chapter Three.docx

WSU College of Business &Economics Dep’t of Economics
Micro Economics Note Page 1
CHAPTER THREE
3. THEORY OF PRODUCTION
3.1 Introduction
In the last two chapters, we focused on the demand side of the market-the preferences and
behavior of consumers under uncertain and certain information. Now we turn to the supply side
and examine the behavior of producers. We will see how firms can organize their production
efficiently and how their costs of production change as input prices and the level of output
change. We will also see that there are strong similarities between the optimizing decisions of
firms and those of consumers-understanding consumer behavior will help us understand
producer behavior.
3.2 Technological Relationship between Inputs and Output
In the production process, firms turn inputs, which are also called factors of production, into
outputs (or products). For example, a bakery uses inputs that include the labor of its workers; raw
materials, such as flour and sugar; and the capital invested in its ovens, mixers, and other
equipment to produce such outputs as bread, cakes, and pastries.
We can divide inputs into the broad categories of labor, materials, and capital, each of which
might include more narrow subdivisions.
Labor inputs include skilled workers (carpenters, engineers) and unskilled workers (agricultural
workers), as well as the entrepreneurial efforts of the firm's managers. Materials include steel,
plastics, electricity, water, and any other goods that the firm buys and transforms into a final
product.
Capital includes buildings, equipment, and inventories.
The relationship between the inputs to the production process and the resulting output is
described by a production function. A production function indicates the output Q that a firm
produces for every specified combination of inputs. For simplicity, we will assume that there are
two inputs, labor Land capital K. We can then write the production function as
Q = F(K, L)
This equation relates the quantity of output to the quantities of the two inputs, capital and labor.
For example, the production function might describe the number of personal computers that can
be produced each year with a 10,000 square-foot plant and a specific amount of assembly-line

labor employed during the year. Or it might describe the crop that a farmer can obtain with a
specific amount of machinery and workers.
The production function allows for inputs to be combined in varying proportions to produce an
output in many ways. For example, wine can be produced in a labor-intensive way by people
stomping the grapes, or in a capital- intensive way by machines squashing the grapes. Note that
equation given above applies to a given technology (i.e., a given state of knowledge about the
various methods that might be used to transform inputs into outputs). As the technology becomes
more advanced and the production function changes, a firm can obtain more output For a given
set of inputs. For example, a new, taster computer chip may allow a hardware manufacturer to
produce more high speed computers in a given period of time.
Production functions describe what is technically feasible when the firm operates efficiently; that
is, when the firm uses each combination of inputs as effectively as possible. Because production
functions describe the maximum output feasible for a given set of inputs in a technically efficient
manner, it follows that inputs will not be used if they decrease output. The presumption that
production is always technically efficient need not always hold, but it is reasonable to expect that
profit seeking firms will not waste resources.
Not all firms will be producing the maximum output Q that is possible from given inputs L and k
at any point in time. There are two main reasons for that to be so:
 Some firms may be inefficient in that they incur undue costs to produce a certain desired
level of output.
 Different firms have machines and equipment of different vintages. Not all firms will
utilize the latest technology.
Thus, the more efficient firms and those using the latest technology will be producing more
output than other firms even for the same levels of measured inputs.
A single production function cannot exist for all firms. We will consider a typical firm and study
the relationship between its inputs and output. We also consider a given state of technology.
We will first consider a production process with a single variable input and then consider two
inputs and substitution between them subsequently.
3.3 Fixed Vs. Variable Inputs and
Short run Vs. Long Run Production Periods

Input are classified into fixed inputs, the supply (quantity) of which cannot be changed over a
short period of time and variable inputs the quantity of which can be varied in the short-run.
The most important fixed inputs in the land, capital, plant and equipment whereas labor and
raw materials are instances of variable in short-run are put in the short-run .The short run refers
to a period of time in which one or more factors of production cannot be changed.
Factors that cannot be varied over this period are called fixed inputs. A firm's capital, for
example, usually requires time to change-a new factory must be planned and built, machinery
and other equipment must be ordered and delivered, all of which can take a year or more. The
long run is the amount of time needed to make all Inputs variable.
In the short run, firms vary the intensity with which they utilize a given plant and machinery; in
the long run, they vary the size of the plant. All fixed inputs in the short run represent the
outcomes of previous long-run decisions based on firms' estimates of what they could profitably
produce and sell.
There is no specific time period, such as one year, that separates the short run from the long run.
Rather, one must distinguish them on a case-by-case basis. For example, the long run can be as
brief as a day or two for a child's lemonade stand, or as long as five or ten years for a
petrochemical producer or an automobile manufacturer.
3.4 Production with a single variable input
Let’s suppose that the production of maize require land, fertilizer, water machinery, etc.--, all
fixed at certain quantities. The only input the producer can adjust is labor. Thus, labor is the
only variable input. If labor input (measured in, say, worker/days) is 0, the output of maize is,
off course 0. As we increase the variable input (labor input), the producer will increase the output
of maize. But, a point will come where increasing labor will not increase the output of wheat at
all and, in fact, might even decrease it. This can be understood form Table 3.1 below

