# 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
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
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
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
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
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
   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
 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
Present value of expected
fixed costs
Fixed
INCREMENTAL
Depreciation of
(RELEVANT)

Oportunity costs
Present value of expected
IMPLICIT            X               Revenue lost by the
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
Present value of
-from operations
- 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