chapter 2 in managerial economic
DESCRIPTION
Basic training-Economic OptimisationTRANSCRIPT
Chapter 2 Economic Optimisation
29
This chapter introduces a number of fundamental principles of economic
analysis.
In chapter one, we defined managerial economics and discussed the various
objectives that managers aim to achieve. The definition specifically states the
application of decision science tools in analysing and evaluating decision
alternatives.
As managers we encounter problems daily. Be it minor or major in nature, each
problem requires serious attention. Of utmost importance is selecting the optimal
course of action in light of available options and objectives. Effective managers
must be able to collect, organise and process these information.
Economists build models to better understand and portray the essential link in
terms of appropriate decision variables, costs and benefits. For very complex
scenario, models are used to breakdown and subdivide aspects of the problem
where necessary. It is easy to build models, but to build a good model requires in
depth knowledge of economic concepts and methodology, which means that
managers must have prior knowledge of basic economics and mathematics.
2 BASIC TRAINING - ECONOMIC OPTIMISATION
Chapter 2 Economic Optimisation
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Even though solutions are easily arrived at via various means, economists place
greater importance on number crunching and precision techniques which offer a
realistic means in dealing with the complexities of goal-oriented managerial
activities. Economists find calculus with specific reference to derivative or
marginal analysis a vital tool.
This chapter places great emphasis on helping students understand marginal
concepts and the rules of differentiation. To reinforce students’ understanding,
chapter texts explicitly illustrate applications of marginal concept (be it
unconstrained or constrained) in optimisation process.
Key terms for review:
Functions
Total, average and
marginal value
Lagrangian multiplier
Aggregate Approach
Marginal Approach
Differentiation
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CHAPTER OVERVIEW
Chapter 2 Economic Optimisation
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Learning Objectives
After reading this chapter, the students should be able to:
1. Write functions to represent your economic problems.
2. Build a model for the economic problem at hand.
3. Relate the relationships that exist between total, average, and
marginal concepts irrespective of whether it is revenue, cost or
product.
4. Perform optimisation analysis graphically via both aggregate and
marginal approach.
5. Conduct first- and second-order derivations.
6. Apply marginal analysis in decision making
7. Distinguish between maximum and minimum values in optimisation
problems.
8. Perform optimisation analysis for multivariate functions.
9. Solve constrained optimisation problems by the Lagrangian
technique.
10. Find an optimal solution to economic problems via
the mathematical format.
Chapter 2 Economic Optimisation
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INTRODUCTION Mathematics scares a lot of people. But we need it in many things we do. This section
intends to show you the practical applications of economic theory because in your
capacity as managers, you will resort to and rely on many concepts, graphs and simple
numerical examples to assist you in your decision making. Moreover, explanations of
economic term, concepts and methods of analysis rely primarily on verbal definitions,
numerical tables, and graphs. Appropriate discussions will centre on the same material
using both algebra and calculus. In addition, exercises and problems will be slotted in to
give students ample opportunity to reinforce their understanding.
Many students have already learnt the mathematics employed in this text, however,
some would have studied this material some time ago and may benefit from a review.
This chapter introduces a number of fundamental principles of economic analysis, basic
economic relations, the tools and techniques of optimisation. First, we will examine the
ways of presenting relationships. Subsequently, we will be examining the relationships
between total, average, and marginal concepts. Then, we will move on to examine
optimisation analysis. To find an optimal solution to complex problems and to facilitate
the above and forth coming discussions, calculus with specific emphasis on rules of
differentiation will be discussed. Finally, we will apply the rules of differentiation to
unconstrained and constrained optimisation problems.
2.1 FUNCTIONAL RELATIONSHIP & ECONOMIC MODELS
Mathematics is an important instructional vehicle in managerial economics. Economists
study variables such as price, output, revenue, cost, and profit. Economists try to
understand how and why the values of these variables change and what conditions will
lead to optimal values.
Chapter 2 Economic Optimisation
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NOTES
Optimal may refer to maximum value (as in the case of profit),or it may
refer to minimum value (as in the case of cost.)
FUNCTION
• Single variable
In mathematics, the relationship of one variable’s value to the value of other variables
is expressed in terms of function. For example;
y = f(x) (Eqn.2.1a)
y is said to be a function of x, where y is the dependent variable, f is the function and
x is the independent variable. The above function (Eqn.2.1a) has only one
independent variable (i.e. x).
• Multivariable
But in most cases, functional relationship involves more than one independent
variable (multivariate function) for example, equation 2.1.1 below:
y = f(x, z,...n) ( Eqn.2.1b)
NOTES
Multivariate function will be discussed further under the partial derivative
topic.
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FUNCTIONAL FORMS Economic data are best presented in the forms of:
1) algebraic equations, 2.1a and 2.1b ;
2) tables ( table 1 – next page ); or
3) graphs (diagram 1- next page)
• Algebraic equation
An equation is an expression of the functional relationships or connection among
economic variables. Five key functions are used in this chapter; demand, total
revenue, total cost, and profit. To illustrate the different ways of expressing a
function, we will try to use as many functions as possible. In economics, the
general functional relationship for total revenue is that it is dependent on the
number of units sold, i.e. TR = f(Q).
RECALL
Total revenue is defined as the unit price of a product (P) multiplied by the
units sold (Q),i.e. TR = PQ
This equation is read, “Total revenue is a function of output.” It merely states some
relation exist between output and total revenue but does not indicate the specific relation
between them. The value of the dependent variable (on the left hand side of the equal
sign) depends on the size of the independent variable (on the right hand side of the
equal sign). As such, a more precise expression would be:
TR = PQ
or TR = 7Q - 0.1Q2 …(Eqn.2.1c)
where P is the price at which each unit of Q is sold.
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Similarly, the algebraic equations of price, cost and profit are respectively
presented as such:
P = 7 - 0.1Q …(Eqn.2.1d)
TC = 10 + 8Q - 0.3Q2+ 0.01Q3 ...(Eqn.2.1e)
π = -10 - Q + 0.2Q2 - 0.01Q3 …(Eqn 2.1f)
Equation 2.1g below is an example of an algebraic equation for a multivariate
function where there are more than 1 independent variable.
Q = 2K + 3L + LK … (Eqn.2.1g)
NOTES
For the purpose of discussion in this chapter, we will adhere consistently to
equations 2.1c to 2.1g (wherever applicable).