Table 3.1 Total Product, Marginal Product, and Average Product for Different Levels of
Units of labour
(Number of workers)
Total product
(Output/unit)
Average product
(Output/unit)
Marginal
Product
0 0 - -
1 100 100 100
2 220 110 120
3 360 120 140
4 480 120 120
5 530 106 70
6 570 95 40
7 595 85 25
8 595 74 0
9 580 64 -15
10 520 52 -60

(b)
Input (Labour)
The above tabular expression of and product concepts are depicted by way of a diagram below:
(a)
MPL
Fig. 3.1 Relationship between TPL, APL, and MPL curves and the three stages of production
From table 3.1 and Fig 3.1 above, the following production concepts need to be defined:
Stage
I
Stage
II
Stage
III
500
400
300
200
100
0
1 2 3 4 5 6 7 8 9 10
A
B C
C TPL
Total, average
and marginal
products
Units of labour
0 3 4 8 9
e
APL
Units of labour
MPL1
APL
0 3 4 8 9
APL
Units of labour
MPL1
APL
0 3 4 8 9
APL
Units of labour
MPL1
APL
0 3 4 8 9
APL
Units of labour
MPL1
APL
0 3 4 8 9
APL
MPL1
APL
Units of labour
MPL,
APL

1) Total physical product (TPPL) is the maximum quantity of output produced and measured
in physical units by the employees of the variable input, holding all other inputs constant.
2) Average physical product (APPL) is total product divided by the units of labour
employed. Mathematically, APP = TPP/L.
3) Marginal physical product (MPPL) is the extra product resulting from one extra units of
labour input, keeping all other inputs constant. That is MPPL = TPP/L.MPPL is the
shape of the TPP curve.
Note that average product and marginal product can be defined for any input into the production
process.
Relationship between the MP and TP Curves
MPL is equal to TP/L. So graphically it is the slope of the TP curve. In general the following
relations can be established between TP and MP curves.
1) When TPP is increasing at an increasing rate, MPP is increasing. That is, if MPP > 0,
TPP will be rising as labour input (L) increases. Here, the additional labour contributes
more to output.
2) When TPP is increasing at decreasing rate, MPPL is falling. Here too, MPP > 0 and the
additional labour odds something to output.
3) When TPP is maximum and contrast (neither increasing nor decreasing mpp is zero. This
means the additional labour does not affect output.
4) When TPP itself declines the MPPL becomes negative. Thus, MPPL< 0, and TPP will be
falling as labour input increase. The additional labour actually reduces output.
Therefore, the total product curve reaches its maximum when MPPL = 0 and then starts
declining when MPPL< 0.
Relationship between MPPL and APPL
1) If MPPL> APPL, the latter will be rising as labour input (L) increases. This is in stage I of
production.
2) If MPPL< APPL the latter will be stage II and III of production (L) increases. This the case
in stages II and III of production.

3) If MPPL = APPL, the latter will be at its maximum. This is point e on Fig 3.2, panel (b).
The three stages of production
Based on the behaviour of MPPL and APPL economists have classified production into three
stages. These stages are marked on Fig 3.2, upper panel of the diagram (panel a).
1. Stage I: It starts from the point where TPP is zero to the maximum point of APPL.
Here, the average product per worker increases. Two or three workers produce more than
twice as much as one worker with the same tools. Here, total product increases at an
increasing rate. This is the stage of increasing returns to labor.
2. Stage II: Total product continues to increase, but at a diminishing, rate. In this stage,
both average and marginal products are declining.
Marginal product, being below the average product, pulls the average product down.
Indeed, marginal product reaches zero at the end of the stage II corresponding to the
maximum total product. This stage is the stage of diminishing return to labor input.
Thus, stage II begins where MPPL and APPL are equal and ends where MPPL is zero.
3. Stage III: Total product declines and this cause the marginal produces to be less than
zero (negative).
This stage is called the stage of negative return to labor input.
Rational Decision of Production
No profit-maximizing producer would produce in stages I or III.
 In stage I by adding one more unit of labor, the producer can increase the average
productivity of all the units. This it would be unwise on the part of the producer to
stop production in this stage. This is so too much fixed inputs triggering
underemployment of the variable input and underutilization of the fixed inputs.
Thus, the rational producer always has an incentive to expand employment of the
variable input and expand the level of output in stage I. Of course,
 MPPL> 0 and MPPL> APPL is stage I.
As for III, it does not pay the producer to operate in this stage because by reducing the labor
input he/she can increase total output and save the cost of a unit of labor. In this stage, there
is too much labor in relation to the fixed inputs. Here, there is over employment of the
variable input and over utilization of the fixed inputs. Therefore, some of the workers also

