managerial concept

1. Topic 2 - Marginal concepts
(1) We have already noted that
- technology defines the possibilities of value addition
- organizational details and management procedures enable us to achieve the possible
value addition.
(2) In general, four types of organizational principles are acknowledged
- the markets and prices
- contracts
- direct control and supervision by management - depends on private benefits and
private costs
- government regulation - based on social benefits and costs
Examples :
- purchase and sale of toothpaste. Identity of sellers and buyers is irrelevant.
- family doctor; Eco 111 teacher vs. Students in the class - identity matters. Trust
necessary in contractual relations.
- CEO (chief executive officer) must guide choice. Secrecy is important. Market
price is inadequate ; Arvind Mills - Benetton tie-up for jeans; Chauhan’s of Parle
products selloff to Pepsi.
- prevent undesirable entry which market cannot - KFC, McDonalds.
(3) Whatever may be the source of value generation and whichever may be the
organizational mechanism used in practice, value maximization is always subject to
choices, conflicts and so on.
Examples :
(a) Choosing where to put your saving
- safely in a bank - low interest rate but low risk
- put in the stock market - high return but high risk also
choice exists.
(b) Arvind Mills is currently the leader in both domestic sales and exports of denim (the
cloth to make jeans). They have entered the value added activity of making jeans and
selling them. (There is a joint venture with Benetton of Italy.) But this means less of
cloth is available for export.
(c) Consider two neighbors in the hostel. One likes to play his music loud while the other
is studying. Both derive value from what they are doing. Both have to accept some
loss
to gain something.
(d) Enron’s Dhabol power project - (1) use of liquified natural gas vs. Naptha (price and
pollution), (2) more of Enron power at 90% capacity (contracted payment) means less

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generation from SEBs, and (3) higher price for more power availability or accept less
supply at a lower price?
(e) P&G and Godrej joint venture : (1) Godrej got a better market exposure through
P&G’s Camy and Ariel but lost on Trilo and Cinthol, and (2) lost some of their
distribution channels through the sale of P&G products though they realized it
only when the breakup was almost inevitable.
Three types of choice problems can therefore be identified
- more than one way of doing something
- more than one use for the same resources
- more than one dimension along which value is defined
(4) What we realize is that there is a uniform structure of choice problems - we consider
that first.
(5) We have one basic concept around which the solution to most choice problems are
built up - The marginal concept. We will develop that also.
(6) To approach the choice problems meaningfully we must have concepts to
- describe the choices available
- define the criteria of choice
- state how these problems can be solved
(7) Notation to be used throughout the course :
- Outputs - Y1, Y2
- Inputs - X1, X2
(8) PP sets (production possibility sets) as a description of the choices available-
Suppose one resource X can be used to produce Y1 and/or Y2. Let R be the amount of X
available. We describe technology in the simplest possible terms first :
- 1 unit of Y1 requires ‘a1’ units of X
- 1 unit of Y2 requires ‘a2’ units of X
The marginal concept implied here is
- MPx = marginal product of X = increase in the output due to the last unit of X used
- it is equal to 1/a1 in the production of Y1 and 1/a2 in the production of Y2
We now have the resource constraint in the production of Y1 and Y2 given by
a1Y1 + a2Y2 ≤ R ; Y1, Y2 ≥ 0
The set of (Y1, Y2) combinations which satisfy these constraints represents the PP set.
Fig. 1 here
Technology is only one of the constraints on the choice. There can be a market constraint
as well. Suppose the maximum Y1 that can be sold is M < R/ a1. Then the PP set should be