* Table
Table is the simplest and most direct form of presenting data. The total revenue
function (2.2) is depicted in table 1, which shows the relationship between total
revenue and quantity over a selected range of output.
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Table 1 Total revenue schedule of a firm
Quantity(Q) Price(P) Total revenue(TR)=P.Q
0 7 0
10 6 60
20 5 100
30 4 120
40 3 120
50 2 100
60 1 60
70 0 0
• Graphs
The following diagrams show algebraic and graphical expressions for demand,
total revenue, cost, and profit.
π
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LINEAR, QUADRATIC, AND CUBIC FUNCTIONS
The demand function in equation 2.1d is in linear form. As a result we will derive a
straight line to relate the relationship. The total revenue is expressed as a quadratic.
This is a result of a second power in the independent variable. This is easily recognised
by its parabolic shape. In addition, the independent variable can be raised to a third
power, and the corresponding relationship is called a cubic function e.g. cost and profit
functions (with 2 loops).
ECONOMIC MODELS - TYPE AND USES In the above discussions, you have encounter jargon, graphs and algebraic equations
and you will keep on seeing and using them later. They are used to model relationship
that exists between and among variables and to facilitate your analysis and decision-
making.
• Type- symbolic model
Models are defined as simplified representation of a complex situation. For example,
symbolic modes use jargons and symbols to represent reality. Managerial economics
are liberal with jargon words (for example, cost, revenue, profit) which are only verbal
models of things and phenomenon. Diagrams and mathematical expressions
similarly model a situation by means of lines or stating the relationship between and
among variables. Where diagram (1.2) are economic models depicted graphically
whilst equation (2.1c - 2.1g) are algebraic models that describe exactly the same
economic phenomenon.
• Uses
All models serve three main purposes:
1) pedagogical purpose - as a device to teach about the operation of a complex
system ,
2) explanatory purpose - as a device to explain the relationship between and among
events in a logical fashion,
3) predictive purpose - to predict future behaviour.
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2.2 TOTAL, AVERAGE AND MARGINAL RELATIONSHIP
In the theory of demand, cost, production and market structures specific functional
relationships called total, average and marginal functions are used. In optimisation
analysis, the relationship between total, average and marginal is important I
understanding the principle of managerial economics. Table 2 shows the relationship
between total, average and marginal revenue of a firm with TR = 7Q - 0.1Q2. The first
and second columns display the relationship between output and total revenue. Column
3 and 4 is the derived average revenue and marginal revenue respectively. Here, we
assume the firm is in imperfect competition.
RECALL
An average relation is the dependent variable divided by the independent
variable. A marginal relation is the change in the dependent variable
caused by a one unit change in the independent variable.
Table 2 Total, average and marginal revenue of a firm
Q TR AR MR
PxQ TR÷Q ∆TR÷∆Q
0 0 - -
10 60 6 6
20 100 5 4
30 120 4 2
40 120 3 0
50 100 2 - 2
60 60 1 - 4
70 0 0 - 6
Remember TR = 7Q-0.1Q2
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AVERAGE REVENUE
Column 3 depicts average revenue. The average revenue of the tenth output is $6,
which is simply a division of total revenue with output. Each subsequent row in the
column is similarly derived.
MARGINAL REVENUE
In column 4, the marginal revenue earned from the tenth unit of output is $6. This is the
change from $0 when there is no output, to $60 earned, when 10 units were sold, thus a
one unit change is equivalent to $6. thus, the change in total revenue as a result of a
change in output is called marginal revenue. Each marginal revenue derived in
subsequent rows is calculated on the same basis.
Diagram 2 Total
revenue function,
Demand function,
and Revenue
maximising price
and quantity
Chapter 2 Economic Optimisation
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SUMMARISING
Let us summarise what we have discussed so far.
The key relationships between total, average and marginal are: 1) The value of the average function at any point is the slope of a ray
drawn from that point to the total function.
2) The value of the marginal function at any point slope of a line drawn
tangent to the total function at that point.
3) When total function increases, both average function and marginal
function is positive
4) When total function is decreasing , average function is still positive but
marginal function is negative
5) When total function is at its maximum, marginal function equals zero.
Example
A firm’s demand function is defined as Q =14 - 2P. Calculate total revenue when price is
equal to 3 and when price is equal to 4. What is marginal revenue equal to between P=3
and P=4?
Explain your answer.
SUGGESTED SOLUTIONS
When P=4, Q = 14 - 2(3) = 6 ∴TR = PQ = $24
P=3, Q = 14 - 2(4) = 8 ∴TR = PQ = $24
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There is no change in TR from P=4 to P=3, thus MR is equal to 0. Thus this firm will
maximise total revenue when output is between 6 units and 8 units.
IMPORTANCE OF CONCEPTS
The concepts of average, marginal and total relations should be thoroughly understood
as they are widely used in short-run optimisation problems (as shown in 2.3, 2.6, 2.7 and
2.8).
NOTES
In diagram 2b, we relate price to quantity in a linear fashion. In a linear
equation, the coefficient (-0.1) is the change in the dependent variable (P)
over the change in the independent variable (Q), (i.e., ∆P/∆Q). In other
words, it represents the slope of the equation, which measures its
steepness.
The slope of a function is critical to economic analysis because it is the essence of marginal analysis. Decisions made by managers involve some sort of change in one
variable relative to a change in other variables. For example, whether or not to raise
price would depend on the resulting change in revenue or profit. Many other economic
decisions rely on marginal analysis, including purchasing additional equipment and hiring
extra staff because of importance to you is the change in cost, productivity, or profit
associated with a change in your resource allocation.
We can use concepts of calculus to clarify the relationship between average, marginal,
and total. Therefore, elementary calculus will be discussed in 2.4.
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2.3 OPTIMISATION ANALYSIS
In economic analysis, optimisation generally means finding either the maximum or
minimum value of a variable of a function, thus a firm determines the output level at
which it maximises total profit or minimises cost.
APPROACH FOR FINDING OPTIMAL VALUE
We can use either the aggregate approach or the marginal approach. Likewise, it can be
illustrated graphically or calculated using calculus. Thus, we will begin by showing
graphically how optimal output is obtained by the aggregate approach followed by the
marginal approach. (In 2.6, we will use calculus to ascertain that the outcome is the
same as shown graphically).