redundant that they must be removed from the job in order that the firm generates more
output. I stage is characterized by MPPL< 0.
Thus, stage II is the only economically meaningful stages from the viewpoint of a rational
producer whose objective is maximizing output from the productive inputs. This because it is
the stage where labor employment brings maximum products.
This stage is characterized or by MPPL> 0 and MPPL< APPL at latter stage of stage II (see Fig.
4.2 panel b).
4. The Law of Diminishing (Marginal) returns.
This is referred to as the law of diminishing marginal productivity (the law of diminishing
marginal returns). It is also called the law of variable proportions. The law of variable
proportions (the law of diminishing marginal returns) states that if the quantity of the variable
input in applied to a fixed quantity of other inputs, output per unit of the variable input will
increase, but beyond some point, the resulting increase will be less and less, with total output
reaching a maximum before it finally begins to decline. The law of diminishing returns
commences to operate with the worker (at point A) in Fig 3.2. This point (Point A) third is
called the point of inflexion because beyond this point, the MPPL commences to drop.
Referring back to Table 3.1 and Fig. 3.2, there are increasing returns to labor for the first three
units of labor employed. The law of diminishing returns sets in with the third worker. In some
cases, the law of diminishing returns is also intuitively plausible. Remember that capital, land
other inputs except labour are fixed. Eventually, additional workers will not have a tractor,
shovel, etc to works with. Hence, they will add loss to output than earlier workers, who had
access to ample quantities of the other inputs.
It is worth nothing that the law of diminishing returns is based on the assumption that the
workers have equal efficiencies and are therefore interchangeable. The law of diminishing
(marginal) returns in exactly symmetrical with the law of diminishing (marginal) utility in
consumer theory.
Long run Production: Production with two variable inputs

3
2
1
1 3
q3
q2
q1
6
Labor
Capital
1
Dear learner, we have completed our analysis of the short-run production function in which the firm uses
one variable input (labor) and one fixed input (capital). Now we turn to the long run analysis of
production. Remember that long run is a period of time (planning horizon) which is sufficient for the firm
to change the quantity of all inputs. For the sake of simplicity, assume that the firm uses two inputs (labor
and capital) and both are variable.
The firm can now produce its output in a variety of ways by combining different amounts of labor and
capital. With both factors variable, a firm can usually produce a given level of output by using a great
deal of labor and very little capital or a great deal of capital and very little labor or moderate amount of
both. In this section, we will see how a firm can choose among combinations of labor and capital that
generate the same output. To do so, we make the use of isoquant. So it is necessary to first see what is
meant by isoquants and their properties.
Isoquants
An isoquant is a curve that shows all possible efficient combinations of inputs that can yield equal level of
output. If both labor and capital are variable inputs, the production function will have the following form.
Q = f (L, K)
Given this production function, the equation of an isoquant, where output is held constant at q is
Q = f (L, K)
Thus, isoquants show the flexibility that firms have when making production decision: they can usually
obtain a particular output (q) by substituting one input for the other.
Isoquant maps: when a number of isoquants are combined in a single graph, we call the graph an
isoquant map. An isoquant map is another way of describing a production function. Each isoquant
represents a different level of output and the level of out puts increases as we move up and to the right.
The following figure shows isoquants and isoquant map.

Capital Capital
Capital
100kg
teff
100kg
wheat
100kg
teff
A
Labor
Labor
5
2
5
Fig 3.1 Isoquant and isoquant map. Isoquants show the fact that long run production process is very
flexible. A firm can produce q1 level of output by using either 3 capital and 1 labor or 2 capital and 3
labor or 1 capital and 6 labor or any other combination of labor and labor on the curve. The set of
isoquant curves q1 q2 & q3 are called isoquant map.
Properties of isoquants
Isoquants have most of the same properties as indifference curves. The biggest difference between them is
that output is constant along an isoquant whereas indifference curves hold utility constant. Most of the
properties of isoquants, results from the word ‘efficient’ in its definition.
1. Isoquants slope down ward. Because isoquants denote efficient combination of inputs that yield
the same output, isoquants always have negative slope. Isoquants can never be horizontal, vertical
or upward sloping. If for example, isoquants have to assume zero slopes (horizontal line) only
one point on the isoquant is efficient. See the following figures.
A B C
A
B
A
An isoquant can never be
horizontal. In this figure, the
firm can produce 100kg of
teff by using either of the
following alternatives: 4
capital and 2 labor,4 capital
and 5 labor or any other
combination of labor and
capital along the curve.
Obviously, only the first
alternative is efficient as it
uses the least possible
combination of inputs. Thus,
all points, except A, are
inefficient and not part of
isoquant.
In this figure, a firm can produce
100kg of wheat by using any
combination of labor and capital
along the isoquant. But only point
A is efficient. For example, point
B shows the same number of
labor as point A, but higher
capital. Thus point B is in
efficient because it shows higher
combination of inputs. Thus,
isoquants can never be vertical
line
In this figure, all points above
point A utilize higher
combination of both inputs to
produce the same output (100 kg
coffee). Point A shows the least
combination of inputs that can
yield 100 kg coffees. Thus all
other points are inefficient and
not part of the isoquants.

