The document discusses production functions and their key concepts. It defines a production function as the relationship between inputs and maximum output. It explains the three stages of production in the short-run and the law of diminishing returns. It also discusses optimal input usage, returns to scale, and the Cobb-Douglas production function.

1. Theory of Production

2. Production Function defined:
• The PF is a statement of the relationship
between a firm’s scarce resources (inputs) and
the output that results from the use of these
resources.
• In mathematical terms, the PF can be
expressed as:
• Q= f (X1, X2…………Xk) where
• Q=output, X1…………Xk=inputs used in
the production process

3. Formal definition of PF
• A PF defines the relationship between inputs
and the maximum amount that can be
produced within a given period of time with a
given level of technology.
• For the purposes of analysis, we write the PF
as follows: Q= f (L, K)
• Where Q=output, L= labour, K=capital

4. A short-Run Analysis of Total, Average
and Marginal Product
• Marginal product of Labor =MPL= ∆Q/∆L,
holding K constant.
• Average product of Labor= APL= Q/L, holding K
constant.

5. Short run changes in production
Units
of K
employ
ed
Output quantit
y
8
7
6
5
4
3
K=2 8 18 29 39 47 52 56 52
1
1 2 3 4 5 6 7 8(Units
of L
employ
ed)

6. Variable input (L) Total Product (TP) Marginal product Average product
1 8 8 8
2 18 10 8.67
3 29 11 9.4
4 39 10 9.75
5 47 8 9.4
6 52 5 8.67
7 56 4 8
8 52 -4 6.5

7. The Three Stages of Production in the
Short-run
• Stage I runs from zero to four units of variable
input (where average product reaches its
maximum and AP and MP are approximately
equal).
• Stage II begins from this point and proceeds to
seven units of input L (to the point where TP is
maximised).
• Stage III continues on from that point.

8. Law of Diminishing Returns
• The key to understanding the pattern of
change in Q, AP and MP is the phenomenon
known as the Law of diminishing returns:
• As additional units of variable input are
combined with a fixed input, at some point
the additional output (the MP) starts to
diminish.

9. Which stage is economical or rational?
• According to economic theory, in the short-
run, rational firms should only be operating in
stage II.
• It is clear why stage III is irrational: the firm
would be using more of its variable input to
produce less output.
• However, it may not be as apparent why stage
I is also considered irrational.

10. • The reason is that if a firm were operating in
stage I, it would be grossly underusing its fixed
capacity.
• That is, it would have so much fixed capacity
relative to its usage of variable inputs that it
could increase the output per unit of variable
input simply by adding more variable inputs to
this capacity.

11. Derived Demand and the optimal level
of variable input case (The case of one
input case)
Optimal decision rule: A profit maximising firm
operating in perfectly competitive output and
input markets will be using the optimal
amount of an input at the point at which the
monetary value of input’s marginal product is
equal to the additional cost of using that
input---in other word’s when MRP=MLC.

12. Optimal Input Usage Rule (one input
Case when price of product=$2 and
labor cost=$10)Labor
Unit (L)
Total
Product
Marginal
Product
Total
Revenue
Product
Marginal
Revenue
Product
Total
labor cost
Marginal
labor cost
1 10 10 20 20 10 10
2 25 15 50 30 20 10
3 45 20 90 40 30 10
4 60 15 120 30 40 10
5 70 10 140 20 50 10
6 75 5 150 10 60 10
7 78 3 156 6 70 10
8 80 2 160 4 80 10

13. Optimal input usage (Multiple input
case)
• In the multiple input case, we must consider
the relationship between the ratio of the MP
of one input and its cost to the ratio of the MP
of other input and its cost. Expressed
mathematically for “k” inputs:
• MP1/W1=MP2/W2=MPk/Wk

14. • Suppose you are the production manager of a
company that makes computer parts and
peripherals in Malaysia and China. At the
current levels of production and input
utilization in two countries, you find that:
• MP of labor in Malaysia (MPmal)=18
• MP of labor in China(MPch)=6
• Wage rate in Malaysia(Wmal)=$6/hr
• Wage rate in China (Wch)= $3/hr

15. • How much would you produce in each
manufacturing facility? Because labor is
cheaper in China you might be tempted to
produce most of your output in that country.
However, a close look at the MP/wage ratio
reveals the opposite conclusion. That is,
• MPmal/Wmal>MPch/Wch
• Or 18/6 > 6/3

16. • This means that at the margin , the last dollar
on a unit of labor in china would yield 2 units
of output (6/$3) while in Malaysia the last
dollar spent would result in 3 additional units
of output (18/$6).
• This inequality implies that the firm should
begin to shift more of its production from
china to Malaysia , until the two ratios are
equalized

17. • Once the implication of the basic model is
understood, other factors can be brought in. If
these factors outweigh the MP-input cost
criteria, a company may well modify its
decision.
• For example, despite Malaysia’s higher
MP/wage ratio, there may be political and
economic risk factors to consider.

