Chapter 7 The Theory and Estimation of Production by w1F3y7

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									The Production
Process and Costs

Andreja Cirman
The Theory and Estimation of
Production

 The Production Function
 The Cost Function
The Production Function
 A production function defines the relationship between
    inputs and the maximum amount that can be produced
    within a given time period with a given technology.

 Mathematically, the production function can be expressed
    as
                   Q=f(K, L)
    Q is the level of output
    K = units of capital
    L = units of labour
   f( ) represents the production technology
The Production Function
 When discussing production, it is important to distinguish
   between two time frames.

 The short-run production function describes the
   maximum quantity of good or service that can be
   produced by a set of inputs, assuming that at least one of
   the inputs is fixed at some level.

 The long-run production function describes the
   maximum quantity of good or service that can be
   produced by a set of inputs, assuming that the firm is free
   to adjust the level of all inputs
Production

 When discussing production, three
  definitions are important.

   •Total Product
   •Marginal Product
   •Average Product
Production in the Short Run

 Total product (TP) is another name for
  output in the short run.
 The marginal product (MP) of a
  variable input is the change in output (or
  TP) resulting from a one unit change in
  the input.
 The average product (AP) of an input is
  the total product divided by the level of
  the input.
        Production in the Short Run
The table below represents a firm’s production function,
Q=f(X,Y):
Units of Y
Employed                Output Quantity (Q)
    8         37   60    83 96 107 117 127 128
    7         42   64    78 90 101 110 119 120
    6         37   52    64 73 82 90 97 104
    5         31   47    58 67 75 82 89 95
    4         24   39    52 60 67 73 79 85
    3         17   29    41 52 58 64 69 73
    2         8    18    29 39 47 52 56 52
    1         4    8     14 20 27 24 21 17
              1    2      3    4    5   6   7 8
                        Units of X Employed
          Production in the Short Run
In the short run, let Y=2. The row highlighted below
   represents the firm’s short run production function.
   Units of Y
   Employed               Output Quantity (Q)
       8        37   60    83 96 107 117 127       128
       7        42   64    78 90 101 110 119       120
       6        37   52    64 73 82 90 97          104
       5        31   47    58 67 75 82 89          95
       4        24   39    52 60 67 73 79          85
       3        17   29    41 52 58 64 69          73
       2        8    18    29 39 47 52 56          52
       1        4     8    14 20 27 24 21          17
                1     2     3    4    5   6   7     8
                          Units of X Employed
       Production in the Short Run
 If MP is positive then
  TP is increasing.
 If MP is negative then
  TP is decreasing.
 TP reaches a maximum
  when MP=0
 Production in the Short Run

 If MP > AP then AP
  is rising.
 If MP < AP then AP
  is falling.
 MP=AP when AP is
  maximized.
The Law of Diminishing
Returns
 Definition
      As additional units of a variable input are
       combined with a fixed input, at some point
       the additional output (i.e., marginal
       product) starts to diminish.
    Diminishing Returns
Variable               Marginal
 Input   Total Product Product
  (X)     (Q or TP)     (MP)
   0            0         8
   1            8        10   Diminishing
   2           18        11     Returns
   3           29        10     Begins
   4           39          8     Here
   5           47
   6           52          5
   7           56          4
   8           52         -4
        The Law of Diminishing
        Returns
 Reasons
         Increasing Returns
         Teamwork and Specialization
      MP                    Diminishing Returns Begins
                            Fewer opportunities for teamwork
                            and specialization




                                           X
                                         MP
Production in the Long Run

 In the long run, all inputs are variable.
 Isoquant defines cominations of inputs
  that yield the same level of product
Isoquant
 K                   E
     5


     4


     3
         A       B       C

     2
                                                  Q3 =90
                                 D         Q2 =75
     1
                                       Q1 =55
             1   2           3       4     5  L
Marginal rate of technical
substitution (MRTS)
K
7

6       ΔK=3


5
                                MRTS  K
4
           ΔL=1
                                                 L
3         ΔK=1
              ΔL=1

