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abel197
Part 1: Theory of Production and Costs
Concept of Production – Production Function –– short run versus long run production function- Law of Variable
Proportions – TP, AP, MP and their interrelationships – Isoquants- Properties- MRTS - Isocost Curve –Producer
Equilibrium- Law of Returns to Scale – Expansion Path – Internal and External Economies- Linearly
Homogeneous Production Function - Cobb-Douglas production function
Part 2: Theory of Costs
Cost function – Cost concepts- Explicit and implicit costs, opportunity cost, private cost, social cost, economic
cost, accounting cost, sunk cost, fixed and variable cost, marginal and average cost -Short run and Long run cost
curves - Modern theory of costs
...

It shows the maximum quantity of output, that can be produced as a function of inputs used in
the production process In functional form we write
Q = f(X1, X2,X3……
...
Xn are inputs
If we consider Labour (L) and (K) are the two factors, then the production function takes the
following form, Q = f (L,K)
Short run and Long run Production Function
The short run is a time frame in which the quantity of one or more resources used in
production is fixed
...
Other resources used
by the firm (such as labor, raw materials, and energy) can be changed in the short run
...
A typical short run production function with one variable factor
is given as
Qmax = f(L, Kconst)

Qmax = f(K, Lconst)

When all inputs are freely variable, we call it long run production function
...
Here both factors are variable
...
In functional form we have
1

Qmax = f(L, Kconst)
A TP curve shows the technological relationship between a variable input and TP(assuming
other inputs remaining the same)
...
From J to L the concave downward indicating that TP
increases at a decreasing rate
...
However, when the technology
improves the TP shift upward to TP1
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AP = TP / No of variable factors
Using the above functional form, APL= Q/L and APK = Q/K
Marginal Product
Marginal Product is defined as the Increment in total output due to the use of an extra
unit of variable factor
...
They are (a) the AP increases if the MP is
greater than the AP , (b) the AP remains constant if the MP is equal to the AP , (a) the AP
decreases if the MP is less than the AP
...
In the graph it is clear that as AP
increases MP>AP
...
When AP got a maxima MP = AP
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The law of variable proportions examines input-output relationship when output is increased
by varying the quantity of one factor (variable input) while other factors are kept constant
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Why does diminishing returns occur? – An example
Think about the effectiveness of extra workers in a small café
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However, there are
only so many chopping boards and space to make sandwiches
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Assumptions of law
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(i) Technique of
production does not change
...
(ii)
Different quantities of one factor can be varied / combined with fixed factors
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(iii) All units of a variable factor are equally efficient
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(v) There is short period of operation
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Phase I
...
In other words, MP rises
...
When more and more units of variable factor are applied to fixed factors, the
underutilized fixed factors get fully utilized raising MP of variable factor
...

Why Increasing Returns in this stage: Because, the quantity of the fixed factor is abundant
relative to the quantity of the variable factor
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4

Phase II
...
i
...
, MP falls but remains positive (+)
...
During this phase
which is called the phase of diminishing returns, both MP and AP fall
...

Phase III
...
MP becomes negative
...
TP starts falling and as a
result slope of TP curve become negative
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Why Negative returns: As the amount of a variable factor continues to be increased to a fixed
quantity of the other factor, a stage is reached when the total product declines and the marginal
product of the variable factor becomes negative
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Stage third is economically meaningless region
...
(
That is too much of labour with too less of capital)
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Thus, stage one is economically meaningless as well
...
Therefore the
producer operate only in the second stage, where, MP of both the factors are positive
...
In the absence of diminishing returns, you could grow the world's food
supply in a flower pot
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PRODUCTION WITH TWO VARIABLE INPUT
ISOQUANT (Equal Product Curve Or Iso Product Curve or production indifference curve)
An isoquant shows the different combinations of two factors (say Labour and Capital) which
produce same level of output
...
Combination A with one dose of labour
and 20 doses of capital produces 100 units
...