OPTIMISATION BY TOTAL REVENUE AND TOTAL COST APPROACH ( AGGREGATE APPROACH)
The aggregate approach looks at aggregate, for example, total profit is the difference
between total revenue and total cost i.e.:
π = TR - TC
If given TR = 7Q - 0.1Q2 and TC = 10 + 8Q - 0.3Q2 + 0.01Q 3
. π = 7Q - 0.1Q2 - 10 - 8Q + 0.3Q2 - 0.01Q 3
= - 10 - Q + 0.2Q2 - 0.01Q 3
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Diagram 3 Total revenue, total cost, marginal revenue, marginal cost and profit
maximisation.
We can show profit is maximised via graph, by plotting the total revenue and total cost
functions for a range of output, Q. Diagram 3a shows the plotted total revenue and total
cost whilst 3b shows profit function. The graphical representation shows that profit is
maximised at Q2* =10 that is when the positive difference between TR and TC is the
greatest (3a) or alternatively when the profit function is at its maximum (3b).
π
π
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OPTIMISATION BY MARGINAL APPROACH (MR=MC)
RECALL
According to the marginal analysis, profit is maximised when MR=MC.
As long as marginal benefit exceed marginal cost, it pays for the firm
to increase output and the firm will continue doing so till marginal
revenue equals marginal cost (at this point the firm would be
maximising profit). It stems from the fact that the distance between
revenue and cost function is maximised at the point where their
slopes are the same. Because the slopes of total revenue and total
cost measures marginal revenue and marginal cost, hence when
these slopes are equal, MR = MC.
The maximum can also be located by finding the derivative or marginal of the function
then determining the value of Q at which the derivative (Marginal Mπ ) is equal to 0. For
our hypothetical example, MR=MC occurs at Q2* = 10 as illustrated in 3a. The relevance
of marginal revenue and marginal cost relations to profit maximisation can be
demonstrated by considering the general expression
π = TR – TC
d π dTR dTC
Marginal profit is M π = ------ = ------ - ------
dQ dQ dQ
given that dTR /dQ is MR and dTC/dQ is MC it follows that
M π = MR – MC
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Because maximization of any function requires that the first derivative equal zero, profit
maximisation occurs where
M π = MR – MC = 0
or where MR = MC. at output Q = 10.
2.4 METHODS OF DIFFERENTIATION CALCULUS - DEFINITION AND PURPOSE
Even though tables and graphs are useful for explaining concepts, equations are more
suitable for problem solving. One reason being that the technique of differential calculus
can be employed to locate maximum and minimum point more precisely.
Calculus is a mathematical technique that enables you to find instantaneous rate of
change of a continuous function. Moreover, it is especially useful in constrained
optimisation problems that often characterise managerial decision making.
CONCEPT OF A DERIVATIVE
A derivative is a precise specification of the marginal relation. As mentioned in earlier
pages, the slope is a measure of the change in y relative to a infinitesimally small change
in x. To find the magnitude of the slope, we need to employ derivatives. Differentiation is
the process of determining the derivatives of a function. Consider the general function y
= f(x).
NOTES
Differentiation means finding the change in y (∆y) for a change in x ( ∆ x)
as the change of x approaches 0.
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Expressed formally:
dy ∆y
----- = lim ----
dx ∆x→0 ∆x
This notation is read: “The derivative of y with respect to x equals the limit of the change
in y relative to the change in x as the change in x approaches zero.” Mathematicians use
d to represent very small changes in a variable. The ratio ∆y/∆x is a general
specification of the marginal concept.
2.5 RULES OF DIFFERENTIATION Let us spend a few minutes reviewing basic differentiation. If you need additional
material, refer to the MAT 153 (Business Calculus) module or any Calculus text
FIRST AND SECOND DERIVATIVES OF FUNCTIONS
Rule for finding derivative
Take for example a function expressed in the general form of
y = bxn
The rule for finding derivative is
dy --- = nbxn-1 dx
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NOTES
Derivative (or first derivative ) is denoted by dy/dx.
Let us try to use the rule..
Thus, suppose we have the equation
y = 5x2
The derivative of equation (2.7) according to this rule is
dy
--- = 2 . 5x 2-1 = 10x
dx
meaning that if x=3, the instantaneous rate of change of y with respect to x is 10(3) or
30.
In economic analysis, there are several other rules for differentiating a function, such as
differentiating a logarithmic function or ‘function of a function’. (For a detailed summary of
rules of differentiation, which will be used extensively in this text, please refer to the table
3 that follows.)
Chapter 2 Economic Optimisation
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Table 3 RULES FOR DIFFERENTIATING FUNCTIONS
Function Derivative
1. Constant Function y = a dy/dx = 0
2. Power function y = axb dy/dx = b.a.xb-1
3. Sums & differences function y = u + v dy/dx = du/dx + dv/dx
y = u - v dy/dx = du/dx - dv/dx
4. Product of two functions y = u . v dy/dx = u. dv/dx + v.du/dx
5. Quotient of two functions y = u/v v(du/dx) - u(dv/dx)
dy/dx = -------------------------
v2
6. Function of a function y = f(u)
where u = g(x) dy/dx = (dy/du) . (du/dx)
Do not worry; do not let the mathematical symbols bother you. The process of
differentiating is rather simple, actually
Examples:
1. y = 3 dy/dx = 0
2. y = 3x3 dy/dx = 3.3 x3-1 = 9x2
The following examples comprise of more than one function. Can you identify and write
down the functions U and V in the box provided?
3. y = 3x3 + (x2 + 2) dy/dx = (3.3x3-1) +(2.x2-1) = 9x2 + 2x
y = 3x3 - (x2 + 2) dy/dx = (3.3x3-1) - (2.x2-1) = 9x2- 2x
Examples 4,5 and 6 take into account the product of two functions
4. y = (3x3 ) ( x2 + 2) dy/dx = (3x3)(2x) + (x2 + 2)(3.3x3-1)
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= (6x4) +(x2 +2)9x2 = 6x4 +9x4 +18x2
=15x4 +18x2
5. y = (3x3 )/( x2 + 2) (x2 + 2)(3.3x3-1) - (3x3)(2.x2-1)
dy/dx = --------------------------------------
( x2 + 2)2
(x2 + 2) 9x2 - (3x3 )( 2x)
= -------------------------------
(x2 + 2)2
(x2 + 2) 9x2 - 6x4
= --------------------
(x2 + 2)2
3x4 + 18x2
= --------------
(x2 + 2)2
6. y = u2 + 5 and u = 3x2 dy/dx = (2u) ( 6x) = 2.3x2(6x)
= 36x3
But, almost all of the mathematical examples and problems involving calculus only
require the rules for constant, powers, sum and differences.