K1
K2
A
B
Capital
q
B
B
2. The further an isoquant lays away from the origin, the greater the level of output it denotes. Higher
isoquants (isoquants further from the origin) denote higher combination of inputs. The more inputs used,
more outputs should be obtained if the firm is producing efficiently. Thus efficiency requires that higher
isoquants must denote higher level of output.
3. Isoquants do not cross each other. This is because such intersections are inconsistent with the
definition of isoquants.
Consider the following figure3.3
Q=20 q=50
K*
L*
This figure shows that the firm can produce at either output level (20 or 50) with the same combination of
labor and capital (L* and K*). The firm must be producing inefficiently if it produces q = 20, because it
could produce q = 50 by the same combination of labor and capital (L* and K*). Thus, efficiency requires
that isoquants do not cross each other.
4. Isoquants must be thin. If isoquants are thick, some points on the isoquant will become inefficient.
Consider the following isoquant.
Capital Fig 3.4Efficiency requires that
isoquants do not cross each other,
because the point of their intersection
implies that there is inefficiency at
this point.

12 15
L
10
K
8
q=100
q3
q2
q1
K1
K2
K
L1
Labor
Fig.3.4: Iso quants can never be thick. Points A and B are on the same iso quant. But point A denotes higher amount
of capital and the same amount of labor as point B. Hence point A denotes inefficient combination of inputs and thus
it lies out of the iso quant. The iso-quant should be thin if point A is to be excluded from the iso quant.
Shape of isoquants
Isoquants can have different shapes (curvature) depending on the degree to which factor inputs can
substitute each other.
1-Linear isoquants
Isoquants would be linear when labor and capital are perfect substitutes for each other. In this case the
slope of an iso quant is constant. As a result, the same output can be produced with only capital or only
labor or an infinite combination of both. Graphically,
Fig.3.5 linear isoquant. Capital
and labor can perfectly substitute
each other so that the same
output (q=100) can be produced
by using either 10k or 8K and
12L or 15L or an infinite
combinations of both inputs.
2. Input output isoquant
It is also called Leontief isoquant. This assumes strict complementarities or zero substitutability of factors
of production. In this case, it is impossible to make any substitution among inputs. Each level of output
requires a specific combination of labor and capital: Additional output cannot be obtained unless more
capital and labor are added in specific proportions. As a result, the isoquants are L-shaped. See following
figure

1 3 5
9
3
5
7
12
Fig. 3.6 L-shaped isoquant. When isoquants are L-shaped, there is only one efficient way of producing a
given level of output: Only one combination of labor and capital can be used to produce a given level of
output. To produce q1 level of output there is only one efficient combination of labor and capital (L1 and
K1). Output cannot be increased by keeping one factor (say labor) constant and increasing the other
(capital). To increase output (say from q1 to q2) both factor inputs should be increased by equal
proportion.
3. Kinked isoquants
This assumes limited substitution between inputs. Inputs can substitute each other only at some points.
Thus, the isoquant is kinked and there are only a few alternative combinations of inputs to produce a
given level of output. These isoquants are also called linear programming isoquants or activity analysis
isoquants. See the figure below.
Fig. 3.7 kinked isoquant in this case labor and capital can substitute each other only at some point at the
kink (A, B, C, and D). Thus, there are only four alternative processes of producing q=100 out put.
4. Smooth, convex isoquants
L
K
A
B
C
D