Isoquant schedule
combination labaour Capital Output
A
1
20
100
B
2
15
100
C
3
11
100
D
4
8
100
E
5
6
100
Plot graph using the table to the right of this

isoquant - Curve showing all
possible combinations of inputs
that yield the same output
...
Along the isoquant the maximum possible output is constant
...
In an isoquant map output increases as we move to higher from isoquants
...
But higher q2 produces 200 and q3 300 units
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Marginal Rate of Technical Substitution (MRTS) or susbstitution among inputs
marginal rate of technical substitution (MRTS)- 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
...
In the case of two factors L and K, MRTSLK is defined as the
amaout of capital that can be replaced by an extra unit of labour without affecting total output
MRTSLK = ΔK/ΔL = dK/dL
...

Since ΔK/ΔL is the slope of the isoquant, the MRTS is the measure of the slope of isoquant
...
Therefore,
MRTSLK= 5
...
Thus MRTS
decreases as the firm moves down an isoquant
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Also there, exist a relationship between MRTS and the marginal product of two factors
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That is
dq = 0
...

( students are advised to draw graph from the above table to examine MRTS)

CHARACTERISTICS OF ISOQUANTS ( Properties): The isoquants have the same general
characteristics of ICs
...

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Isoquants are negatively slopped: This is because increase in the quantity of one factor is
followed by decreased in the quantity of other factor
...
This is given by the Marginal Rate of Technical Substitution (MRTS)
...
That is as we move
down along an isoquant, the absolute value of its slope or MRTS declines and the isoquant is
convex
...
That is
the quantity of labour increases and quantity of capital reduces
...
Hence we need only less and less capital for
compensating the loss of labour
...
( draw graph)
( that is d2K/dL2> 0 Second derivative is more than zero)
Two isoquants will never interest
Higher isoquants shows higher levels of output
In between two isoquants there are large number of isoquants
Isoquant specify cardinal measure of output
Isoquants never touch either X or Y Axis

TYPES OF ISOQUANTS

Linear isoquant - This type assumes perfect substitutability of factors of production: a given
commodity may be produced by using only capital, or only labour, or by an infinite
combination of K and L
...
The isoquant take the shape of a right angle
...

Kinked Isoquant – This assumes limited substitutability of K and L
...
Substitutability of factors is possible only at the
kinks
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Smooth Convex Isoquant - This form assumes continuous substitutability of K and L only
over a certain range, beyond which factors cannot substitute each other
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9

RETURNS TO SCALE - LONG RUN PRODUCTION
[ change in output as all the factors change by same proportion]
The laws of returns to scale explain the behavior of output in response to a proportional and
simultaneous change in inputs
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When a firm increases both the inputs proportionately, there are three possibilities Total output
may increase more than proportionately Total output may increase proportionately Total output may
increase less than proportionately Accordingly, there are three kinds of return to scale - Increasing
returns to scale (IRS) Constant returns to Scale (CRS) and Decreasing returns to scale (DRS)
Let a production function Q = f(L,K)
If we increase both the factors by λ (Lambda) times , we have Q = f(Lλ,Kλ)
...
Assume the scalar λ got a power n
...
When the quantity of all inputs is increased by 10%, and output also
increases exactly by 10%, then we say that constant returns to scale are operating
...
For example, when the quantity of all inputs are increased by 10%,
and output increases by 5%, then we say that diminishing returns to scale is operating
...
For example, when the quantity of all inputs are increased by 10%, and
output increases by 15%,then we say that increasing returns to scale is operating
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increasing returns to scale Situation in which output more than doubles when
all inputs are doubled
...

decreasing returns to scale Situation in which output less than doubles when all
inputs are doubled
...
(the gap between isoquants remains the
same)
When there are decreasing returns to scale as shown in (b), the isoquants move farther as
inputs are increased along the line
...
(the gap between isoquants reduces)
ISOCOST LINE
An isocost line includes all possible combinations of labor and capital that can be purchased for a
given total cost
...

The slope of isocost is given by the ratio of prices of two factors, that is, w/r
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PRODUCERS EQUILIBRIUM ( Optimal Input combination of factors of production ) or
Least Cost Condition
A firm is in equilibrium when it produces a given level of output at the lowest possible cost
...
It occurs when the slop of isocost is tangent to the highest isoquant
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The condition for equilibrium is therefore occur
when MRTS is equal to the prices of two factors of production
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(1) Maximisation of profit subject to a cost constraint
...
Here we opt the lowest isocost from several
isocost for a given level of output
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Isoquant: An isoquant shows a given amount of output produced by various combinations of two
inputs
...
Where Q* is fixed
...
See a typical isoquant below
...