18. • This was indeed the case when the Malaysian
government imposed foreign exchange controls
in 1998 by requiring foreign investors to keep
their profits in Malaysia for at least 1 year before
they could be repartriated .
• In contrast, China is a fairly stable economy with
leaders who do not seem to want to impose any
such trade restrictions. Its proximity to Indian
markets would also would also reduce
transportation costs.

19. Call centers: Applying the Production
Function to a Service
• Let us consider the example of a call center
represented by the following production
function:
• Q = f (X,Y) where
• Q= number of calls
• X= variable input (this includes call center
representatives and complementary hardware
such as PCs, desks, and software)
• Y= fixed input (this includes call center building,
hardware such as servers and
telecommunications etc)

20. Three Stages of production
• Stage I could be a situation in which there is so
much fixed capacity relative to number of
variable inputsthat many representatives sit
around idle, waiting for calls to come in.
• Stage II could be a situation in which
representatives are constantly occupied and
callers are connected to representatives
immediately after the call is answered or are
kept waiting for no more than a certain amount
of time (3 min).

21. • Stage III could be a situation in which callers
begin to experience a busy signal on a more
frequent basis or all call representatives may
begin to experience a slower computer
response or more frequent computer ‘down
times’.

22. The Long Run Production Function
• In the long run, a firm has time enough to
change the amount of all inputs. The following
Table illustrates what happens to total output
as both inputs L and K increase one unit at a
time.

23. Returns to Scale
Units
of K
emplo
yed
Outpu
t
Quant
ity
8 125
7 119
6 90
5 75
4 60
3 41
2 18
1 4
1 2 3 4 5 6 7 8 Units
of L
emplo
yed

24. • The resulting increase in output as both inputs
vary is known as Returns to Scale.
• Returns to scale are of three types:
• Increasing returns to scale
• Decreasing returns to scale
• Constant returns to scale

25. • If an increase in a firm’s input by some
proportion results in an increase in output by
a greater proportion, the firm experiences
IRTS.
• If output increases by the same proportion as
the inputs increase, the firm experiences
CRTS.
• A less than proportional increase in output is
called decreasing returns to scale.

26. • One way to measure RTS is to use the
coefficient of output elasticity:
• EQ= % change in Q/% change in inputs
• If EQ>1, we have IRTS
• If EQ<1, we have DRTS
• If EQ=1, we have CRTS

27. • Another way of looking at the concept of RTS is
based on the production function:
Q = f (L, K)
• Now if we increase both inputs by r times and
output increases by t times, that is
• tQ= f(rL, rK) then
• If t>r, we have IRTS
• If t<r, we have DRTS
• If t=r, we have CRTS.

28. The Cobb-Douglas Production
Function
• The C-D production function was introduced
in 1928 and it is still a common functional
form in economic studies today.
• It has been used extensively to estimate both
individual firm and aggregate production
function.
• The formula for production function which
was suggested by Cobb, was of the following
form: Q= aLbK1-b

29. Why is this production function so
useful?
• 1. To make this equation useful, both inputs
• must exist for Q to be a positive number.
• This makes sense because total product is
• a result of combining two or more factors.
• 2.The function can exhibit increasing,
decreasing or constant returns. Originally,
cobb-douglas assumed RTS are constant. Later
they relaxed this assumption and rewrote the
equation as follows: Q= a LbKc

30. • Under this assumption if b+c>1, RTS are
increasing, if b+c<1, RTS are decreasing and if
b+c=1, RTS are constant.
• 3. The function permits us to investigate the MP
for any factor while holding all others constant.
MP of labor turns out to be MPL=bQ/L and MP of
capital is MPk=cQ/K.
• In the C-D function, the elasticities of the factors
are equal to their exponents, in this case b and c.

31. • 4.Because a power function by using
logarithms, it can be estimated by linear
regression analysis, which makes for a
relatively easy calculation with any software
package.
• 5. Cobb-Douglas can accommodate any
number of independent variables as follows:
• Q=aXb
1Xc
2Xd
3..Xm
n

32. • 7. A theoretical production function assumes
technology is constant. However, the data
fitted by the researcher may span a period
over which technology has progressed. One of
the independent variables in the previous
equation could represent technological
change and thus adjust the function to take
any technology into consideration.

33. Shortcomings of C-D production
Function
• This function cannot show the MP going
through all three stages of production in one
specification.
• Similarly, it cannot show a firm or ndustry
passing through increasing, constant and
decreasing returns to scale.