Chapter 8
Production Function
&
Cost Function
© Pilot Publishing Company Ltd. 2005
Contents:
• Short Run and Long Run
• Production Function
• Cost Accounting of Factor Inputs
• Cost Function
• Short Run Cost Curves
• Long Run Cost Curves
• Theory of Production Cost by A. Alchian
• Advanced Material 8.1: Costs Related to an
Owned Durable Equipment
© Pilot Publishing Company Ltd. 2005
Short Run & Long Run
© Pilot Publishing Company Ltd. 2005
Reasons for the existence of different production runs:
Whenever a market situation changes a firm has to
make a new decision  so as to maximize wealth.
At the beginning
 The firm is uncertain if the change in the market situation
is temporary or permanent.
 It will make only the minimal and necessary change
in factors to minimize cost.
© Pilot Publishing Company Ltd. 2005
Reasons for the existence of different production runs:
Afterwards
Even if the change is certain to be permanent,
 the adjustment in factors should still be slow and
gradual because hasty change involves a larger cost.
© Pilot Publishing Company Ltd. 2005
Reasons for the existence of different production runs:
Since adjustment is gradual, according to the completeness
in the adjustment in factors, three different production runs
are classified.
© Pilot Publishing Company Ltd. 2005
Classification of production runs
Very short run (VSR)
all factors are fixed (remains unchanged).
Short run (SR)
some factors are varied but some are fixed.
Long run (LR)
all factors are variable and
all required variations have been made.
© Pilot Publishing Company Ltd. 2005
Variable factors versus fixed factors
Variable factors: are factors of which the
employment varies with output.
Fixed factors: are factors of which the
employment does not vary with output.
© Pilot Publishing Company Ltd. 2005
Q8.1 Fill in the table.
VSR
Any change in
factor
employment?
Any change in
output level?
Reasons
No
No
SR
LR
All factors are
Some factors are
variable and all
varied but some
required changes are
are fixed
made
Yes
Yes
No adjustment —Temporary adjust.
Time is needed —Time is needed Final adjustment –
to recognize the to identify if the Time is long enough
change, make change is permanentfor the final adjust.
decision &
& to make gradual to be determined &
implemented.
implement
adjustment to
adjustment
minimize cost
© Pilot Publishing Company Ltd. 2005
Q8.2:
“Some economists define the very long run as
the period over which the technology changes.”
Comment..
© Pilot Publishing Company Ltd. 2005
Production Function
© Pilot Publishing Company Ltd. 2005
Production function (生產函數) describes
the relationship between inputs (投入量) and
output (產量).
Inputs
© Pilot Publishing Company Ltd. 2005
Output
Production function in the short run
Assumptions:
 only two factors are involved  capital & labour
Capital
Variable
factor
Fixed
Factor
© Pilot Publishing Company Ltd. 2005
Three variables are defined to measure the output:
Total product (TP)
____________________:
is the whole amount of output
produced by all the factors employed.
Average product (AP) is the output per unit of the
____________________:
variable factor employed.
____________________:
Marginal
product (MP) the change in output resulting
from employing an additional unit of the variable factor.
TP = Q
TP Q
AP 