In equation form the total cost is
C* = wL + rK,
where C = Total cost or budget level, w= the wage rate,
L = the amount of labor taken,
r = the rental price of capital, and
K = the amount of capital taken
...
Maximisation of output subject to Cost Constraint
Given the production function Q* = f(L,K), given by the isoquant and Isocost C* = wL + rK
firm maximize the profit by producing a given level of output at the lowest possible cost
...
The firm is in equilibrium
when it maximiszes its output given its total cost outlay and the prices of the factors
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Producer Equilibrium is attained at the point where the isocost line is tangent to the isoquant
curve
...
It doesn’t intersect
the isocost line
...
The optimal
output is IQ3
...
Minimisation of cost for a given level of output
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In this case we have one isoquant and several
isocost
...
The least cost of production Qo output is Co
...


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EXPANSION PATH
An expansion path shows the locus of the least cost input combinations for producing various
levels of output assuming that input prices remain constant
...
Along expansion path, input-price
ratio is constant & equal to the marginal rate of technical substitution
...
( see graph)
expansion path Curve passing
through points of tangency
between a firm’s isocost lines and
its isoquants
...
Thus, an expansion path is an isoclines
...
In general an expansion path may take many
different shapes, concave upward, downward, linear or some combination of different shapes
...
( Draw all shapes and practice)
The Expansion Path and Long-Run Costs or Cost Function From Production Function
Cost function is derived functions
...
Thus a mathematical expression showing the relationship
between volume of output and its total cost of production is called cost function
The total cost curve is determined by the locus of points of tangency of successive isocost line
with the higher isoquants
...

Cost function can be derived from the expansion path as given below
...
It is formed from the point of tangency of the isocosts (wL
+rK = C) and the isoquants Q = f(L,K) at different levels of inputs
...
Plotting these points of expansion
path at different levels of output on a two dimensional diagram we obtain TC curve for the
long run
...
Now using this
information we can derive long run total
cost (LTC)
...
With
this information, directly we can derive
LTC
...
In the graph the points X,Y and
Z on the LTC curve correspond directly to
points X, Y and Z in the expansion path
graph
...
Then find the point of tangency
of that isoquant with an isocost line
...

Graph the output-cost combination
...
If the production function is given as Q = f(L,K) and if we increase both the
factors by λ times , we have Q = f(L λ,K λ)
...

Assume the scalar λ got a power n
...
Such
function shows CRS
...

COBB DOUGLAS PRODUCTION FUNCTION – Developed by two economists CW
Cobb and DH Douglas

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The Cobb–Douglas production function is a particular functional form of the production
function, widely used to represent the technological relationship between the amounts of two
or more inputs, particularly physical capital and labor, and the amount of output that can be
produced
...
APL or α Q/L
Marginal Product of Capital MPK= 1-α
...
Mathematically, C = f(q) where q is the optimal output level
...
Symbolically a cost function is given as
C = f(Q,T,Pt)
C= cost, Q = output, T = technology, Pt = price of factors of production
Cost function is a derived function as it depends on the level of output
...
These includes the wages to hire labor, the rental
price of capital, equipment, and buildings, and the purchase price of raw materials and semi
finished products
...

Implicit costs refers to the value of the inputs that are owned and used by the firm in its own
production activity
...
Implicit costs are also called as "Imputed costs"
...
Economic costs relate to future
...
Since the only costs that
matter for business decisions are the future costs, it is the economic costs that are used for
decision-making

Accounting Costs: Accounting costs are the actual or outlay costs
...

Since these costs relate to the past, these are generally sunk costs
...

Accounting costs refer only to the firm’s actual expenditures, or explicit cost, incurred for
purchased or hired inputs
...
g
...
Sunk costs are a part of the outlay costs
...
Sunk costs are also called as "Non-Avoidable costs" or "Inescapable costs"
...
The best example is amortization
of past expenses, like depreciation
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Opportunity cost or Alternative cost: Opportunity cost of anything is the next best
alternative foregone
...
Opportunity cost is concerned with
the cost of forgone opportunities
...
The concept of opportunity cost focuses attention
on the net revenue that could be generated in the next best use of a scarce input
...
For Example, if the firm owns land there is no cost of
using the land (i
...
, the rent) in the firm's account
...