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represented by the constraints
a1Y1 + a2 Y2 ≤ R
Y1 ≤ M
Y1, Y2 ≥ 0
Fig. 2 here
Similarly, there may be more than one resource needed in the production process.
Suppose there are two such resources : X1, X2. Let the technology be defined by
- 1 unit of Y1 requires a1 of X1 and b1 of X2
- 1 unit of Y2 requires a2 of X1 and b2 of X2
Let the resource availability be denoted by
- X1 available is R1 units, and X2 available is R2 units
The PP set is now the combinations (Y1, Y2) which satisfy the inequalities
a1Y1 + a2Y2 ≤ R1
b1Y1 + b2Y2 ≤ R2
Y1, Y2 ≥ 0
Consider a numerical example :
Y1 + Y2 ≤ 6
2Y1 + Y2 ≤ 10
Y1, Y2 ≥ 0
For this example the PP set is given in fig. 3.
Fig. 3 here
(9) We now attempt a more general description of technology. Suppose the production
process involves processing a raw material on one machine. Then there are five elementary
operations in the production process.
Input → M1 → goods in progress → M2 → final output
We generally expect efficiency in the operations if there is a division of labor and each of
the elementary operations is entrusted to a separate person. The following examples are
adequate
- when you are at your study the first half an hour will not be as productive as the next
because it takes some time to reach the best level of concentration
- if too many tubewells are dug in the same area and share the same aquifier the additional
output from the new tubewell decreases.
- given that there is only a finite market for a commodity increasing the number of sales
personnel will decrease the addition to sales even it increases it initially.
We represent by MPx the incremental output from the additional unit of X and call it the
marginal product of X. Normally the general description of the technology consists in
stating that the marginal product increases at first but will decrease after the resource use
exceeds a certain amount of X.
See Fig.4

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Fig. 4 here
Suppose, for the sake of convenience, that the input
required to produce Y1 units of output is represented by Y1β. Then, the PP set is
given by the constraints
Y1β + bY2 ≤ R ; Y1, Y2 ≥ 0
Fig. 5 here
(10) The objective of the firm can now be stated as maximizing the revenue from the sale
of Y1, and Y2 for a given amount of resource. In general, the problem can be stated as
Max p1Y1 + p2Y2
Subject to (Y1 ,Y2) ∈ PP set
Before we proceed to the solution procedure the following concepts are important :
- Every point on the boundary of the PP set represents the maximum (Y 1 , Y2) that can be
produced given the technology and the resource constraints. It is not possible to
increase the production of either of the commodities without reducing that of the other
if the current combinations are on the boundary. The engineer calls every combination
on the boundary technically efficient.
- Only one of these combinations will be economically efficient in the sense that it
maximizes the objective function specified above.
(11) Consider the problem
Maximize p1Y1 + p2Y2
Subject to
a1Y1 + a2Y2 ≤ R
Y1, Y2 ≥ 0
The PP set is depicted in Fig. 6.
Fig. 6 here
One solution procedure now proceeds along these lines :
- The point P in the interior of the PP set cannot be optimal. For, any point along PQ, PR,
or in the triangle PQR would be superior. Hence, the optimal Y1, Y2 must satisfy the
equation
a1Y1 + a2Y2 = R
- Consider any point on this line It would have the following coordinates
(Y1, Y2) = [ α (0, R/a2) + (1 - α) (R/a1, 0)]
= [ (1 - α) (R/a1), α (R/a2)] ; 0 ≤ α ≤ 1
Hence, p1Y1 + p2Y2 = (p1R/a1) + αR [(p2/a2) - (p1/a1)]
That is, α = 1 is optimal if (p2/a2) > (p1/a1) and so on. This can be justified by using
certain standard marginal concepts.
Since Y1 = x1/a1 (where x1 is the amount of input used to produce Y1) MP1 = 1/a1,
and the revenue earned by selling this on the market would be VMP1 = p1/a1.
Similarly, VMP2 = p2/a2. Clearly, VMP2 > VMP1 would suggest that the last unit of
resource should be used to produce Y2 in preference to Y1.