L
L
© Pilot Publishing Company Ltd. 2005
MP 
TP
L

L 1
Q
L
L 1
The law of diminishing marginal productivity
Law of diminishing marginal productivity (邊際產量遞減定律)
[or the law of diminishing returns (邊際回報遞減定律)
or the law of variable proportions (可變比例定律)]
states that if a _________
variable factor is added continuously to a
given amount of _________
fixed factors, the marginal product
(and the average product) of the _________
variable factor must
finally decrease, ceteris paribus.
© Pilot Publishing Company Ltd. 2005
Derivation of the law:
With a given amount of fixed factors, when one worker is
employed, he can use only some of the fixed factors each
time.
When more workers are employed, they can specialize
and raise the productivity. (Both MP & AP ).
However, after all the fixed factors have been efficiently
used, additional workers can help the preceding workers
only. Hence MP  which will finally drag down AP (and
even TP).
© Pilot Publishing Company Ltd. 2005
Graphical
illustration:
MP
Variable factors Fixed factor
MP
AP 
© Pilot Publishing Company Ltd. 2005
AP
Once the MP curve
passes through the AP
curve and lies below it,
the AP curve will also
be dragged down. Why?
When AP rises, MP must lie above AP.
When AP falls, MP must lie below AP.
How can MP lying
above AP become lying
below it? (If MP curve is
continuous, it must pass
through the maximum
point of AP curve.)
© Pilot Publishing Company Ltd. 2005
Features:
The slope of TP
curve is MP.
The slope of the
line joining the
origin and a point
on TP is AP.
Notice the points
where MP = maxi.;
MP=AP &
MP = 0.
© Pilot Publishing Company Ltd. 2005
Implications of the law (if the law is violated)
By adding units of fertilizer or worker continuously to a
given plot of land, no matter how small its size is, TP can be
increased continuously. Enough food can be produced to
feed all the people in the world.
An infinitesimal piece of land is adequate to supply the
amount of food required. Hence the supply of land is no
longer scarce.  Land price would drop to zero.
MP of workers cultivating superior land does not fall and is
always larger than MP of workers cultivating inferior land.
 No inferior land would be cultivated.
© Pilot Publishing Company Ltd. 2005
Implications of the law (if the law holds)
If the amount of land is fixed, TP cannot be
increased significantly (∵ MP & AP ) to feed all
the people in the world.
Hence to raise production, more land is needed.
However, as the supply of land is limited and scarce,
under competition, land price must be positive.
Once MP of superior land falls below MP of inferior
land, inferior land will also be cultivated.
© Pilot Publishing Company Ltd. 2005
Production Function in the long run
Returns to scale (規模回報)
 refers to the change in output when all
factors are increased by the same proportion.
© Pilot Publishing Company Ltd. 2005
Production Function in the long run
There are three kinds of returns to scale:
 Increasing returns to scale
 Constant returns to scale
 Decreasing returns to scale
© Pilot Publishing Company Ltd. 2005
Increasing returns to scale (規模回報遞增)
when all factors are increased by the same proportion,
the total product increases more than proportionately.
© Pilot Publishing Company Ltd. 2005
Constant returns to scale (規模回報不變)
 when the production scale increases,
the total product increases proportionately.
© Pilot Publishing Company Ltd. 2005
Decreasing return to scale (規模回報遞減)
when the production scale increases,
the total product increases less than proportionately.
© Pilot Publishing Company Ltd. 2005
Assumptions (in the LR):
At the beginning
 the production function has increasing returns to scale
At the end
 the production function has decreasing returns to scale
© Pilot Publishing Company Ltd. 2005
Cost Accounting
of
Factor Inputs
© Pilot Publishing Company Ltd. 2005
For factors hired or employed by a firm:
The costs are (the value of) the highestvalued alternative use of the money spent in
hiring them.
They are called explicit costs (顯性成本),
as they involve a transfer of money.
© Pilot Publishing Company Ltd. 2005
For factors owned by a firm:
The costs of using these factors are (the value
of) the highest-valued alternative uses of the
factors.
They are called implicit costs (隱含成本)
or imputed costs (設算成本),
as they do not involve
a transfer of money.
© Pilot Publishing Company Ltd. 2005
Q8.4:
What is the cost to a firm of using an owned
property as an office for a year,
if
the premise can be leased at $20 000 a month or
sold at $5 million, given a market interest rate of
4% per annum?
© Pilot Publishing Company Ltd. 2005
Classification of costs of different factor inputs
Sunk cost (historical cost)
 The cost of a past act.
 As past options are not available at present, sunk cost
cannot be avoided now. Sunk cost is not a (present or
future) cost.
 Bygone is bygone. It should have no effect on
any present or future decisions.
© Pilot Publishing Company Ltd. 2005
Fixed cost
 The cost of employing fixed factors.
 It does not change with output.
 It has no effect on MC & no effect on the determination
of the wealth-maximizing output level.
 It is a present cost paid for the use of fixed factors
and hence it affects the net receipt.
© Pilot Publishing Company Ltd. 2005
Variable cost
 The cost of employing variable factors.
 It changes with output.
 It affects marginal cost & hence it affects the wealthmaximizing output.
 It is a present cost paid for the use of variable factors
& hence it affects the net receipt.
© Pilot Publishing Company Ltd. 2005
Q8.6:
A restaurant is making a short run decision for its production
next month. Identify if the following costs are sunk costs (SC),
fixed costs (FC) or variable costs (VC).
(a) Rent of the restaurant under a 2-year contract ( )
(b) Wage payments
( )
(c) Expenditure on meat and vegetables
( )
(d) Water charges
( )
(e) Electricity charges
( )
(f) Acquisition cost of machines
( )
(g) Continuing possession cost of machines
( )
(h) Operating cost of machines
( )
© Pilot Publishing Company Ltd. 2005
Cost Function
© Pilot Publishing Company Ltd. 2005
Cost function (成本函數) describes the
relationship between output and cost.
Output
= ???
© Pilot Publishing Company Ltd. 2005
Short-run Cost Curves
© Pilot Publishing Company Ltd. 2005
Measure of costs
Output changes  Cost changes
Change in cost can be expressed in three ways:
Total cost (TC)
Average cost (AC)
Marginal cost (MC)
© Pilot Publishing Company Ltd. 2005
Total cost
 is the whole amount of payments to all factors used
in producing a given amount of output (Q), composed of:
 Total fixed cost (TFC): is the whole amount of
payments to fixed factors.
 Total variable cost (TVC): is the whole amount of
payments to variable factors.
© Pilot Publishing Company Ltd. 2005
Formula:
Assume two factors only:
 Capital (fixed factor) and labour (variable factor)
 L units of labour are employed at a wage rate of w.
Total Cost:
• total fixed cost:
• total variable cost:
© Pilot Publishing Company Ltd. 2005
TC = TFC +TVC
a constant independent of output
TVC = w x L
Average cost/average total cost (ATC)
 is the cost per unit of output, composed of :
•average fixed cost (AFC):
the fixed cost per unit of output.
•average variable cost (AVC):
the variable cost per unit of output.
© Pilot Publishing Company Ltd. 2005
Formula:
Average Total Cost:
ATC 
TC TFC  TVC