Take this example that involves all the above.
π = - 10 - Q + 0.2Q2 - 0.01Q3
dπ/dQ = -1 + 0.4Q - 0.03Q2
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Example Determine the derivative of this function:
y = 2000 - 200x2 + 3x3
SUGGESTED SOLUTIONS
dy/dx = - 400x + 9x2
FIRST DERIVATIVE
As mentioned earlier, the first derivative of a non-linear function is also called the
marginal function.
Turning now to the total revenue function;
TR = 7Q - 0.1Q2
When we take the first derivative of the above total function;
d(TR)
------ = 7 - 0.2Q = MR
dQ
the outcome is also known as the marginal revenue (MR) function.
As opposed to the above power function, take the case of a linear function such as the
demand equation P = 7 - 0.1Q.
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The derivative is
dP
----- = - 0.1 dQ
NOTES
(dP/dQ is equal to a constant value - 0.1). Here, we see that the first derivative of
a linear function is simply the value of the b coefficient, - 0.1 (i.e. the slope of the
linear function itself).
SUMMARISING
First derivative of 1) a linear function is the slope and is a constant
2) a quadratic function is the marginal function
SECOND DERIVATIVE- DEFINITION
Sometimes, mathematical calculations require the use of second derivative. Generally,
the second derivative is the derivative of its first derivative. Confusing? Not to worry. It
simply means that the second derivative of a function is a measure of the rate of change
of the first derivative.
PROCEDURE FOR FINDING SECOND DERIVATIVE
The procedure for finding the second derivative is quite simple. All the rules for finding
the first derivative also apply to obtaining the second derivative.
For example, using the same TR function
Chapter 2 Economic Optimisation
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TR = 7Q - 0.1Q2
d(TR)
first derivative: --------- = 7 - 0.2Q
dQ
When we derive it for the second time
d2 (TR)
------- = - 0.2
dQ2
NOTES
Second derivative is denoted with a superscript 2.
Second order derivative is very important in optimisation problems. (We will deal with this
in more detail in 2.6 .)
Example
Determine the first- and second- order derivatives of this function:
C = 2000 - 200x2 + 3x 3
Solution
First derivative: dC/dx = - 400x +9x2
Second derivative: d2C/dx = - 400 + 18X
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2.6 OPTIMISATION BY CALCULUS
MARGINAL ANALYSIS IN DECISION MAKING
Managerial decision making requires one to find minimum or maximum value of a
function. When a function is at a minimum or maximum, its slope or marginal value is
equal to zero, thus the derivative must be equated to zero. To illustrate we will discuss
both aggregate and marginal approaches.
AGGREGATE APPROACH
A primary objective of managerial economics is finding optimal values of key variables.
Beside being done graphically, the optimisation analysis is conducted much more
expediently with the use of calculus - using marginal analysis and derivative concepts.
For our hypothetical example, the profit function is obtained as follows:
π = TR - TC
= 7Q - 0.1Q2 - 10 - 8Q + 0.3Q2 - 0.01Q 3
= -10 - Q + 0.2Q2 - 0.01Q 3
Let us now employ calculus to determine the point of profit maximisation. We begin by
i) finding the first derivative of the profit function,
ii) setting it equal to zero, and
iii) solving for the value of Q that satisfies this condition.
d π/dQ = - 1 + 0.4Q - 0.03Q 2 = 0
(-10 +Q)(10 - 3Q) = 0
Q2* = 10 or Q1* =10/3
MARGINAL APPROACH
Alternatively, you can use the marginal approach. First,
i) find the derivative of total revenue and total cost, then
ii) set them equal to each other and finally
As expected, Q2*
coincides with the
point shown in
diagram 3a.
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55
iii) solve for Q*
MR = dTR/dQ = 7 - 0.2Q and MC = dTC/dQ = 8 - 0.6Q + 0.03Q2
By equating MR = MC = Mπ = 0
7 - 0.2Q = 8 - 0.6Q + 0.03Q2
-1 + 0.4Q - 0.03Q2 = 0
Q2* = 10 or Q1* =10/3
Marginal profit, the derivative of total profit is Mπ = dTR/dπ
Mπ = dTR/dQ - dTC/dQ .
Given that dTR/dQ by definition is marginal revenue, MR and dTC/dQ is marginal cost,
MC, it follows that Mπ = MR = MC. Since maximisation of any function requires that the
first derivative = 0, profit maximisation occurs at Mπ = MR = MC = 0 or where MR = MC =
0.
Example
A firm’s demand function is Q = 16-P and its total cost function is defined as
TC = 3 + Q + 0.25Q2.
Form the firm’s profit function and then determine the level of output that yields the
maximum profit. What is the level of profit at the optimum?
Chapter 2 Economic Optimisation
56
SUGGESTED SOLUTIONS
TR = PxQ
given Q = 16 - P
∴ P = 16 - Q
hence TR = 16Q - Q2 and with TC = 3 + Q + 0.25Q2
π = 16Q - Q2 - 3 - Q - 0.25Q2
= - 3 + 15Q - 1.25Q2
Using calculus:
first derivative: dπ/dQ = 15 - 2.5Q = 0
implies Q = 6
and the second derivative d2π/ dQ2 = - 2.5, which implies that Q = 6 is a maximum.(see
page 15 for second-order condition)
Alternatively set MR = MC
hence MR = 16 -2Q
MC = 1 + 0.5Q
16 - 2Q = 1 + 0.5Q
2.5Q = 15
Q = 6.
To find profit, substitute Q = 6 into the profit function, hence
profit = - 3 + 15(6) - 1.25 (36)
= 42
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DISTINGUISHING MAXIMUM AND MINIMUM VALUES IN THE OPTIMISATION PROBLEMS
In economic analysis, finding optimum values generally means finding the maximum or
minimum value of a variable, depending on the type of function discussed. For example,
when a profit or total revenue function is analysed, the maximum value is the focus,
whilst the minimum value would be the main concern if the total cost function is analysed
SECOND DERIVATIVE- REVISITED
However, there are instances when the function has both a maximum and a minimum
value, (diagram1d - when a function is in cubic form). Since marginal value equals zero
for both maximum and minimum values of a function, that means, the method described
previously cannot tell us whether the optimum is a maximum or minimum. Hence further
analysis is required. A formal mathematical procedure to distinguish between maximum
or minimum value, requires the use of second derivative, which is, differentiation of the
first derivative. As mentioned in our earlier discussions, rules for finding first derivative
also apply to obtaining the second derivative.
