functions have fkl < 0
over some input ranges
   when we assume diminishing RTS we are
assuming that MPl and MPk diminish quickly
enough to compensate for any possible negative
cross-productivity effects
Lee, Junqing                     Department of Economics , Nankai University
Returns to Scale
Returns to Scale

   If the production function is given by q = f(k,l)
and all inputs are multiplied by the same
positive constant (t >1), then

Effect on Output Returns to Scale
f(tk,tl) = tf(k,l)   Constant (CRS)
f(tk,tl) < tf(k,l)   Decreasing(DRS)
f(tk,tl) > tf(k,l)   Increasing(IRS)

Lee, Junqing                    Department of Economics , Nankai University
Returns to Scale

   It is possible for a production function to
exhibit constant returns to scale for some
levels of input usage and increasing or
decreasing returns for other levels
   economists refer to the degree of returns to
scale with the implicit notion that only a fairly
narrow range of variation in input usage and
the related level of output is being considered

Lee, Junqing                      Department of Economics , Nankai University
Returns to Scale

Lee, Junqing            Department of Economics , Nankai University
increasing Returns to Scale

Lee, Junqing         Department of Economics , Nankai University
Reason Increasing Returns to Scale

Lee, Junqing       Department of Economics , Nankai University
Decreasing Returns to Scale

Lee, Junqing              Department of Economics , Nankai University
Reason Decreasing Returns to Scale

Lee, Junqing        Department of Economics , Nankai University
Constant Returns to Scale

Labor(hours)

Lee, Junqing                 Department of Economics , Nankai University
Constant Returns to Scale
   Constant returns-to-scale production functions are
homogeneous of degree one in inputs
f(tk,tl) = t1f(k,l) = tq
   This implies that the marginal productivity functions are
homogeneous of degree zero
   if a function is homogeneous of degree k, its derivatives are
homogeneous of degree k-1
k
f ( ,1)
f (k , l ) f (tk , tl )        l
MPk                                         (t  1 / l )
k           k              k
k
f ( ,1)
f (k , l ) f (tk , tl )        l
MPl                           
l           l              l
Lee, Junqing                             Department of Economics , Nankai University
Constant Returns to Scale

   The marginal productivity of any input depends on
the ratio of capital and labor (not on the absolute
levels of these inputs)
   The RTS between k and l depends only on the
ratio of k to l, not the scale of operation
k
f ( ,1)
l
RTS 
MPl
     l
MPk        k
f ( ,1)
l
k
Lee, Junqing                    Department of Economics , Nankai University
Constant Returns to Scale
   The production function will be homothetic
   Geometrically, all of the isoquants are
   Along a ray from the origin (constant k/l), the
RTS will be the same on all isoquants
k per period
The isoquants are equally
spaced as output expands

q=3

q=2
q=1
l per period
Lee, Junqing                  Department of Economics , Nankai University
homothetic production function
   The IRS and DRS also have a homothetic
difference indifference curve , because the
property of homotheticity is retained by any
monotonic transformation of homogeneous
function

F (k , l )  [ f (k , l )]r ,   0
F (tk , tl )  [ f (tk , tl )]r  [tf (k , l )]  t  [ f (k , l )]  t  F (k , l )  tF (k , l )
  1, t  1  IRS
  1, t  1  DRS

Lee, Junqing                                     Department of Economics , Nankai University
Returns to Scale

   Returns to scale can be generalized to a
production function with n inputs
q = f(x1,x2,…,xn)
   If all inputs are multiplied by a positive
constant t, we have
f(tx1,tx2,…,txn) = tkf(x1,x2,…,xn)=tkq
   If k = 1, we have constant returns to scale
   If k < 1, we have decreasing returns to scale
   If k > 1, we have increasing returns to scale

Lee, Junqing                        Department of Economics , Nankai University
Elasticity of Substitution
Properties of Production Function

When   MPL> APL, then the labor elasticity,

eL> 1。1 percent increase in labor will increase output by more than 1 percent.

When   MPL< APL, then the labor inelasticity,

eL< 1。 1 percent increase in labor will increase output by less than 1 percent.