∆K=4
∆L=1
∆K=2
∆L=1
∆L=1
∆K=1/2
This shape of isoquant assumes continuous substitution of capital and labor over a certain range, beyond
which factors cannot substitute each other. Basically, kinked isoquants are more realistic: There is often
limited (not infinite) method of producing a given level of output. However, traditional economic theory
mostly adopted the continuous isoquants because they are mathematically simple to handle by the simple
rule of calculus, and they are approximation of the more realistic isoquants (the kidded isoquants). From
now on we use the smooth and convex isoquants to analyze the long run production.
Fig: 3.8 the smooth and convex isoquant. This type of isoquant is the limiting case of the kinked isoquant
when the number kink is infinite. The slope of the iso quant decrease as we move from the top (left) to the
right (bottom) along the isoquant. This indicates that the amount by which the quantity of one input
(capital)can be reduced when one extra unit of another inputs(labor)is used ( so that out put remains
constant) decreases as more of the latter input (labor)is used.
The slope of an isoquant: marginal rate of technical substitution (MRTS)
The slope of an isoquant (-K/L) indicates how the quantity of one input can be traded off against the
quantity of the other, while out put is held constant. The absolute value of the slope of an isoquant is
called marginal rate of technical substitution (MRTS). The MRTS shows the amount by which the
quantity of one input can be reduced when one extra unit of another input is used, so that output remains
constant. MRTS of labor for capital, denoted as MRTS L, K shows the amount by which the input of
capital can be reduced when one extra unit of labor is used, so that output remains constant.
L
K
Q

This is analogues to the marginal rate of substitution (MRS) in consumer theory.
MRTS L,K decreases as the firm continues to substitute labor for capital (or as more of labor is used). In
fig.2.9 to increase the amount of labor from 1 to 2, the firm reduces 4 units of capital (K=4), to increase
labor from 2 to 3, the firm reduce 2 unit of capital (K=2), and so on. Hence, the firm reduces lower and
lower number of capital for the successive one unit of labor. Dear learner, why does this happen?
The reason is that when the number of capital is large and that of labor is low, the productivity of capital
is relatively lower and that of labor is higher (due to the low of diminishing marginal returns). Thus, at
this point relatively large amount of capital is required to replace one unit of labor (or one unit of labor
can replace relatively large amount of capital). As the employment of labor increases and that of capital
decreases (as we move down ward along the isoquant), quite the reverse will happen. That is, productivity
of capital increases and that of labor decreases. Hence, the amount of capital that needs to be reduced
increase when one extra labor is used decreases. The fact that the slope of an isoquant is decreasing
makes an isoquant convex to the origin.
MRTS L, K (the slope of isoquant) can also be given by the ratio of marginal products of factors. That is,
MPK
MPL
L
K
MRTS K
L 




,
This can be shown algebraically as follows:
Let the production function is given as:
q= f (L, K)
Given this production function, the equation of a specific isoquant can be obtained by equating the
production function with a given level of output, say q .
q = f (L, K) = q
Total differential of q measures the total change in q that happens as a result of a simultaneous change
in L and K. i.e,
q
d
dk
k
f
dL
L
q
dq 





 .
.
But since q is constant, dq is zero (dq =0)
So, 0
. 





dk
k
q
dL
L
q
(But, MPl
L
q



and MPk
k
q



)
Where q- is output
L- is unit of labor employed
K-is the amount of capital employed.

Thus, the above equation can be written as:
MPL. dL + MPK.dk = 0

dL
dK
MPk
MPL 

Therefore, the slope of an isoquant can be given as the ratio of marginal products of inputs.
Elasticity of substitution
MRTS as a measure of the degree of substitutability of factors has a serious defect. It depends on the units
of measurement of factors. A better measure of the ease of factor substitution is provided by the elasticity
of substitution, δ. The elasticity of substitution is defined as
MRTS
L
K



%
%
 =
MPK
MPL
L
K
%
%
=
)
(
)
(
MPK
MPL
MPK
MPL
d
L
K
L
K
d






The elasticity of substitution is a pure number independent of the units of measurement of K and L, since
the numerator and the denominator are measured in the same units and be cancelled.
Factor intensity
A process of production can be labor intensive or capital intensive or neutral process. A process of
production is called labor intensive if it uses many labors and relatively few capitals. If it uses many
capitals and relatively few labor it is called capital intensive technology. On the other hand, if the process
uses equal proportion of both it is called neutral technology. The factor intensity of any process is
measured by the slope of the line through the origin representing the particular process. Thus, the factor
intensity is the capital-labor ratio. The higher the capital-labor ratio is the higher the capital intensity but
the lower the capital-labor ratio is the higher labor intensity of the process.
Capital

A
K1
K2
L1 L2
X
O
Fig 3.9 Process A uses k1 and L1 units of labor and capital to produce x amount of output. The factor
intensity of this process can be measured by the slope of OA, which equals AL1/OL1 =
1
1
1
1
L
K
OL
OK

Similarly, factor intensity of process B is given by
2
2
L
K
Since ,
2
2
1
1
L
K
L
K
 process A is more capital intensive than process B or B is more labor intensive than A.
The upper part of the isoquant includes more capital intensive processes and the lower part, labor
intensive techniques.
Now let’s illustrate the above concepts with the most popular and applicable form of production function,
Cobb-Douglas production function
The Cobb- Douglas production function is of the form
2
1
0 b
b
K
L
b
x 
From this production function
1. MPL =
2
1
1
0
1
2
2 b
b
K
L
b
b
L
X 