The rule is that if the second derivative is positive,
we have a minimum, and if the second derivative is
negative, we have a maximum.
Using mathematical notation, we can then state that the first-and second-order
conditions for determining maximum and minimum values of a function are as follows.
First order Second order
condition condition
Maximum value dy/dx = 0 d2 /dx 2 < 0
Minimum value dy/dx = 0 d2 /dx2 > 0
−.
RULES
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USE OF MARGINALS TO MAXIMISE THE DIFFERENCE BETWEEN TWO FUNCTIONS
Now to illustrate,
Suppose the firm has the following revenue and cost function (note: we use a cubic
function)
TR = 7Q - 0.1Q2
TC = 10 + 8Q - 0.3Q2 + 0.01Q3
Based on these equations, the firm’s total profit function is
π = - 10 - Q + 0.2Q2 - 0.01Q3
Let us now employ calculus along with the first- and second- order conditions to
determine the point at which the firm maximises its profit.
d π
--- = -1 + 0.4Q - 0.03Q2 = marginal profit
d Q
= (-10 + Q) (10-3Q) = 0
Q1* = 10/3 Q*2 = 10
Although both Q1 * and Q2 * fulfil the first-order condition, only one satisfies the second-
order condition. To find out which one, let us find the second derivative of the function by
taking the derivative of the marginal profit function or second derivative of the profit
function:
d2π
---- = 0.4 - 0.06Q
dQ2
Chapter 2 Economic Optimisation
59
By substitution, we see that when Q= 10/3, the value of the second derivative is a
positive number:
d2π/ dQ2 = 0.4 - 0.06(10/3) = 0.2
On the other hand, when Q=10, the value of the second derivative is a negative number:
d2π/ dQ2 = 0.4 - 0.06(10) = - 0.2
Thus, we see that only Q2* enables us to adhere to the second-order condition of
d2π /dQ2 < 0.
As shown by graph (3a, 3b) and calculus above, only Q2* is THE optimal output. Our
example also shows that MR=MC at profit maximising output level, the converse does
not hold true. Profit is not necessarily maximised at any point where MR=MC (for
example Q2*.=10/3)
NOTES
Optimal output is derived when total profit function reaches a maximum. This
corresponds to the point when Mπ = 0 at Q2*. To be the optimal output, it must
satisfy both the F.O.C. and S.O.C that is
dπ/dQ = 0 and d2π/ dQ2 < 0.
Chapter 2 Economic Optimisation
60
Example
Given the demand function Q = 1000 - 40P, where Q is quantity and P is price,
determine the quantity that results in maximum total revenue.
SUGGESTED SOLUTIONS
Given Q = 1000 - 40P
∴ P = 25 - 0.025Q
and TR = PQ
= 25Q - 0.025Q2
To determine the quantity that maximises TR, take the first derivative, set it equal to
zero, and solve for Q.
d(TR)
------ = MR = 25 - 0.05Q = 0
dQ
Q = 500
That Q=500 is a maximum is clear because the second derivative is negative, that is,
d2(TR)
------- = - 0.05 < 0
dQ2
Chapter 2 Economic Optimisation
61
CHECKLIST
Take a few minutes to check if you are now able to:
Write functions to represent your economic problems.
Build a model for the economic problem at hand.
Relate the relationships that exist between total, average, and
marginal concepts irrespective of whether it is revenue, cost or
product.
Perform optimisation analysis, graphically via both aggregate and
marginal approach.
Conduct first- and second-order derivations.
Apply marginal analysis in decision making
Distinguish between maximum and minimum values in
optimisation problems.
2.7 MULTIVARIATE OPTIMISATION
In this section, we determine maximum and minimum values of a function with more than
one variable. First, we introduce the concept of partial derivative and then use it to
examine the maximisation process of a multivariate function
PARTIAL DERIVATIVE
The functions discussed so far involve only one variable. Many functional relationships
which you will encounter later involve more than one independent variable. For a
multivariate function, for example y = f(x,z), the concept of partial derivative is employed.
Chapter 2 Economic Optimisation
62
SOME EXAMPLES OF MULTIVARIATE FUNCTIONS
Most economic relationship involves more than one variable, for example;
demand of a product (Qdx) is influenced by the price of the product itself (Px ),
price of a related product (Py, where y could be a substitute or complement),
income (I), taste(T) and others, that is {Qdx = f(Px, Py, I,T,…)}; total revenue (TR)
depends on output (Q), advertisement (A), and taste factors (T), that is {TR=
f(Q,A,T)} and output (Q)is a function of labour (L) and capital inputs(K). Hence, it
is important to determine the marginal effect of each independent variable
separately. To do so, we use partial derivative.
For example, given y = f (x, z)
y = 3x2 -xz + z2
When analysing multivariate function, we first take the partial of y with respect to
x, i.e. ∂y/∂x. This indicates the slope relationship between y and x when z is held
constant. Similarly, the partial derivative of y with respect to z (∂y/∂z) is derived by
assuming x to be constant, and taking the first derivative of y with respect to z.
Thus,
∂y/∂x = 6x - z
and ∂y/∂z = 2z - x
The partial derivative ∂y/∂x means that a small change in x is associated with y
changing at the rate of (6x - z) when z is held constant. If z =2, the slope
associated with y and x is (6x - 2). Similarly, ∂y/∂z means a small change in z
is associated with y changing at the rate of (2z - x), when x is held constant.
NOTES
Partial derivative is denoted by ∂y/∂x.
Chapter 2 Economic Optimisation
63
MAXIMISING A MULTIVARIATE FUNCTION
To maximise or minimise a multivariate function, we have to set each partial derivative
equal to zero and solve simultaneously for the optimal value of the independent
variable.
For example, to maximise total profit function
π = xz -3x2 - z2 + 11x
we find partial derivative, set it equal to 0 and solve for x and z.
∂ π/∂ x = z - 6x + 11 = 0 …( i )
∂ π/ ∂z = x - 2z = 0 …( ii )
Multiplying ( i ) by 2 and add to ( ii ), we get
2z-12z + 22 = 0
x - 2z = 0
-------------------------------------------
11x = 22
x = 2
∴ z = 1
Substituting x = 2 into ( i ), we get z = 1 and substituting these values into the
profit function , we get total profit:
π = (2)(1)-3 (2)2 - (1)2 + 11(2) = 11
Thus when this firm maximises its
profit amounting to $11 when it
sells 2 units of x with 1 unit of y.