Lee, Junqing                                    Department of Economics , Nankai University
Properties of Production Function

Lee, Junqing             Department of Economics , Nankai University
Elasticity of Substitution
   The elasticity of substitution () measures the
proportionate change in k/l relative to the
proportionate change in the RTS along an
isoquant

   The value of  will always be positive
because k/l and RTS move in the same
direction
Lee, Junqing                   Department of Economics , Nankai University
Elasticity of Substitution

   Both RTS and k/l will change as we move
from point A to point B

k per period                                 is the ratio of these
proportional changes

RTSA                       measures the
A
RTSB
curvature of the
isoquant
(k/l)A              q = q0
B
(k/l)B
l per period
Lee, Junqing                            Department of Economics , Nankai University
Elasticity of Substitution

Lee, Junqing             Department of Economics , Nankai University
Elasticity of Substitution

   If  is high, the RTS will not change much
relative to k/l
   the isoquant will be relatively flat
   If  is low, the RTS will change by a
substantial amount as k/l changes
   the isoquant will be sharply curved
   It is possible for  to change along an
isoquant or as the scale of production
changes
Lee, Junqing                        Department of Economics , Nankai University
Elasticity of Substitution

   Generalizing the elasticity of substitution to
the many-input case raises several
complications
   if we define the elasticity of substitution
between two inputs to be the proportionate
change in the ratio of the two inputs to the
proportionate change in RTS, we need to hold
output and the levels of other inputs constant

Lee, Junqing                     Department of Economics , Nankai University
Four simple production functions
The Linear Production Function
   Suppose that the production function is (Man vs.
women labor)
q = f(k,l) = ak + bl
   This production function exhibits constant
returns to scale
f(tk,tl) = atk + btl = t(ak + bl) = tf(k,l)
   All isoquants are straight lines
   RTS is constant
   =

Lee, Junqing                          Department of Economics , Nankai University
The Linear Production Function

Capital and labor are perfect substitutes

k per period
RTS is constant as k/l changes

slope = -b/a
=

q1   q2        q3
l per period
Lee, Junqing                             Department of Economics , Nankai University
Fixed Proportions

   Suppose that the production function is
q = min (ak,bl) a,b > 0
   Capital and labor must always be used in
a fixed ratio
   the firm will always operate along a ray where
k/l is constant
   Because k/l is constant,  = 0

Lee, Junqing                     Department of Economics , Nankai University
Fixed Proportions

No substitution between labor and capital
is possible
k per period                     k/l is fixed at b/a

=0
q3/a                         q3

q2

q1

l per period
q3/b
Lee, Junqing                      Department of Economics , Nankai University
Fixed Proportions

Lee, Junqing            Department of Economics , Nankai University
Cobb-Douglas Production
Function

   Suppose that the production function is
q = f(k,l) = Akalb A,a,b > 0
   This production function can exhibit any
returns to scale
f(tk,tl) = A(tk)a(tl)b = Ata+b kalb = ta+bf(k,l)
   if a + b = 1  constant returns to scale
   if a + b > 1  increasing returns to scale
   if a + b < 1  decreasing returns to scale

Lee, Junqing                           Department of Economics , Nankai University
Cobb-Douglas Production
Function
   The Cobb-Douglas production function is linear in
logarithms
ln q = ln A + a ln k + b ln l

   a is the elasticity of output with respect to k
   b is the elasticity of output with respect to l

Lee, Junqing                           Department of Economics , Nankai University
Cobb-Douglas Production
Function
The elasticity of substitution of Cobb-Douglas
production function is one

fl  Ak  l  1  k
RTS                 
 1 
f k  Ak l        l
       k
ln( RTS )  ln( )  ln( )
       l
k
 ln( )
        l 1
ln( RTS )

Lee, Junqing                        Department of Economics , Nankai University
CES Production Function

   Suppose that the production function is
q = f(k,l) = [k + l] /   1,   0,  > 0
    > 1  increasing returns to scale
    < 1  decreasing returns to scale

q = f(tk,tl) = [t  k +t  l] / = t  [ k + l] / =t  f(k,l)
  1,   0,  > 0

Lee, Junqing                            Department of Economics , Nankai University
CES Production Function
   For this production function
 = 1/(1-)
    = 1  linear production function (prefect substitution)
    = -  fixed proportions production function
    = 0  Cobb-Douglas production function
 (   ) /   1
q             l
fl                             k (1  )
RTS                               ( )
f k  q (   ) /   k  1   l

k
ln( RTS )  (1   ) ln( )
l
k
 ln( )
          l  1
ln( RTS )        (1   )
Lee, Junqing                                   Department of Economics , Nankai University
Technical Progress