= b0 b1
L
X
bo
K
L
L b
b

2
1
= b0.APL
MPK =
1
2
1
0
2
2
2 
 b
b
K
L
b
b
L
K
= bo 
L
X
b0.APK
2. Marginal rate of technical substitution
B
Labor

(MRT SLK) =
L
K
b
b
K
X
b
L
X
b
MPK
MPL
.
2
1
2
1


3. The elasticity of substitution
MPl
MPk
MPk
MPl
d
k
l
l
k
d 







= 1
2
/
1
)
2
1
(
/
)
(

l
b
K
b
l
b
k
b
d
l
k
l
k
d
4. Factor intensity is measured by the ratio b1/b2. The higher the ratio, the more labor intensive the
technique. Similarly, the lower the ratio
2
1
b
b
the more capital intensive the technique
5. The efficiency of production. This is measured by the coefficient b0. Obviously it is clear that if two
firms have the same K, L, b1 and b2 and still produce different quantities of output, the difference could
be due to the superior organization and entrepreneurship of one of the firms, which results in different
efficiencies. The more efficient firm will have a larger b0 than the less efficient one.
3.8 The efficient region of production: long run
In principle the marginal product of a factor may assume any value, positive, zero or negative. However,
the basic production theory concentrates only on the efficient part of the production function, i.e. over the
range of out put over which the marginal product of factors are positive and declining. In the short run
production function efficient region of production prevails in stage two (stage II), where MPL >O, but
L
MPL


< 0.
Similarly, efficient region of production in the long run prevails when the marginal product of all variable
inputs is positive but decreasing. Graphically this can be represented by the negatively slopped part of an
isoquant. The locus of points of isoquants where the marginal products of factors are zero form the ridge
lines. The upper ridge line implies that the MP of capital is zero. MPk is negative for all points above the
upper ridge line and positive for points below the ridge line. The lower ridge line implies that the MPL is
zero. For all points below the lower ridge line the MPL is negative and positive for points above the line.

q3
q2
q1
Lower ridge line
Upper ridge line
Labor
Capital
Production techniques are technically efficient inside the ridge lines symbolically; in the long run efficient
production region can be illustrated as:
MPL >0, but
L
MPL


<0
MPk >0, but
K
MPk


<0
Graphically, efficient region of production is shown as follow:
Fig 3.10: Thus efficient region of production is defined by the range of isoquants over which they are
convex to the origin.
EQUILIBRIUM OF THE FIRM: CHOICE OF OPTIMAL COMBINATION OF FACTORS OF
PRODUCTION
A firm is said to be in equilibrium when it employs those levels of inputs that will maximize its profit.
This means the goal of the firm is profit maximization (maximizing the difference between revenue and
cost). Thus the problem facing the firm is that of constrained profit maximization, which may take one of
the following forms:
a) Maximizing profit subject to a cost constraint. In this case total cost and prices are given and the
problem may be stated as follows

Max П = R – C
П = PxX – C
Clearly maximization of П is achieved in this case if X is maximized, since C and Px are constants.
b) Maximize profit for a given level of output.
Max П = R- C
П = PxX –C
Clearly in this case maximization of profit is achieved by minimizing cost, since X and Px are given.
To derive graphically the equilibrium point of the firm, we will use the isoquant map and the isocost line.
An isoquant is a curve that shows the various combinations of K and L that will give the same level of
output. It is convex to the origin whose slope is defined as:
- ∂K/∂L = MRSL,K = MPL/MPK = ∂X/∂L
∂X/∂K
The isocost line is defined by the cost equation
C = rK + wL
Where w=wage rate, and r=price of capital services.
The isocost line is the locus of all combinations of factors that the firm can purchase with a given
monetary cost outlay. The slope of the isocost line is equal to the ratio of the prices of the factors of
production, w/r.
K the isocost equation is given by C=wL + rK
C/r => rK = C - wL
=> K = C/r – w/r L
From this the slope is –w/r or it is the vertical change
over the horizontal change.
=> Slope = C/r
C/w
=> Slope = C/r.w/C
=> Slope = w/r.
O C/w L
Case 1: Maximization of output subject to a cost constraint.
Given the level of cost and the price of the factors and output, the firm will be in equilibrium when it
maximizes its output. This is at the point of tangency of the isocost line to the highest possible isoquant
curve. In the following graph, it is at point e where the firm produces X2 with K1 and L1 units of the two
inputs. Higher levels of output to the right of e are desirable but not attainable due to the cost constraint.
Other points below the isocost line lie on a lower isoquant than X2. Hence X2 is the maximum output that
can be achieved given the above assumptions (C, w, r, & Px being constant).
K
A