Chapter 2 Economic Optimisation
64
Example
Given a firm’s total profit depends on the sale of product x and product z is π = f(x,z) =
100x - x2 - xz - 2z2 + 190z,
Find partial derivative of π with respect to x and z. Find the combined output of x and z
that maximises profit and profit, at that combination.
SUGGESTED SOLUTIONS
∂π/∂x = 100 - 2x - z …( i )
and ∂π/∂z = -x - 4z + 190 …( ii )
To solve,
( i ) x 4 400 - 8x - 4z = 0 …( iii )
( iii) - (ii) 190 - x - 4z = 0
---------------------
210 - 7x = 0
x = 30
∴ z = 40
Substituting x =30 and z = 40 into the profit function gives profit amounting to $5300
π = 100(30) - (30)(30) - 30(40) - 2(40)(40)
= $5300
Chapter 2 Economic Optimisation
65
2.8 CONSTRAINED OPTIMISATION
All the above discussions are illustrations of unconstrained optimisation, but that is not a
realistic scenario.
SOLVING BY SUBSTITUTION METHOD
In chapter 1, we mentioned that in attempting to achieve goals, managers face
constrains. This would limit their options, hence the maximisation or minimisation of an
objective function is subjected to constrain. One way to solve a constrained optimisation
problem is the substitution method.
Suppose, a firm operates with this total cost function:
TC = 3x2 + z2 - xz
where x and z represent two products to be produced.
The manager is requested to determine the least-cost combination of x and z, subject to
the constraint that total output of both products is 10 units. Therefore, the constrained
optimisation problem is:
Minimise TC = 3x2 + z2 - xz,
subject to x + z = 10.
Solving the constraint for z and substituting this value into the objective function results
in
z = 10 - x,
and TC = 3x2 + (10-x)2 - x(10-x)
= 3x2 + 100 - 20x + x2 - 10x + x2
= 5x2 - 30x + 100
Now, it is possible to treat the above equation as an unconstrained minimisation
problem. To solve, we find the first derivative, set it equal to zero. And solve for value of
x:
Chapter 2 Economic Optimisation
66
dTC
------ = 10x -30 = 0
dx
10x = 30
x = 3
A check of the sign of the second derivative at that point has to be made to ensure a
minimum:
d2TC
----- = 10
dx2
Since the second derivative is positive, x = 3 is indeed a minimum.
When we substitute x = 3 into the constraint equation, we can determine the optimal
output for z.
x + z = 10
z = 10 -3
= 7
Thus a production of 3 units of x and 7 units of z is the least-cost combination for
producing 10 units. The total cost of this combination is
TC = 3(9) - (3)(7) + 49
= $55.
SOLVING BY LAGRANGIAN MULTIPLIER TECHNIQUE
Fallacy of the substitution technique
Unfortunately, the substitution technique discussed in the preceding section is not
always feasible. Some decision problems involve numerous and complex constraint
conditions. To overcome this, we have to resort to the Lagrangian multiplier technique.
Chapter 2 Economic Optimisation
67
LAGRANGIAN TECHNIQUE
This method optimises a function which incorporates the original objective function and
the constraint function. This combined equation called the Lagragian function is created
in such a way that when it is maximised or minimised, the original objective function is
also maximised or minimised and all constraints are satisfied.
Lagrangian function
The first step in this method is to form a Lagrangian function. This is simply done by
writing the objective function that the firm wishes to seek; (either maximises or
minimises) plus λ (lambda - denoting the Lagragian multiplier ) times the constraint
function and set equal to zero−.
For our hypothetical example;
Min.x, z TC subject to total output
(Objective function) (Constraint function)
Thus L TC = 3x2 + z2 - xz + λ( x + z - 10)
L TC is defined as the Lagrangian function for the constrained optimisation under
consideration.
The partial derivative of the above functions with respect to the three unknown, x,
z, λ and are as follows:
∂LTC
----- = 6x - z + λ ,
∂x
Chapter 2 Economic Optimisation
68
∂LTC
------ = 2z - x + λ
∂z
∂LTC
and ------ = x + z - 20
∂λ
At a minimum point on a multivariate function, all partial derivatives are equal to zero.
Setting these partial derivatives equal to zero, gives three equations and three unknown.
6x - z + λ = 0 … (i)
2z - x + λ = 0 … (ii)
x + z - 10 = 0 …(Iii)
The above can be solved simultaneously.
First we get rid of one unknown, ( λ ) by subtracting ( ii ) from ( i )
6x - z + λ - 2z + x - λ
gives 7x - 3z = 0 … ( iv )
Next, to get rid of z, multiply ( iii) by 3 and add to ( iv ) to give the solution for x:
3 x + 3z - 30 = 0
7x - 3z = 0
---------------------------------------
10x - 30 = 0
10x = 30
x = 3
Substituting x = 3 into equation ( iii ) yields z = 7;
Chapter 2 Economic Optimisation
69
3 + z = 10
z = 7
and substituting x = 3 and z = 7 into equation ( i ), gives the λ value:
6(3) - 7 +λ = 0
λ = -11
Here, λ = -11 is interpreted as the marginal cost of producing 10 units; meaning that if
the firm reduces output from 10 to 9 units, total cost will fall by approximately $11.Thus,
λ is the marginal effect on the objective function associated with per unit change
(increase or decrease) in the constraint. The marginal relation described by the multiplier
provides managers with economic data to evaluate the potential benefits or costs of
relaxing constraints.
NOTES
The use of Lagrangian multiplier method is further examined in chapter 4 to find
consumer’s maximum satisfaction from consumption. Again, in chapter 5, the
Lagrangian technique for constrained optimisation is used to develop the optimal
input proportion rule.
SUMMARISING
Steps to follow:
1. Form the Lagrangian function, (L ( ) )
2. Find partial derivative (L ( ) ) with respect to the first variable,
3. Find partial derivative (L ( ) ) with respect to the second variable,
4. Find partial derivate (L ( ) ) with respect to λ,
5. Finally, solve simultaneously to get values of the first and second variable
and λ.
Chapter 2 Economic Optimisation
70
The exercise below provides another perspective of the usefulness of the Lagrangian
method.