Example: Suppose that a businessman can buy either a lathe machine or a paper pressing
machine with his limited resources and he can earn annually Rs
...
70,000respectively from the two alternatives
...
But in the process of
earningRs
...
50,000 annually from the lathe
machine
...
50,000 is his opportunity cost or Alternative cost
...
Social costs, on the other hand, are the
total costs to the society on account of production of a good
...
Thus, the economic costs include both the private and
social costs
...
The LRC functions describe costs when all inputs can be varied freely
...
Short run Cost
Total Costs – TC of producing any output is defined as the minimum cost that must be
incurred to produce that output
...
C = f(q) +b
Fixed costs are those costs that do not vary with the quantity of output produced
...
FC includes Salaries of administrative staff,
Depreciation, expenses of building etc
...
It is
dependent on the level of output
...

TC = TFC + TVC
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TFC

TVC

TFC,TVC and TC

Total Fixed Costs (TFC), Total Variable Costs (TVC),Total Costs (TC)
Average fixed cost (AFC) is total fixed cost per unit of output
...
That is TVC/Q or f(q)/q
Average Costs (AC): Average costs can be determined by dividing the firm’s costs by the
quantity of output it produces
...

Average total cost (ATC) is total cost per unit of output
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It is the slope of the TC curve
...


MC =

(change in total cost) TC
=
(change in quantity)
Q

MC = ∂TC/∂q = ∂(TVC+TFC)/∂q = ∂TVC/∂q
AFC

AC and AVC

Rectangular Hyperbola

U shape

MC

U shape

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Use the table given below and draw various cost curves and practice
Q
0
1
2
3
4
5

TFC
$60
60
60
60
60
60

TVC
$0
20
30
45
80
135

TC
$60
80
90
105
140
195

AFC
$60
30
20
15
12

AVC
$20
15
15
20
27

ATC
$80
45
35
35
39

MC
$20
10
15
35
55

Relationship between TC, TFC and TVC

Total fixed cost is the same at each output level
...
Total cost, which is the sum of TFC and TVC also increases as output increases
...
Also remember that
the gap between TC and TVC is TFC
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The AVC curve is Ushaped
...
The
ATC curve is also U-shaped
...
Where AVC is falling, MC is
below AVC
...
At the minimum AVC, MC equals AVC
...
Where ATC is rising, MC is above ATC
...


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Relationship between AC and MC
Define both terms and give separate graph
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The marginal cost and average cost curves are related
...
When MC exceeds AC, average cost must be rising
...
This relationship explains why marginal cost curves always
intersect average cost curves at the minimum of the average cost curve
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On the other hand, if the marginal cost (MC) is below the average cost
(AC); average cost falls
...
At the point of intersection L where MC is equal to AC, AC is
neither falling nor rising
...
Thus, marginal cost curve
cuts the average cost curve at the latter’s minimum point
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LONG RUN COST – ENVELOP CURVE OR PLANNING CURVE
In the long-run there are no fixed inputs, and therefore no fixed costs
...

In the Long-run firm has therefore no fixed factors and therefore, no fixed costs
...
in order to expand or contract output
...
Longrun average cost curve depicts the least possible average cost for producing all possible levels
of output
...

The long-run average cost of production is the least possible average cost
of production of producing any given level of output when all inputs are
variable
...


Long run Average cost Curve (LAC) – (How to derive) The LAC curve is the locus of
points denoting the least cost of producing the corresponding output
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It is called envelop curve because it envelops the
short run cost curves (SRC)
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Reason for U shape is economies and
diseconomies of scale
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It is seen in the graph that that up to OB amount of output, the firm will operate on the shortrun average cost curve SAC1 though it could also produce with short-run average cost curve
SAC2
...
AL is smaller than AH
...

It is thus clear that in the long run the firm will produce any output up to OB on SAC1
...
Therefore, the output larger than OB but less then OD, can
be produced at a lower cost per unit on SAC2 than on SAC1
...
Therefore, if the firm plans to produce between output OB and OD, it will employ
the plant corresponding to short-run average cost curve SAC2
...
Therefore, for output larger than OD, the firm will employ plant
corresponding to the short-run average cost curve SAC3
...
The
long-run average cost curve depicts the least possible average cost for producing various
levels of output when all factors including the size of the plant have been adjusted
...
In that case, the long-run average cost curve will be a smooth and
continuous curve without any zigzag
...
There will be infinite short-run
average cost curves in such a case, though only 11 have been shown in the figure
...