   Methods of production change over time
   Following the development of superior
production techniques, the same level of
output can be produced with fewer inputs
   the isoquant shifts in

Lee, Junqing                       Department of Economics , Nankai University
Technical Progress

   Suppose that the production function is
q = A(t)f(k,l)
where A(t) represents all influences that go
into determining q other than k and l
   changes in A over time represent technical
progress
 A is shown as a function of time (t)
 dA/dt > 0

Lee, Junqing                         Department of Economics , Nankai University
Technical Progress
solow(1959), data :1909  1949
q = A(t)f(k,l)
Gq  2.75 per year
dq dA                      df (k , l )
 1.00 , l )  year
Gk  dt  f (kper A  dt
dt
Gl dA 75 per year dk f dl 
dq      1. q        q  f
                   k  dt  l  dt 
eq , k  0.65 f (k , l ) 
dt     dt A                                        
dq / dt dA / dt f / k dk f / l dl
eq ,l   35 
0.                                       
q        A       f (k , l ) dt        f (k , l ) dt
dq / dt dA / dt f             k         dk / dt f       l    dl / dt
                                              

G A  Gq A eq , k Gk f  eq ,l Glk  1.l 0 f (erl )year
q                k       (k , l )                5 p k,      l
Gq  GA  eq ,k Gk  eq ,l Gl

Lee, Junqing                         Department of Economics , Nankai University
Production vs. Utility
Production vs. Utility

Lee, Junqing               Department of Economics , Nankai University
The Optimal Choice In the Long Run
   The optimal choices includes:
   1.Maximization of output for a given cost
   2.Minimization of cost for a given output

Lee, Junqing                          Department of Economics , Nankai University
The Optimal Choice In the Long Run

Lee, Junqing         Department of Economics , Nankai University
The Optimal Choice In the Long Run

Lee, Junqing         Department of Economics , Nankai University
Contents

   Marginal productivity
   Isoquant Maps and the rate of technical
substitution
   Returns to Scale
   The elasticity of substitution
   Four simple production function
   Technical progress

Lee, Junqing               Department of Economics , Nankai University
Important Points to Note:

   If all but one of the inputs are held
constant, a relationship between the
single variable input and output can be
derived
   the marginal physical productivity is the
change in output resulting from a one-unit
increase in the use of the input
   assumed to decline as use of the input
increases

Lee, Junqing                             Department of Economics , Nankai University
Important Points to Note:

   The entire production function can be
illustrated by an isoquant map
   the slope of an isoquant is the marginal rate of
technical substitution (RTS)
 it shows how one input can be substituted for
another while holding output constant
 it is the ratio of the marginal physical
productivities of the two inputs

Lee, Junqing                           Department of Economics , Nankai University
Important Points to Note:

   Isoquants are usually assumed to be
convex
   they obey the assumption of a diminishing
RTS
 this assumption cannot be derived exclusively
from the assumption of diminishing marginal
productivity
 one must be concerned with the effect of changes
in one input on the marginal productivity of other
inputs

Lee, Junqing                           Department of Economics , Nankai University
Important Points to Note:

   The returns to scale exhibited by a
production function record how output
responds to proportionate increases in all
inputs
   if output increases proportionately with input
use, there are constant returns to scale

Lee, Junqing                      Department of Economics , Nankai University
Important Points to Note:

   The elasticity of substitution () provides
a measure of how easy it is to substitute
one input for another in production
   a high  implies nearly straight isoquants
   a low  implies that isoquants are nearly L-
shaped

Lee, Junqing                      Department of Economics , Nankai University
Important Points to Note:

   Technical progress shifts the entire
production function and isoquant map
   technical improvements may arise from the
use of more productive inputs or better
methods of economic organization

Lee, Junqing                    Department of Economics , Nankai University
Chapter 7
PRODUCTION FUNCTIONS

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