Example
Given the utility function as follows
u = 2x 0.75 - y 0.25
and the constraint function
1000 = 2x + 4y
a) Formulate a Lagrangian function
b) What are the values of x and y that will maximise utility?
c) Find the value of λ and interpret it.
SUGGESTED SOLUTIONS
a) L( U*) = 2x0.75 y0.25 - λ (1000 - 2x - 4y)
b) F.O.C.
∂ U*
---- = 0.75(2x -0.25 y0.25 ) + 2 λ = 0 …(1)
∂ x
∂ U*
---- = 0.25(2x0.75 y -0..75) + 4λ = 0 …(2)
∂ y
It would be great if you attempt the
question before looking at the
answer. Just follow the steps above
Chapter 2 Economic Optimisation
71
∂ U*
------ = - 1000 + 2x + 4y = 0 …(3)
∂λ
Divide equation (1) by (2)
1.5x -0.25 y0.25 - 2λ
---------------- = -----
0.5x0.75 y -0.75 - 4λ
3 y 2
---- = ----
x 4
2x = 12y
x = 6y …(4)
Substitute equation (4) into (3)
2 (6y) + 4y = 1000
y = 62.5
x = 6(62.5)
= 375
A combination of x = 375 and y = 62.5 will maximises utility.
c) Substitute x = 375 and y = 62.5 into equation (1)
5(375)-0.25 (62.5)0.25 + 2 λ = 0
λ = - 0.479
λ is the marginal effect of the objective function associated with the per unit change in
the constraint function. A λ of - 0.479 means that total utility will fall by 0.479 when the
income falls by one unit.
Chapter 2 Economic Optimisation
72
QUESTIONS
Q1. Assume a firm produces its product in a market described with the following
production function and price data.
Q = 2x + 3y + 6xy
Px = $4, Py = $6
where x and y are two variable input factors in the production of output Q, and Px
and Py are prices of x and y respectively.
a) What is the optimal input combination for x and y in this production if the firm
is operating with a $1000 budget constraint? (use the Lagrangian multiplier
method)
b) What is the increase in output that could be obtained from an additional
expenditure of $1?
Q2. A firm’s production function is given by
Q = L2 + 5LK + 4K2
The price of a unit of labour service and a unit of capital is $5 and $10
respectively. The firm has a cost limitation of $1000 per time period.
a) Formulate the Lagrangian function.
b) What should be the combination of K and L that will maximise output Q?
c) How much is the maximum output?
Chapter 2 Economic Optimisation
73
d) Interpret the meaning of the Lagrangian multiplier.
Q3. A. Einstein invented a new alarm system for which he receives royalty of 20% of
total revenue from the sales of the alarm system. If the total cost and demand
function is given by
TC = 200 000 + 30Q + 0.002Q
Q = 12 500 - 50P
a) Calculate the total revenue maximising price and output. What is the amount
of royalty Einstein would receive?
b) Calculate the profit maximising output and price. What is the amount of
royalty Einstein would receive at this output?
Q4. A company produces according to the following production function:
Q = 500L - 0.5L2 + 1000K -0.25K2 + 100E - 0.1E2
where Q is output per month,
L is number of workers employed per month,
K is machines used per month
and E is electricity used per month.
All output is sold at $10 per unit. L costs $2000 per month, machines cost $9000,
electricity costs $0.10 per kilowatt/hour.
a) Formulate the profit function.
b) What is the profit maximising level of labour, machine and electricity per
month?
c) What is the profit maximising level of output?
Chapter 2 Economic Optimisation
74
d) What is total profit per month?
Q5. The demand function faced by john associate is
P = 45 – 0.5Q
where Q represents output and p represents price
Its total cost is given by the equation
TC = Q 3- 8Q 2 + 57Q + 2
a) derive the equation of the total profit function
b) determine the output level that maximises total profit
c) calculate the maximum total profit
Q6. Given Q = 1500 –50P
a) determine the TR function
b) determine the rate of output that maximise TR
c) calculate maximum TR
Chapter 2 Economic Optimisation
75
SUGGESTED SOLUTIONS
Q1 Steps to follow:
1 ) Form the Lagrangian function, (L ( ) )
2 ) Partial derivate (L ( ) ) with respect to x,
3 ) Partial derivate (L ( ) ) with respect to y,
4 ) Partial derivate (L ( ) ) with respect to λ,
5 ) Finally, solve simultaneously to get values of x, y, and λ.
Objective function ⇒ Q = 2x + 3y + 6xy
Constraint function ⇒ 4x + 6y = 1000
a) LQ = 2x + 3y + 6xy + λ (4x + 6y - 1000)
∂LQ/ ∂x = 2 + 6y + 4λ = 0 …(1)
∂LQ/ ∂y = 3 + 6x + 6λ = 0 …(2)
∂LQ/ ∂λ = 4x + 6y - 1000 = 0 …(3)
(1) x 3 ⇒ 6 + 18y + 12λ = 0 …(4)
(2) x 2 ⇒ 6 + 12x + 12λ = 0 …(5)
(3) - (5) ⇒ 18y -12x = 0
y = 2x /3
Substitute into (3)
4x + 6 (2/3)x = 1000
4x + 4x = 1000
x = 125
y = 125(2/3)
= 83.33
Chapter 2 Economic Optimisation
76
Therefore optimal combination is x = 125 and y = 83.33 units
b) Substitute x = 125 and y =83.33 into (1)
2 + 6(83.33) = 4λ
λ = 502/4
= 125.5
When expenditures increase by $1, output will increase by 125.5 units.
Q2. See Q1 for steps to follow.
a) Objective function ⇒ Q = L2 + 5LK + 4K2
Constraint function ⇒ 5L + 10K = 1000
∴ LQ = L2 + 5LK + 4K2 + λ (5L +10K -1000)
b) ∂LQ / ∂L = 2L + 5K + 5λ = 0 …(1)
∂LQ/ ∂K = 5L + 8K + 10λ = 0 …(2)
∂LQ/ ∂λ = 5L + 10K - 1000 = 0 …(3)
(1) x 2 ⇒ 4L + 10K + 10λ = 0 …(4)
(4)- (2) ⇒ -L + 2K = 0 …(5)
From (5) ⇒ L = 2K …(6)
Substitute (6) into (3) ⇒ 5(2K) + 10K = 1000
20K = 1000
K = 50
∴ L = 100
Output is maximised when K=50 and L =100 .
