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1
Introduction to
Microeconomics
Lecture 8
Maths: Functions
Ian Crawford
Outline
z
Function notation and some common functions
z
Linear, polynomial
z
Inverse functions
z
Composite function
Composite functions
z
Exponential and logarithmic functions
z
Functions of several variables
z
Homogeneous Functions, and Returns to Scale
Functions
z
y =
5
x
–8
A
linear function
z
y
depends on
x.
We often write
y
(
x
) to emphasize dependence
z
C(q)
= 3 + 2
q
2
Total cost,
C
, depends on output,
q
z
For general functions it is common to use
f
(
x
) in place of
y
:
x
is called the
argument
.
z
The key feature of a function is that for each value of
x
, the input, there is a
unique
value of
f
(
x
), the output
z
Ideally we should specify the
domain
– the set of inputs – and the range
(the set of outputs). Usually in economics these are fairly obvious.
z
If the graph of
f
(
x
) is upwardsloping then
f
(
x
) is an
increasing function
(or
a “
monotonic increasing function
”), e.g. many supply functions
z
If the graph of
f
(
x
) is downwardsloping the
f
(
x
) is a
decreasing function
(or
a “
monotonic decreasing function
”), e.g. most demand functions
z
A monotonic function is either increasing or decreasing
Graph of
y
= 5
x
–8for
x
≥
0
30
40
50
8
20
10
0
10
20
024681
0
1
2
x
y = 5x 
NB: sometimes we are interested in negative values of
x
and
y
Graph of
C
= 3 + 2
q
2
for
q
≥
0
150
200
250
0
50
100
0
1
2
q
C
Polynomials and related
functions
z
A
polynomial
function takes the form
f
(
x
) =
a
n
x
n
+ a
n
1
x
n
1
+ …+
a
0
z
A linear function is a polynomial of degree 1
z
A quadratic function is a polynomial of degree 2
z
E.g. the cost function:
C
= 10 +
q
+ 3
q
2
z
A useful function is
f
(
x
) =
x
n
for
x
> 0
z
If
n
is an integer this is a polynomial of degree
n
z
n
can also be negative or a fraction
z
Q
=
p
2
is a demand function with
n
=
−
2
z
f
(
x
) =
x
0.3
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 Spring '10
 Vines
 Economics, Microeconomics

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