2

1                            ΔK=1/3
                                      ΔL=1
0
    0      1   2     3   4     5      6      7        L
    Perfect Substitution
K

           A




                         B



               Q1   Q2       Q3


       0                          C
                                      L
No substitution (Leontief isoquants)
   K

                      C
                          Q3


                B
                              Q2



   K1      A
                              Q1



   0
          L1              L
Producion with Variable
Input Usage
MRTS and marginal productivity
    Change in total product due to change in units of
    labour employed:



    Change  in total product due to change in units of
    capital employed:
Producion with Variable
Input Usage
 MRTS and marginal productivity
       with constant total product:
Production in the Long Run

 The long run production process is
  described by the concept of returns to
  scale.
 Returns to scale describes what
  happens to total output as all of the
  inputs are changed by the same
  proportion.
Production in the Long Run

 If all inputs into the production process
  are doubled, three things can happen:
      output can more than double
         increasing   returns to scale (IRTS)
      output can exactly double
         constant   returns to scale (CRTS)
      output can less than double
         decreasing   returns to scale (DRTS)
    Production in the Long Run
One way to measure returns to scale is
to use a coefficient of output elasticity:
             Percentage change in Q
     EQ 
          Percentage change in all inputs
 If E>1 then IRTS
 If E=1 then CRTS
 If E<1 then DRTS
        Production in the Long Run
 Economists hypothesize
  that a firm’s long run
  production function may
  exhibit at first increasing
  returns, then constant
  returns, and finally
  decreasing returns to scale.
The Theory and
Estimation of Cost
          The Theory and
          Estimation of Cost

 The Short Run Relationship Between Production and Cost
 The Short Run Cost Function
 The Long Run Relationship Between Production and Cost
 The Long Run Cost Function
 The Learning Curve
 Economies of Scope
 Other Methods to Reduce Costs
   SR Relationship Between
   Production and Cost
 A firm’s cost structure is intimately
  related to its production process.
 Costs are determined by the production
  technology and input prices.
 Assume the firm is a “price taker” in the
  input market.
             SR Relationship Between
             Production and Cost
                               Total
 In order to illustrate the   Input
  relationship, consider        (L)    Q (TP)   MP
  the production process         0        0
                                 1      1.000   1.000
  described in the table.        2      3.000   2.000
                                 3      6.000   3.000
                                 4      8.000   2.000
                                 5      9.000   1.000
                                 6      9.500    500
                                 7      9.850    350
                                 8     10.000    150
                                 9      9.850   -150
          SR Relationship Between
          Production and Cost
 Total variable cost (TVC)     Total
  is the cost associated with   Input                    TVC
  the variable input, in this    (L)    Q (TP)   MP      (wL)
  case labor. Assume that         0        0               0
  labor can be hired at a         1      1.000   1.000    500
  price of w=$500 per unit.       2      3.000   2.000   1.000
  TVC has been added to
                                  3      6.000   3.000   1.500
  the table.
                                  4      8.000   2.000   2.000
                                  5      9.000   1.000   2.500
                                  6      9.500    500    3.000
                                  7      9.850    350    3.500
                                  8     10.000    150    4.000
                                  9      9.850   -150    4.500
    SR Relationship Between
    Production and Cost
 Plotting TP and TVC illustrates that they are mirror
  images of each other.
 When TP increases at an increasing rate, TVC
  increases at a decreasing rate.
SR Relationship Between
Production and Cost
 Total fixed cost (TFC) is the cost
  associated with the fixed inputs.
 Total cost (TC) is the cost associated
  with all of the inputs. It is the sum of
  TVC and TFC.
 TC=TFC+TVC
           SR Relationship Between
           Production and Cost
 Marginal cost (MC) is the change in total
  cost associated a change in output.
                       TC
                  MC 
                       Q
  •MC can also be expressed as the change in
  TVC associated with a change in output.
     TC (TFC  TVC ) TFC TVC      TVC
MC                            0
     Q      Q         Q   Q        Q
          SR Relationship Between
          Production and Cost
 Marginal Cost has   Total
  been added to the
                      Input                    TVC
  table.
                       (L)      Q      MP      (wL)    MC
 When MP is
  increasing, MC is     0       0                0
  decreasing.           1     1.000    1.000    500    0,50
 When MP is            2     3.000    2.000   1.000   0,25
  decreasing, MC is     3     6.000    3.000   1.500   0,17
  increasing.
                        4     8.000    2.000   2.000   0,25
                        5     9.000    1.000   2.500   0,50
                        6     9.500     500    3.000   1,00
                        7     9.850     350    3.500   1,43
                        8     10.000    150    4.000   3,33
                        9     9.850    -150    4.500
              The Short Run Cost
              Function
 A firm’s short run cost function tells us the minimum cost
  necessary to produce a particular output level.
 For simplicity the following assumptions are made:
       the firm employs two inputs, labor and capital
       labor is variable, capital is fixed
       the firm produces a single product
       technology is fixed
       the firm operates efficiently
       the firm operates in competitive input markets
       the law of diminishing returns holds
The Short Run Cost Function