Chapter 2 Economic Optimisation
77
c) Substitute K=50 and L =100 into the output function.
Q = L2 + 5LK + 4K2
= 100(100) + 5(100)(50) + 4(50)(50)
= 45 000
d) Substitute K =50, L=100 into equation (1)
2(100) + 5(50) + 5λ = 0
λ = -90
When production budget falls by a unit, the output falls by 90 units.
Q3.
Steps to follow:
1) Find TR
2) Conduct F.O.C on TR to get Q* and P*
3) Calculate 20% of TR
4) Conduct S.O.C. to reaffirm maximising.
a) TR = PxQ
given Q = 12500 - 50P
∴P = 250 - 0.02Q
and TR = 250Q - 0.02Q2
F.O.C. dTR/ dQ = 250 - 0.04Q = 0
Q* = 6250
Substitute Q*=6250 into P function: P* = 250 - 0.02(6250)
= $ 125
therefore TR = 125 x 6250
= $ 781 250
therefore the amount of royalty = 20%( 781 250)
= $ 156 250
Chapter 2 Economic Optimisation
78
b) Steps to follow:
1) Find profit
2) Conduct F.O.C on π function to get Q* and P*
3) Conduct S.O.C. to reaffirm maximising
4) Find TR by substitute Q* and P* into TR
5) Calculate 20% of royalty
π = TR - TC
= 250Q - 0.02Q2 - { 200 000 + 30Q + 0.0002Q2 }
= - 200 000 + 230Q - 0.022Q2
∂ π/∂Q = 230 - 0.044Q =0
Q* = 5 000
P* = 250 - 0.02(5000)
= $150
TR = PxQ
= 150(5 000)
= $ 750 000
Royalty = 20%(750 000) = $ 150 000
Q4. Steps to follow:
1) Find profit by subtracting TC from TR
2) Partial derivate π with respect to L, K, E
3) Solve for Q* for respective L, K,E
4) Conduct S.O.C. to reaffirm maximising
5) Find π by substituting into answer a)
a) π = TR - TC
TR = PxQ
given P = $10
and Q = 500L - 0.5L2 + 1000K -0.25K2 + 100E - 0.1E2
Chapter 2 Economic Optimisation
79
∴ TR = 10(500L - 0.5L2 + 1000K -0.25K2 + 100E - 0.1E2) given PL = 2000, PK = 9000, and PE = 0.1
∴ TC = 2000L + 9000K + 0.1E
∴ π = 10(500L - 0.5L2 + 1000K -0.25K2 + 100E - 0.1E2)
- 2000L + 9000K + 0.1E
b)To maximise π , input is employed to the point where the change in π
associated with hiring one additional unit of input is equal to zero.
Labour : ∂π/∂L = 10( 500 - L) - 2000 = 0
L* = 300
Machines: ∂π/∂K = 10( 1000 - 0.5K) - 9000 = 0
K* = 200
Electricity: ∂π/∂E = 10( 100 - 0.2E) - 0.1 = 0
E* = 499.95
c) Q = 500(300) - 0.5(300)2 + 1000(200) - 0.25(200)2 +100(499.95) –
0.1(499.95)2
= 320 000 units
d) π = TR - TC
TR ⇒ 10( 320 000) 3 200 000
TC ⇒ Labour : 2000(300) = 600 000
Machines: 9000(200) = 1 800 000
Electricity: 0.1(499.95) = 50
2 400 050
---------------
Total profit $799 950
---------------
Well, you have reached the end of the chapter. The math in this chapter has been
carefully explained step by step. Please make sure you have understood the steps used
in the calculation. It has also been a long chapter; if you find difficulties in understanding
Chapter 2 Economic Optimisation
80
the material, please go through the sections again carefully and note down questions
that you need to discuss with your lecturer or “classmates”
SUMMARY
The functional relationship y = f(x) means there is a systematic relationship
between the dependent variable y and the independent variable x.
Functional relationships can be expressed in algebraic equations, tables or
graphs.
Functions can be linear, quadratic or cubic.
Economic models are used to illustrate an economic principle, explain an economic
phenomenon, or to predict implications as a result of changes that affect the
functional relationship.
The key relationships between total, average, and marginal are:
⎩ The value of the average function at any point is the slope of a ray drawn from
that point to the total function.
⎩ The value of the marginal function at any point is the slope of a line drawn
tangent to the total function at that point.
⎩ When total function increases, both average function and marginal function is
positive.
⎩ When total function decreases, average function is still positive but marginal
function is negative.
⎩ When total function is at its maximum, marginal function equals zero.
Optimisation means finding either the maximum or minimum value of a variable of a
function.
Chapter 2 Economic Optimisation
81
The slope of a function y = f(x) is the change in y (∆y) divided by the corresponding
change in x (∆x). The derivative of this function dy/dx or f ’(x) is the slope of a straight
line drawn tangent to the function at that point.
The rules of differentiation contained in page will help you find derivatives of
functions encountered in managerial economics.
Higher order derivatives are found by taking the first derivative of each resultant
derivative, for example, second derivative is the derivative of the first derivative.
The maximum or minimum value of a function y = f(x) can be found by setting the
first derivative equal to zero (dy/dx = 0 ) and solving for the value of x.
A function is at a maximum if dy/dx = 0 and d2y/dx2 < 0. Conversely,
A function is at a minimum if dy/dx = 0 and d2y/dx2 > 0.
Profit is maximised when MR=MC or Mπ = 0 and dMπ /dQ< 0 (or alternatively
dπ/dQ = 0 and d2π/dQ2 < 0).
Revenue is maximised when MR = 0
Average cost is minimised when MC = AC and average cost is increasing as
output expands.
For multivariate function (function with more than one independent variable -
y = f (x z) ), the concept of partial derivative ( ∂y /∂x) has to be employed. When we
partial derivate y with respect to x, we hold z constant meaning that we isolate the
marginal effect on y from changes in x only.
To maximise or minimise a multivariate function, we set each partial derivative
equal to zero and solve the resulting set of simultaneous equations for the optimal
value of the independent variables.
Chapter 2 Economic Optimisation
82
For constrained optimisation problems, the Lagrangian technique has to be
adopted. The Lagrangian function incorporates the objective function and the
constraint conditions. The Lagrangian multiplier, λ, indicates the marginal effect on
the objective function as a result of a unit increase or decrease in the constraint
function.
STUDY NOTES