K1 e X3
X2
X1
0 L1
At the point of tangency:
a. slope of isoquant = slope of isocost
w/r = MPL/MPK = MRSL,K. this is a necessary condition.
b. the isoquant is convex to the origin. This is the sufficient condition.
NOTE: If the isoquant is concave to the origin, the point of tangency does not define the equilibrium
position.
K
e1
e
X2
O e2 L
Output X2 depicted by the concave isoquant can be produced with lower cost at e2 which lies on a lower
isocost curve than e (corner solution).
Mathematical derivation of the equilibrium of the firm.
A rational producer seeks the maximization of its output, given total cost outlay and the prices of factors.
That means:
Maximize X = f (K, L)
Subject to C = wL + rK
This is a constrained optimization which can be solved by using the lagrangean method. The steps are:
a. rewrite the constraint in the form
wL + rK – C = 0
b. multiply the constraint by a constant which is the lagrangian multiplier
(wL + rK – C) = 0

c. form the composite function
Z = X - (wL + rK – C)
d. partially derivate the function and then equate to zero
∂Z = ∂X - w = 0
∂L ∂L
 MPL = w
  = MPL-----------------------------------------------------------------------(1)
w
∂Z = ∂X - r = 0
∂K ∂K
 MPK = r
  = MPK----------------------------------------------(2)
r
∂Z = rL + rK – C = 0------------------------------------------- (3)
∂
From equation (1) and (2) we understand that
MPL = MPK
w r
=> MPL = w
MPK r
This shows that the firm is in equilibrium when it equates the ratio of the marginal productivities of
factors to the ratio of their prices. It can be shown that the second order conditions for the equilibrium of
the firm require that the marginal product curves of the two factors have a negative slope.
Slope of MPL = ∂2
X
∂L2
Slope of MPK = ∂2
X
∂K2
=> ∂2
X < 0 and ∂2
X < 0
∂L2
∂K2
2
And ∂2
X . ∂2
X > ∂2
X
∂L2
∂K2
∂L∂K
Case 2: minimization of cost for a given level of output
The condition for the equilibrium of the firm is formally the same as in case 1. That is, there must be
tangency of the given isoquant and the lowest possible isocost line, and the isoquant must be convex.
However, in this case we have a single isoquant which denotes the desired level of output, but we have a
set of isocost lines. Curves closer to the origin show a lower total cost outlay. Since isocosts are drawn on
the assumption of constant prices of factors, they are parallel to each other and their slopes (w/r) are
equal. Thus the firm minimizes its cost by employing the combination of K and L determined by the point
of tangency of X isoquant with the lowest possible isocost line. Points below e are desirable because they

show lower cost but are unattainable for output X. points above e show higher costs. Hence point e is the
least cost point.
K
e
K1
0 L1 L
In this case also the lagrangian method can be followed to derive the equilibrium point mathematically.
But the problem is different. That is,
Minimize C = wL + rK
Subject to X = f(K,L)
The lagrangian function will be:
Z = (wL + rK) + [X-f(K,L)]
Partially derivate Z w.r.t L, K, &  and equate to zero.
∂Z = w -  ∂f(K,L) = 0
∂L ∂L
=> w -  ∂X = 0
∂L
=> w =  MPL
=>  = MPL ----------------------------------------- (1)
w
∂Z = r -  ∂f(K,L) = 0
∂K ∂K
=> r -  ∂X = 0
∂K
=> r =  MPK
=>  = MPK --------------------------------------- (2)
r
∂Z = X – f(K,L) = 0 ------------------------------------(3)
∂
From equation (1) and (2):
MPL = MPK
w r
=> w = MPL = MRSL,K
r MPK
This is the same as the condition in case one. In a similar way, the second condition will be:
=> ∂2
X < 0 and ∂2
X < 0 and
∂L2
∂K2
2
∂2
X . ∂2
X > ∂2
X