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- In general, the optimal solution in such simple problems will be one of the corners of
the PP set. (A corner of a convex set cannot be on the line joining any two other
points of the set.)
- Under the assumption made above the marginal value of the resource is (p 2/a2).
Consider the following numerical example :
Max Y1 + 5Y2
Sub Y1 + 2Y2 ≤ 10
Y1, Y2 ≥ 0
If one unit of input goes to Y1 - MP1 = 1.0, VMP1 = 1, whereas if it is used to produce
Y2 we have MP2 = 0.5, VMP2 = 2.5. Hence, every unit of the resource should be used
to produce Y2. The optimal solution is Y1 = 0, Y2 = 5. Also the marginal value of the
resource is 2.5.
One qualification is in order before we proceed further. Upto this point we did not
consider the price of a unit of resource. If px > 2.5 it would not be feasible to produce Y2.
Only values less than 2.5 make the above solution feasible.
A second solution procedure is the Lagrange multiplier method. This is also based on
some very simple economic commonsense of marginal value comparisons. Define
H = p1Y1 + p2Y2 + λ (R - a1Y1 - a2Y2)
Here λ = value of the resource not used in the production of either of the commodities.
This value should be interpreted as the value in the production of these commodities. It
need not be equal to the market price. Then we say that the marginal value of the resource
is zero if it is not able to generate additional revenue by producing either Y1 or Y2.
Otherwise λ > 0. Rewrite the H function as
H = Y1 (p1 - λa1) + Y2 (p2 - λa2) + λR
If (p1 - λa1) > 0 and (p2 - λa2) < 0 then the optimal solution is Y1 = R/a1, and λ = p1/a1.
The other cases can be dealt with in a similar fashion.
(12) We now consider the case where there are diminishing returns to factor use. Consider
the problem
Max 2Y1 + Y2
Sub Y12 + Y2 ≤ 9
Y1, Y2 ≥ 0
Three solution procedures have been identified. Each of these generates the solution on
the basis of a new marginal concept.
(a) Observe that, as before, the optimal solution is always on the boundary of the PP set if
p1, p2 are constants (i.e., independent of the levels of output). Then, clearly, the optimal
solution is the point of tangency between the PP set and the line 2Y1 + Y2 = constant. See
fig. 7. Hence, it follows that
Fig. 7 here
dY1/dY2 = -2 = - 2Y1
Hence, the optimal solution is Y1 = 1, Y2 = 8
(b) The solution procedure can be related to marginal products. Note that

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X1 = Y12, so that dY1/dX1 = 1/2Y1, and VMP1 = 1/Y1. Similarly, MP2 = 1, and VMP2 = 1.
Hence, referring to Fig. 8, the optimal choice of Y1 satisfies the equation
Fig. 8 here
1/Y1 = 1, or Y1 = 1. Hence, the optimal Y2 is 8.
(c) The Lagrange multiplier method uses a similar marginal value concept. For, as usual
we construct the Hamiltonian
H = 2Y1 + Y2 + λ ( 9 - Y12 - Y2)
where λ is the marginal value of the resource to the firm in the sense that it represents the
revenue generating potential of the last unit of resource in the process of converting it to
one of the outputs. Maximizing H to choose Y1, Y2 we have
2 - 2λ Y1 = 0, and 1 - λ = 0.
Hence, λ = 1, and Y1 = 1. It follows that Y2 = 8.
(13) For the sake of completeness we now note that several exceptions to the above
mentioned approaches are likely. Caution is needed in using the techniques and
interpreting the marginal concepts appropriately.
(a) Corner solutions may be optimal even under diminishing returns. Let the problem be
Max 25 Y1 + Y2
Sub Y12 + Y2 ≤ 9
Y1, Y2 ≥ 0
It can be readily verified that Y1 = 3, and Y2 = 0 is the optimal solution.
(b) Solutions in the interior of the PP set are possible if p1, and/or p2 turn out to be
functions of Y1, and Y2. For instance, suppose the problem is
Max (2 - Y1) Y1 + (4 - Y2)Y2
Sub Y1 + Y2 ≤ 10
Y1,, Y2 ≥ 0
It can be readily verified that production beyond Y1 = 1, and Y2 = 2 is not feasible. The
entire input available will not be used.
(c) Increasing returns to scale also create corner solutions. Consider the problem
Max Y1 + Y2
Sub Y1.5 + Y2 ≤ 10
Y1, Y2 ≥ o
Since X1 = Y1.5, MP1 = 2X1 = 2Y1.5, and it is an increasing function of Y1. Hence, it is
never optimal to produce Y2. Hence, the optimal solution is Y1 = 100, Y2 = 0.
(14) The basic conclusion from this topic is that there are several useful marginal concepts
in economics which will assist us in understanding the choice problems better. We will use
them generously throughout to provide some economic content to the algebraic solutions
which we will offer while dealing with many of the topics that follow.
I would also like to alert you to the idea that the use of algebra in a course like this is a
convenience both from your point of view as well as mine. It is not the ultimate goal as
such. So we do not want to be slavish.

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