. Since the same relation holds true for a variable
y
, if
y = f
(
x
),
we have a further way of defining the elasticity of
y
with respect to
x
:
E =
(
dy/y
)/(
dx/x
)
=
d
(
ln y
)
/d
(
ln x
)
.
The second application of exponential functions concerns growth rates.
In Module 9, we shall see how use of the exponential growth function
X
t
= X
0
e
gt
(M.8.1)
can actually simplify some calculations!
Finding the Roots of Quadratic Equations
:
This note is a reminder that the roots of a
quadratic equation in the form
ax
2
+
bx + c
= 0 may be calculated using the following
formula:
x =
–
b ±
2
b
a
2
– 4
ac
.
(M.8.2)
Recall that if the expression under the square root sign is negative, the roots of the equa
tion have an imaginary component.
M82
MATH MODULE 8: SOME SPECIAL FUNCTIONS AND FORMULAS
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1.2 SOME SPECIAL FUNCTIONS
The text contains extended discussions of some important functions that are widely
used in consumption and production theory: the CobbDouglas, perfect complement,
and perfect substitute functions. CobbDouglas functions are analyzed in Appendixes
3 and 9, especially on pages 5923 and 6368. Perfect complement and perfect substitute
functions are discussed on pages 5879 and 639. The importance of these functions is
not primarily their realism, but rather the fact that they provide some simple yet strong
cases that indicate the widely different outcomes that can occur in economic processes
as a result of the nature of differing utility and production functions.
Another function that rests on special assumptions is the constantelasticity demand
function, which is discussed on pages 5946 of Appendix 4. In this section we provide
the parallel analysis of constantelasticity
supply
functions. The constantelasticity sup
ply curve has
exactly
the same form as the constantelasticity demand curve:
Q = KP
, or equivalently
P = kQ
1/
,
(M.8.3)
where
k =
(1/
K
)
1/
is a positive number and
is the constant price elasticity of supply.
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 Fall '12
 Danvo
 Supply And Demand, Natural logarithm, Special functions, Math Module

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