 AFC  AVC
Q
Q
TFC
AFC 
Q
average fixed cost:
average variable cost:
AVC 
TVC
Q

w L
Q
w L

L
Q
L
© Pilot Publishing Company Ltd. 2005
w
w


Q
AP
L
Features:
AFC curve drops
continuously. (∵AFC
= TFC/Q)
AVC curve is Ushaped. (∵ AVC =
w/AP and AP is
inverted-U shaped.)
ATC curve and AVC curve will come closer and closer as
the amount of output increases (∵ATC = AFC + AVC and
AFC drops continuously).
© Pilot Publishing Company Ltd. 2005
The turning point of ATC curve (b) occurs at a larger
output than the turning point of AVC curve (a). Why?
∵ At (a), the fall in AFC is > the rise in AVC initially
but at (b), the fall in AFC is < the rise in AVC eventually
(a) (b)
© Pilot Publishing Company Ltd. 2005
Marginal Cost
is the change in total cost for producing an additional
unit of output, composed of :
• marginal fixed cost (MFC): is the change in fixed cost
for producing an additional unit of output
• marginal variable cost (MVC): is the change in variable cost
for producing an additional unit of output.
© Pilot Publishing Company Ltd. 2005
Formula:
Marginal cost:
MC 
TC TFC  TVC

 MFC  MVC
Q
Q
marginal fixed cost:
TFC
MFC 
0
Q
marginal variable cost:
w  L
MVC 
TVC
Q
© Pilot Publishing Company Ltd. 2005
w  L
w
w

