Download Economic Techniques 102 Week 3 Lecture DCDM BUSINESS

Economic Techniques 102
Week 3 Lecture
DCDM BUSINESS SCHOOL
FACULTY OF MANAGEMENT
ECONOMIC TECHNIQUES 102
LECTURE 3
NON-LINEAR FUNCTIONS
0. Preliminaries
The following functions will be discussed briefly first:
•
Quadratic functions and their solutions
•
Cubic functions
•
Other Polynomial Functions
1. Non-linear Functions
Linear functions are of limited applicability to real-life situations, hence the need for
more versatile non-linear models. For example, a revenue function of the type
TR = PQ
(1)
is reasonable for a perfectively competitive firm that faces a constant price. Should
the firm be a monopolist then (1) is not appropriate, as total revenue depends on Q. In
that case,
TR = P (Q ) Q .
(2)
Further, with an inverse demand curve,
P = 50 − 2Q ,
(3)
the corresponding revenue function is:
TR = (50 – 2Q) Q
(4)
= 50 Q – 2Q2
which is non-linear. In particular, the revenue function is quadratic in Q, and has the
shape:
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Economic Techniques 102
Week 3 Lecture
TR = __________
0
25
Q
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Economic Techniques 102
Week 3 Lecture
Equation (4) is a member of the class of equations
y = ax 2 + bx + c
(5)
where x = Q, a = −2, b = 50 and c = 0 .
The roots (or solutions) of a quadratic equation are the values of x for which the
quadratic function equals zero. The roots to (5) are found using the formula:
x1 , x2 =
−b ± b 2 − 4ac
.
2a
(6)
EXAMPLE 1
A firm's total cost function is given by the equation TC = 200 + 3Q, while the demand
function is given by the equation P =107 – 2Q.
(a) Write down the equation for total revenue (TR) function.
(b) Graph the TR function for 0<Q<60. Hence, estimate the output Q, and total
revenue when TR is maximum.
(c) Plot the total cost function on the diagram in (b). Estimate the break-even point
from the graph. Confirm your answer algebraically.
(d) State the range of values for which the company makes a profit.
EXAMPLE 2
The demand and supply functions for a good are given by the equations:
Pd = −(Q + 4) 2 + 100, Ps = (Q + 2) 2 .
(a) Sketch each function on the same diagram; hence, estimate the equilibrium price
and quantity.
(b) Confirm the equilibrium using algebra.
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Economic Techniques 102
Week 3 Lecture
3. Polynomial Functions
Quadratic functions are members of a larger class of functions called polynomial
(multi-term) functions. Their general form is:
y = a0 x 0 + a1 x1 + a2 x 2 + .... + an x n
(7)
although (7) is more commonly written,
y = a0 + a1 x + a2 x 2 + .... + an x n
(8)
as x 0 = 1 and x1 = x .
Specific subclasses of polynomials include:
Constant Function:
y = a0
(9)
where a1 = a2 = .... = an = 0 .
Linear Function:
y = a0 + a1 x
(10)
y = a0 + a1 x + a2 x 2
(11)
y = a0 + a1 x + a2 x 2 + a3 x 3
(12)
where a2 = a3 = .... = an = 0 .
Quadratic Function:
where a3 = a4 = .... = an = 0 .
Cubic Function:
should a4 = a5 = .... = an = 0 .
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Economic Techniques 102
Week 3 Lecture
Quartic Function:
y = a0 + a1 x + a2 x 2 + a3 x 3 + a4 x 4
(13)
where a5 = a6 = .... = an = 0 .
4. Rational Functions
Rational functions are defined as the ratio of one (or more) polynomial functions. For
example,
y=
x −1
x + x2 + 4
(14)
3
is a rational function. An important rational function is the rectangular hyperbola,
which has the mathematical form,
y=
α
x
α >0
(15)
is used in economics to represent demand and supply curves. Two other classes of
functions important in explaining growth and decay are logarithmic and exponential
functions.
5. Logarithm to the Base b
The logarithm to the base b of a number x , is the power to which b must be raised
to yield x . It is denoted log b x .
EXAMPLE 3
When b = 5 the following values are defined:
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Economic Techniques 102
Week 3 Lecture
log 5 125 = 3
as
53 = 125
log 5 25 = ____
as
5__ = 25
log 5 5
= ____
as
5__ = __
log 5 1
= ____
as
50 = __
as
5__ =
as
5__
1
= ____
5
1
log 5
= ____
25
log 5
1
5
1
=
25
It should be noted that:
(a)
the log of x < 0 does not exist and
(b)
the log of 0 < x < 1 is always negative.
6. Log Rules
Several rules for manipulating logarithms are useful. They are:
(a) log of a product
log( xy ) = log x + log y
log 5 (25 × 5) = ______________
(17)
= ______________
= ______________
(b) log of a ratio
x
log( ) = log x − log y
y
log 5 (
25
) = _____________
5
= _____________
(18)
= _____________
(c) log of an exponent
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Economic Techniques 102
Week 3 Lecture
log x k = k log x
log 5 53 = ___________
(19)
= ___________
= ___________
The definition of logarithm is applicable for any b > 0 . The natural (or ln) logarithm
has the natural constant e (=2.7183….) as its base. The natural log is frequently used
by economists because it is easy to differentiate.
It is denoted:
y = log e x ≡ ln x .
(20)
7. Exponential Functions
Another important class of functions has the form:
f ( x) = b x
(21)
where b > 0. Note here the index (x) is changing while the base remains constant.
The corresponding graph will pass through the ordered pairs:
x
f ( x)
3
8
2
4
1
2
0
1
-1
1/2
-2
1/4
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Economic Techniques 102
Week 3 Lecture
As x > 0 increases, f ( x ) increases very rapidly. However, as x < 0 becomes more
negative f ( x ) declines slowly. When b = e , the base of the natural logarithm, the
function is the natural exponential function. Any economic variable that exhibits a
constant proportional growth will behave like an exponential function, e.g., compound
interest.
EXAMPLE 4
The demand and supply functions for a brand of tennis shoes are:
Pd =
500
and Ps = 16 + 2Q , where P is the price per pair, Q is the quantity in
Q +1
thousands of pairs.
(a) Calculate the equilibrium price and quantity.
(b) Graph the supply and demand functions; hence confirm your answer
graphically.
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