The LAC curve first falls and then beyond a certain point it rises, that is, the long-run average
cost curve is U-shaped ( but flatter than short run AC)
...

The LAC curve is not tangent to the minimum points of the short-run average cost curves at
all levels
...

On the other hand, when the LAC curve is rising, it will be tangent to the rising portions of
the SAC curves
...

Derive LAC – See lecture note – three graphs may be given

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Derivation of LMC Curve

Long run marginal cost (LMC) shows the increase in LTC per unit change in output
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If the output OA is produced in the long run, then it must be produced on the LAC at point H
which is a tangency point with the short-run average cost curve SAC1
...
Corresponding to the tangency point H
there is a point N on the short run marginal cost curve SMC
...
Therefore point N must lie on the
long-run marginal cost curve corresponding to output OA
...

Q is also the point on the short-run marginal cost curve SMC2, corresponding to output OB
...
Thus Q must also lie on the long-run marginal cost
curve corresponding to output OB
...

By connecting points N, Q and K we obtain the long-run marginal cost curve LMC
...

It should also be remembered that the relationship between the long-run marginal cost curve
LMC and the long-run average cost curve LAC is the same as that between the short-run
marginal cost curve and the short -run average cost curve
...

When the long- run marginal cost is equal to the long- run average cost, the latter will be neither rising nor falling
...
Like the traditional theory, modern microeconomics
distinguishes between short-run and long-run costs
...

The average fixed cost:
The fixed costs include the costs for
1
...
The wear and tear of machinery (standard depreciation allowances)
3
...
The expenses for the maintenance of land on which the plant is installed and operates
...
Raw materials
2
...

The SAVC in modern theory has a saucer-type shape, that is, it is broadly U-shaped but has a flat stretch
over a range of output (figure 2)
Figure 2

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The flat stretch corresponds to the built-in-the-plant reserve capacity
...
To the left of the flat stretch, MC lies below SAVC, while
to the right of the flat stretch the MC rises above the SAVC
...
With better skills the wastes in raw materials are also being
reduced and a better utilization of the whole plant is reached
...
The traditional theory
assumes that each plant is designed (without any flexibility) to produce optimally only a single level of
output
...
In figure 2, the range of output X1 X2
reflects the planned reserve capacity which does not lead to increases in costs and the entrepreneur expects
to operate his plant within the X1 X2 range
...
The ATC is shown in figure 3

Figure 3

The ATC curve falls continuously up to the level of output (X2) at which the reserve capacity is
exhausted
...
The MC will intersect the average total-cost curve at its
minimum point (which occurs to the right of the level of output X2, at which the flat stretch of the AVC
ends)
...

Long-run costs are distinguished into production costs and managerial costs:
Production costs:
Production costs fall sharply to begin with and then gradually as the scale of production increases
...

Initially these economies are substantial but after a certain level of output is reached all or most of these
economies are attained and the firm is said to have reached the minimum optimal scale, given the
technology of the industry
...
The cost of different techniques of
management first fall up to a certain plant size
...

In summary:
Production costs fall smoothly at very large scales, while managerial costs rise only slowly at very large
scales
...

Assuming that a plant is used ‘normally’ when it operates at two -third of its full capacity, we may
draw the LAC curve by joining the points on the SATC curves corresponding to the two-third of the
capacity of each plant size
...


30

If the LAC falls continuously (though smoothly at very large scales of output), the LMC will lie below the
LAC at all scales as shown in figure 5
...
Economies of scale lead to reductions in unit costs as the scale of operation increase
...
This include
Technical: Possible through (1) Specialisation: Large organisations can employ specialised
labour (2) Indivisibility of plant – machines can’t be broken down to do smaller jobs!
Commercial: Large firms can negotiate favourable prices as a result
of buying in bulk
...

Financial: Large firms able to negotiate cheaper finance deals
...
Large firms able to utilise skills of
merchant banks to arrange finance
Managerial: Use of specialists – accountants, marketing, lawyers, production, human
resources, etc

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