L




Q
Q
Q
MP
L
L
As TFC is a constant, MFC = 0. So MC = MVC.
MC = MVC = w/MP. As MP curve is inverted-U shaped, MC
or MVC curve is U-shaped.
MC curve passes through the minimum points of AVC curve
and ATC curve.
MC or MVC
curve is
U-shaped
© Pilot Publishing Company Ltd. 2005
Derivation of total cost
curves:
MC curve (= MVC
curve) = Slope of TC
curve & TVC curve.
Notice the points
where MC = mini.;
MC = AVC and MC
= ATC.
© Pilot Publishing Company Ltd. 2005
Q8.7: The following table is composed of product items and cost
items of a firm. Suppose the unit cost of capital and labour are
$10 and $20 respectively. Fill in the missing columns..
Units Units
of
of
capital labour
4
4
4
4
4
4
1
2
3
4
5
6
TP
AP
2
5
10
14
14
12
© Pilot Publishing Company Ltd. 2005
MP
TFC
TVC
TC
ATC
Q8.8
(a) When output increases, if AP of a variable factor rises,
what will happen to AVC and ATC?
(b) When output increases, if AP of a variable factor falls,
what will happen to AVC and ATC?
© Pilot Publishing Company Ltd. 2005
Long-run Cost Curves
© Pilot Publishing Company Ltd. 2005
The firm enjoys economies of
scale at the beginning
LRAC & LRMC 
As the scale of production 
further, the firm suffers
diseconomies of scale
LRAC & LRMC
© Pilot Publishing Company Ltd. 2005
Optimum scale
The production scale (combination of factors) with
the lowest LRAC.
U-shaped LRAC curve
Optimum scale
© Pilot Publishing Company Ltd. 2005
LRAC curve with
a horizontal region
Derivation of total
cost curves:
LRMC = slope of LRTC
Notice the points
where LRMC = mini.
and LRMC = LRAC.
© Pilot Publishing Company Ltd. 2005
Slope
=LRMC
=LRAC
Theory of Production Cost
by A. Alchian
© Pilot Publishing Company Ltd. 2005
Volume = Rate x Time Duration
Rate effect: To produce the same volume of output,
the faster the rate, the higher the average cost. Why?
Volume effect: At the same rate of production,
the larger the volume, the lower the average cost. Why?
Proportionate increase in both the rate and volume:
If both the rate and the volume are increased
proportionately, the average cost may fall at the beginning
but it must rise eventually. Why?
© Pilot Publishing Company Ltd. 2005
Advanced Material 8.1
Cost related to an owned durable equipment
1. Present value and future value
Present value
Future value
Computing the present value
(PV) of a future sum ($X) is
called discounting (折現).
Computing the future value
(FV) of a present sum ($Y) is
called compounding (複算).
PV = $X / (1+r)t
FVt = $Y (1+r)t
© Pilot Publishing Company Ltd. 2005
2. Classification of costs related to an owned durable
equipment
Acquisition cost (取得成本) is the amount forgone in
acquiring the ownership of an asset.
Continuing possession cost (繼續擁有成本) is the
amount forgone in keeping an asset without using it.
Operating cost (操作成本) is the amount forgone in
using an asset.
© Pilot Publishing Company Ltd. 2005
Worked example:
-Purchase price of machine A = $10 000
-Immediate resale value = $9 600
-Resale value if machine A has been laid idle for a year = $8 000
-Resale value if machine A has been operated for a year = $4 000
-Additional expense for operating machine A (e.g., labour cost) = $1 000
-Market interest rate = 10%
(a) Acquisition cost = ?
(b) Continuing possession cost = ?
(c) Operating cost = ?
© Pilot Publishing Company Ltd. 2005
Correcting Misconceptions:
1. In the long run, all factors are variable and
are varied gradually so as to minimize cost.
2. The law of diminishing returns states that
when all inputs increase, output will increase
at a decreasing rate eventually.
3. If the law of diminishing marginal productivity
does not hold, scarcity no longer exists and
production involves zero cost.
© Pilot Publishing Company Ltd. 2005
Correcting Misconceptions:
4. The cost of using a factor is equal to zero if
no explicit payment is involved.
5. The cost of using an owned asset is equal to
the purchase price of the asset.
6. Fixed cost is the same as sunk cost.
7. When output increases, if AP of the variable
factor falls, the AC rises.
© Pilot Publishing Company Ltd. 2005
Survival Kit in Exam
Question 8.1:
In a recession, an employer can earn enough revenue to
cover the rent, labour cost, water and electricity charges
and the maintenance cost of machines. However, the
remaining revenue cannot cover their repayment to
bank loans for buying machines. Suppose the machines
have zero resale value. Is it irrational for the employer
(a) to continue the running of the firm, and
(b) to buy the machines?
© Pilot Publishing Company Ltd. 2005