 The following average cost functions will be
  useful in our analysis.
 Average total cost (AC) is the average per-unit
  cost of using all of the firm’s inputs.
 Average variable cost (AVC) is the average
  per-unit cost of using the firm’s variable inputs.
 Average fixed cost (AFC) is the average per-
  unit cost of using the firm’s fixed inputs.
The Short Run Cost Function

 Mathematically,


     AVC = TVC/Q
     AFC = TFC/Q

 ATC=TC/Q=(TFC+TVC)/Q=AFC+AVC
The Short Run Cost
Function
  The Short Run Cost Function
 Graphically, these results are be depicted in
  the figure below.
        The Short Run Cost Function
 Important Observations
    AFC declines steadily over the range of production.
    In general, AVC, AC, and MC are u-shaped.
    MC measures the rate of change of TC
    When MC<AVC, AVC is falling
     When MC>AVC, AVC is rising
     When MC=AVC, AVC is at its minimum
    The distance between AC and AVC represents AFC
   The LR Relationship Between
   Production and Cost

 In the long run, all inputs are variable.
 In the long run, there are no fixed costs
 The long run cost structure of a firm is related
  to the firm’s long run production process.
 The firm’s long run production process is
  described by the concept of returns to scale.
        The LR Relationship Between
        Production and Cost
 Economists hypothesize that a firm’s long-run production
  function may exhibit at first increasing returns, then
  constant returns, and finally decreasing returns to scale.
 When a firm experiences increasing returns to scale
      A proportional increase in all inputs increases output by a
       greater percentage than costs.
      Costs increase at a decreasing rate
        The LR Relationship Between
        Production and Cost
 When a firm experiences constant returns to scale
   A proportional increase in all inputs increases output by the
    same percentage as costs.
   Costs increase at a constant rate

 When a firm experiences decreasing returns to scale
   A proportional increase in all inputs increases output by a
    smaller percentage than costs.
   Costs increase at an increasing rate
LR Costs

 Cost of inputs


     C = wL + rK

     Cost line: combination of inputs (K & L) the
      company can buy by keeping costs at a
      constant level.
 Optimal Level of Variable
 Input Usage
Capital/year


          K2




          K1             B

                    A
          K3                                D
                                                 Q1
                             C0   C1            C2
          0    L2       L1             L3       Labour/year
 Chage in Prices of Inputs
Capital/year

                                   The slope of the cost line
                                         –(w/L) changes


          K2    B


                     A
          K1


                                      Q1
                         C2   C1
          0    L2   L1                Labour/year
LR Costs


 Isoquant, the cost line and the
  production function
LR Costs

  When the firm employs multiple variable inputs, the firm
    should choose the level of the inputs which equates the
    marginal product per dollar across each of the inputs.
    Mathematically,




        .
Example
 A firm uses two variable inputs, labor (L) and raw
 material (M), in producing its output. At current level of
 output:

     CL = 10 USD/unit
     CM = 2 USD/unit
     MPL = 25
     MPM = 4

 1.   Determine whether the firm is operating efficiently,
      given that its objective is to minimize the cost of
      production at given level of output
 2.   Determine what changes (if any) in the relative
      proportions of labor and raw materials need to be made
      to operate efficiently.
        The LR Relationship Between
        Production and Cost
 This graph illustrates
  the relationship
  between the long-run
  production function and
  the long-run cost
  function.
The Long-Run Cost Function

 Long run marginal cost (LRMC)
  measures the change in long run costs
  associated with a change in output.
 Long run average cost (LRAC)
  measures the average per-unit cost of
  production when all inputs are variable.
 In general, the LRAC is u-shaped.
The Long-Run Cost Function

 When LRAC is declining we say that the firm is
  experiencing economies of scale.
 Economies of scale implies that per-unit costs
  are falling.
 When LRAC is increasing we say that the firm
  is experiencing diseconomies of scale.
 Diseconomies of scale implies that per-unit
  costs are rising.
      The Long-Run Cost Function
 The figure
  illustrates the
  general shape
  of the LRAC.
        The Long-Run Cost Function

 Reasons for Economies of Scale
     Increasing returns to scale
     Specialization in the use of labor and capital
     Indivisible nature of many types of capital equipment
     Productive capacity of capital equipment rises faster than
      purchase price
     The Long-Run Cost Function
 Reasons for Economies of Scale
   • Economies in maintaining inventory of
     replacement parts and maintenance personnel
   • Discounts from bulk purchases
   • Lower cost of raising capital funds
   • Spreading promotional and R&D costs
   • Management efficiencies
 The Long-Run Cost Function
 Reasons for Diseconomies of Scale
 • Decreasing returns to scale
 • Disproportionate rise in transportation costs
 • Input market imperfections
 • Management coordination and control
   problems
 •Disproportionate rise in staff and indirect
   labor
           The Long-Run Cost Function
 In the short run, the firm has a
  fixed level of capital equipment or
  plant size.
 The figure illustrates the SRAC
  curves for various plant sizes.
 Once a plant size is chosen, per-
  unit production costs are found by
  moving along that particular SRAC
  curve.
The Long-Run Cost Function

 In the long run the firm is able to adjust
  its plant size.
 LRAC tells us the lowest possible per-
  unit cost when all inputs are variable.
 What is the LRAC in the graph?
The Long-Run Cost Function

 The LRAC is the lower envelope of all of
  the SRAC curves.
 Minimum efficient scale is the lowest
  output level for which LRAC is
  minimized.
           The Learning Curve
 Measures the percentage
  decrease in additional labor cost
  each time output doubles.
 An “80 percent” learning curve
  implies that each time output
  doubles, the labor costs
  associated with the incremental
  output will decrease to 80% of
  their previous level.
 The figure illustrates an 80-
  percent learning curve.
The Learning Curve

 A downward slope in the learning curve
  indicates the presence of the learning
  curve effect.
      workers improve their productivity with
       practice
 The learning curve effect acts to shift the
  SRAC downward.
       Economies of scale and learning curve

cost per unit




                                      Economies of scale
                                  A
                                      B
                                           AC1
                learning effect   C
                                        AC2

                                      production in units
     Economies of Scope

 The reduction of a firm’s unit cost by
  producing two or more goods or services
  jointly rather than separately.
Other Methods to Reduce
Costs
 Reduction in the Cost of Materials
 Using IT to Reduce Costs
 Reduction of Process Costs
 Relocation to Lower-Wage Countries or
  Regions
 Mergers, Consolidation, and Downsizing
 Layoffs and Plant Closings
Estimating Cost
Functions
Estimating Cost Functions


   Statistical Techniques
   Engineering Cost Techniques
Statistical techniques

 using multiple regression analysis
 linear, polynomial, logarithmic functions
  etc.
Statistical techniques-example
   Quarter of the   Units produced   Total costs
       year                              ( EUR)

