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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
     radial expansions of one another
    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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