∂L2
∂K2
∂L∂K
Laws of Returns to scale: Long-run Analysis of Production
In the long-run, the expansion of output may be achieved by varying all factors. In the long-run,
all factors of production and variable. Thus, in the long-run the firm expands production (expands the
scale of its operations) by using more of all inputs-more labour, more equipment, more space, etc.
The long-run output may be increased by changing all factors by the same proportion or by
different proportions. Traditional theory of production concentrates on the first case-that is, the study of
output as all inputs change by the same proportion. When firm increases output by using more of all
inputs, the input-output relation is one of scale. Returns to scale refer to a property of the production
function that indicates the relationship between the proportionate change in inputs and the resulting
change in output. Returns to scale refer to the long-run analysis of production and show how output
responds to an equi-proportionale change in all inputs.
It is crucial to note that the law of diminishing marginal returns and the law of returns to scale are
two different things. The law of diminishing marginal returns studies the behaviour of production when
only some inputs change while other inputs are fixed in the short-run. However, the law of returns to
scale examines the effect of changes of all inputs on the level of output. Thus, while the law of
diminishing marginal returns is the sort-run phenomenon, the law of returns to scale is the long-run
scenario.
The law of returns to scale, which refers to the changes in output as all factors change by the
same proportion, can be studied under three major headings:
a) Increasing Returns to Scale
A production function is said to exhibit increasing returns to scale if a certain percentage
change in combination of inputs is accompanied by a higher percentage change in the resulting
output. For instance, if when labour and capital double, the resulting output more than doubles,
the production function displays increasing returns to scale. Increasing returns to scale occur if a
10% increase in labour and capital triggers more than 10% increase in output.
Causes of Increasing Returns to Scale
One possible cause of increasing returns to scale comes from higher degrees of specialization, as
Adam smith pointed out. With more labour, the firm can subdivide tasks and thus gain efficiency of
labour. Thus, because of division of labour and specialization each worker specializes in performing a
particular task rather than engaging in different assignments and, consequently labour productivity
(average product of labour) boosts.

Another possible cause of increasing returns to scale may be technical and/or managerial
indivisibilities. Usually, most processes can be duplicated, but it may not be possible to halve them.
One of the basic characteristics of advanced industrial; technology is the existence of mass-
production methods over large sections of manufacturing industry. Mass production methods (like the
assembly line in the motor car industry) are processes available only when the level of output is large.
For example, assume that we have three production Processes:
L(men) K(machines) X (in tons)
A: Small-scale process 1 1 1
B: Medium-scale process 50 50 100
C: Large-scale process 100 100 100
The K/L ratio is the same for all processes and each process can be duplicated (but not halved). Each
process has a different level of output. The large-scale processes are technically more productive than the
small-scale processes. Clearly, if the large-scale processes were equally productive as the small-scale
methods, no firm would use them: the firm would prefer to duplicate the smaller scale already in we, with
which it is already familiar.
For X < 50 the small scale process would be used but for 50 < X < 100 the medium scale process
would be used. The switch from the small-scale to the medium scale process gives a discontinuous
increase in output (from 49 tons produced with 49 units of L and 49 units of K, to 100 tons produced with
50men and 50 machines). If the demand in the market required only 80 tons, the firm would still use the
medium-scale process, producing 100 units of X, selling 80 units, and throwing away 20 units (assuming
zero disposal costs). This is one of the cases in which a process might be used inefficiently, but still
relatively efficient compared to the small scale process. Similarly, the switch from the medium-scale to
the large-scale process gives a discontinuous increase in output from 99 tons (produced with 99 men and
99 machines) to 400 tons (produced with 100 men and 100 machines). If the demand absorbs only 350
tons, the firm would use the large-scale process inefficiently (producing 400 units, but selling 350 units
and throwing away the 50 units). The large-scale process, even though inefficiently used, is still more
[productive (relatively efficient) compared with the medium-scale process.
b) Decreasing Returns to scale
A production function (technology) is said to display decreasing returns to scale when a certain
percentage change in the combination or inputs brings about a lower percentage change in output. For
example if when labour and capital scale up by 15% and output less than doubles, decreasing returns to

scale occurs. If a firm gets less than twice as much output from having twice as much of each input, it is
doing something wrong.
Possible causes of Decreasing Returns to Scale
The most common causes of decreasing returns to scale are the mounting difficulties of coordination and
control as scale of operation increases. A plant with a very large number of workers more difficult to
manage than a smaller one. The larger the number of workers, the larger becomes the number of foremen
and supervisors between the responsible plant manager and the people who actually do the job. The
‘management’ is responsible for the coordination of the activities of the various sections of the firm. Even
authority is delegated to individual managers (production manager scales manager, etc), the final decisions
have to be taken from the “top management” (Board of Directors). As the output (firm) grows top
management becomes eventually overburdened and hence less efficient in its rule as coordinator and
ultimate decision maker. It is a commonly observed fact that as firms grow beyond the appropriate optimal
plateau, management diseconomies cheep in-the firm becomes unmanageable as the span of control is
widened.
Another possible cause of decreasing return to scale may be sound in the exhaustible natural
resources: doubling the fishing fleet may not lead to a doubling of fish catch.
C. Constant Returns to Scale
A production function dip day’s constant returns to s scale if a certain percentage change in the
combinative of inputs is associated with the some proportionate change in the resulting output. Here,
doubling or quadrupling of inputs causes doubling or quadrupling of output.

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