     Q1 1998                 1.943    1.785.473
     Q2 1998                 1.196    1.237.884
     Q3 1998                 1.642    1.669.551
     Q4 1998                 1.796    1.728.314
     Q1 1999                 1.193    1.265.707
     Q2 1999                 1.471    1.599.034
     Q3 1999                 1.150    1.479.660
     Q4 1999                 1.355    1.257.092
     Q1 2000                 1.777    1.647.861
     Q2 2000                 1.066    1.253.829
     Q3 2000                 1.063    1.234.464
     Q42000                  1.112    1.404.944
     Q1 2001                 1.321    1.530.458
     Q2 2001                 1.928    2.187.350
     Q3 2001                 1.970    1.692.745
     Q4 2001                 1.953    1.771.476
     Q1 2002                 1.532    1.813.423
     Q2 2002                 1.196    1.468.879
     Q3 2002                 1.510    1.677.987
     Q4 2002                 1.502    1.520.320
                        Statistical techniques-example
                2.400.000

                2.300.000
                                   TC = 635,83*Q + 617.880
                                        R2 = 68,85%
                2.200.000

                2.100.000

                2.000.000

                1.900.000

                1.800.000
       ( EUR)
[TC]




                1.700.000

                1.600.000

                1.500.000

                1.400.000

                1.300.000

                1.200.000

                1.100.000

                1.000.000
                        1.000   1.100    1.200   1.300       1.400    1.500      1.600   1.700   1.800   1.900   2.000
                                                                     Units [Q]
Engineering Cost
Techniques
 Estimating cost function using
  knowledge of production technology
 Attempts to determine the lowest cost
  combination of labor, capital equipment
  and raw materials required to produce
  various levels of output
        Engineering Cost
        Techniques
                  Variable costs               Fixed costs                Average costs
Units        Raw     Services   Labor       Labor       Other       AVC       AFC         ATC
            material


        0     6.655       350           0   4.500.000   9.500.000         -         -           -
 10.000       6.655       350           0   4.500.000   9.500.000    7.005      1.400      8.405
 20.000       5.830       350           0   4.500.000   9.500.000    6.180        700      6.880
 30.000       5.830       350           0   4.500.000   9.500.000    6.180        467      6.647
 40.000       5.500       350           0   4.500.000   9.500.000    5.850        350      6.200
 50.000       5.500       350           0   4.500.000   9.500.000    5.850        280      6.130
 60.000       5.280       350           0   4.500.000   9.500.000    5.630        233      5.863
 70.000       5.280       350       94      4.500.000   9.500.000    5.724        200      5.924
 80.000       5.005       350       94      4.500.000   9.500.000    5.449        175      5.624
 90.000       5.005       350       94      4.500.000   9.500.000    5.449        156      5.605
100.000       5.005       350       94      4.500.000   9.500.000    5.449        140      5.589
Engineering Cost Techniques -
Example
       9.000


       8.500


       8.000
Average costs




       7.500


       7.000


       6.500


       6.000


       5.500


       5.000
                0   10.000   20.000   30.000   40.000   50.000   60.000   70.000   80.000   90.000   100.000
                                                        UNITS
Profit Maximization and
Competitive Supply
Do Firms Maximize Profits

 profit is likely to dominate desicions in
  owner managed firms
 managers in larger companies may be
  more concerned with goals such as
      revenue maximization
      dividend pay-out
      on the long run they must have profit as
       one of their highest priorities
           Profit maximization condition MR =
           MC
                                              C(q)

Costs, Revenue, Profit
                                                     R(q)
                                  A

                                  B




                         0   q0    q*
                                                 π(q)
                                        Units/year
        Profit is maximized, when MR = MC.
Competitive Firm
                                           MC
    60
 Price
    50

    40                                               P=MR=AR
                                               ATC
    30                                         AVC

    20

    10

     0   1    2   3   4   5   6   7   8    9   10    11 Units
         q0                           q*
Competitive Firm
Incurring Losses

Price                   MC         ATC

                   B
   C
   D                          P = MR
                    A
                             AVC
   F                     Should they
                    E
                         continue their
                         operatios?

                   q*     Units
           Adequate Condition for Profit
           Maximization: P >= AVCmin
 Even if P=MR=MC, the company may operate with profit/loss:
      if P>ATCmin -> PROFIT;
      if P=ATCmin, -> ZERO PROFIT;
      if AVCmin<P<ATCmin, -> LOSS, however lower than it would be if the
       firm had not operated;
      if P=AVCmin, the company is indifferent to stay in business or to
       shut down, loss is equal to TFC
      if P<AVCmin, shut down
 Firm’s Supply Curve

 Since firm decides upon P=MC,and considers
  P>=AVCmin, its individual supply curve equals
  its MC curve above AVCmin.
Firm’s Supply Curve
   Price
           S = MC over AVC

                               MC
     P2                                ATC
     P1                                AVC


P = AVC


                             q1 q2   Units
Market Supply
 sum of individual supply curves
 price



                 MC1       MC2        MC3
                                                      S
   P2




   P1

  P3




  0      2   4   5     7   8     10         15   21       units
Costs for Decision
Making
      Economic costs

 Accounting costs are historic costs
     Historical cost is the cost incurred at the time of
      procurement.
     do not incorporate opportunity costs
Economic costs
 For business decision making we use economic costs
       explicit cost
       implicit cost
 Historic costs match to some extent explicit costs, implicit
    costs are opportunity costs
       Opportunity cost is the value that is forgone in
        choosing one activity over the next best alternative.



   Economic Profit = Accounting Profit – Opportunity Costs
Costs for Decisin Making

 A cost is relevant if it is affected by a
  management decision.

 A cost is irrelevant if it is not affected by
  a management decision.
Costs for Decisin Making

 Sunk cost does not vary with decision
  options
Incremental Analysis

 Incremental analysis is used to analyze
  business opportunities .
 Incremental cost varies with the range
  of options available in the decision
  making process.
 Incremental analysis uses only decision
  relevant revenues and cost
Incremental Analysis Process

 Incremental Analysis Process :
     Define relevant revenues and costs
     Define incremental revenues and costs
     If incremental revenues exceed
      incremental costs, take the decision,
      otherwise reject it
Classification of Costs
                                                               Time

                                Past                   Present                       Future

                                                        Variable
                                                  Additional material
                                                   Additional labor
                                                Additional general costs
                                                                           Present value of expected
              EXPLICIT            X                                         additional variable and
                                                                                  fixed costs
                                                         Fixed
INCREMENTAL
                                                   Depreciation of
 (RELEVANT)
                                                 additional equipment
                                                   Additional labor


                                                   Oportunity costs
                                                                           Present value of expected
              IMPLICIT            X               Revenue lost by the
                                                                           additional opportunity costs
                                                  taking the decision


                                                 Predetermined costs
                                                     Fixed salaries
                                                   Interest on loans
                             Sunk costs                  Rentals
                                                                           Present value of expected
              EXPLICIT   Costs resulting from      Other contractual
                                                                             predetermined costs
                            past decision              obligations
 IRRELEVANT
                                                 Other predetermined
                                                 costs, independent of
                                                      the decision


              IMPLICIT            X                        X                            X
    Classification of Revenues
                                                     Time

                               Past                 Present                Future
                                               Additional revenue
                                                                      Present value of
                                                -from operations
              EXPLICIT           X                                   expected additional
                                                 - from financing
                                                                          revenue
                                                 - from investing
INCREMENTAL
 (RELEVANT)
                                              Opportunity revenue     Present value of
              IMPLICIT           X            The cost avoided by         expected
                                                 this decision.      opportunity revenue


                                                 Predetermined        Present value of
                          Historic revenue
                                                    revenue               expected
              EXPLICIT     Result of past
                                              Independent       of     predetermined
                         business decisions
IRRELEVANT                                    business decision           revenue


              IMPLICIT           X                     X                     X
Examples of Incremental
Analysis
 Outsourcing opportunities for small
 businesses: A quantitative analysis

								
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