30C00300 Mathematical Methods for Economists (6 cr)
4) Univariate and multivariate functions
Simon & Blume chapters: 13, 15
Slides originally by: Timo Kuosmanen
Slides amended by: Anna Lukkarinen
Lecture held by: Anna Lukkarinen
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How much more rope?
1m
1m
1m
1m
1m
1m
1m
1m
Outline
1. Function as a mapping
2. Inverse function
3. Implicit function and correspondence
4. Microeconomic content
– Production function
– Utility function
– Cost, revenue and profit functions
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Function
Definition (S&B, Ch. 13.1): A function from a set A to a
set B is a rule that assigns to each object in A, one and
only one object in B. In this case, we write
f: A → B.
• Set A is called the domain of f.
• Set B is called the target or target space.
• y=f(x) is the image (or range) of x under f.
Note: it is incorrect to write ”f(x)” to denote the function
itself. It is the value of function f at point x.
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Function – In other words
A function is a rule that assigns a unique object to
each object in the function’s domain
Input
Function
Output
Function - Basic definitions
𝒚 = 𝒇(𝒙)
y
x
• Depends on x
• Dependent variable
• Endogenous variable
• “Fixed outside model”
• Independent variable
• Exogenous variable
Image or range
Domain
• Set of all output
elements
• Set of all input
elements
Polynomial functions
𝒇 𝒙 = 𝒂𝒏 𝒙𝒏 + 𝒂𝒏−𝟏 𝒙𝒏−𝟏 + ⋯ + 𝒂𝟏 𝒙𝟏 + 𝒂𝟎
(𝒂𝒏 ≠ 𝟎)
Polynomial functions
Linear
functions
Cubic
functions
Quadratic
functions
Etc.
Exponential functions
𝒇 𝒙 = 𝑨𝒂𝒙
Source of picture: Matti Karvonen, lecture notes for Mathematics and Statistics for Managers, 2008
(𝒂 > 𝟎, 𝒂 ≠ 𝟏)
Firm’s production function
• Production function 𝑓: 𝐑𝑘+ → 𝐑 + indicates the maximum
output that can be produced with given input vector x.
Definition: f (x)  max  y (x, y )  T
• Note: production function represents Eff(T).
• Example: Cobb-Douglas production function
k
f (x)   x11 x22 x33 ...xkk    xii
i 1
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Consumer’s utility function
• Utility function u can be used for modeling choices.
• Utility function indicates the level of satisfaction
obtained from consumption of commodity basket
x = (x1, x2, … , xm)
• Example: Cobb-Douglas utility function
u : R m  R  ,
u (x)    x11  x22  x33  ...  xmm
m
   xii
i 1
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Graph of a function
• The graph of a function can help to visualize functions
of one or two variables.
• The graph is a set that contains pairs of points
consisting of the elements of domain A and the
corresponding values of f:
( x, f ( x) x  A
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Outline
1. Function as a mapping
2. Inverse function
3. Implicit function and correspondence
4. Microeconomic content
– Production function
– Utility function
– Cost, revenue and profit functions
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Inverse function
• If f: A→B is an injection, the inverse function f – 1: B→A
exists.
• Note: The domain of the inverse function is the target
of the original function, and vice versa.
• Given equation y = f(x), the inverse function of f is
obtained by solving x.
• Example: f(x) = 2x + 3
f –1(y) = ½(y – 3).
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Examples of inverse functions
• The inverse of the power function f(x) = xn is the nth
root f(x)-1 = x1/n .
• The inverse of the exponential function f(x) = exp(x) is
the logarithm function f(x)-1 = lnx.
• Examples in economics:
Demand function:
Inverse demand:
q = D(p)
p = D-1(q)
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Exponential and logarithmic functions
Source of picture: Matti Karvonen, lecture notes for Mathematics and Statistics for Managers, 2008
Increasing and decreasing functions
In 𝐑𝟐 , if the graph of a function rises (drops) from left to
right on an interval I, it is increasing (decreasing) on I
Function f is
increasing on
[a,b]
Function g is
decreasing on
[a,b]
Modified from: Matti Karvonen, lecture notes for Mathematics and Statistics for Managers, 2008
Function h is neither
increasing nor
decreasing on R
Increasing and decreasing functions
𝒚 = 𝒇(𝒙)
Increasing
Decreasing
• 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) ≤ 𝒇(𝒙𝟐 )
• 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) ≥ 𝒇(𝒙𝟐 )
Strictly increasing
• 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) < 𝒇(𝒙𝟐 )
Strictly decreasing
• 𝒙𝟏 < 𝒙𝟐 ⇒ 𝒇(𝒙𝟏 ) > 𝒇(𝒙𝟐 )
(-2,2)
y
y
(1,1)
(-2, 2)
(1,-1)
x
x
Inverse function
• Note: Not all functions are invertible: the inverse
function does not necessarily exist.
• To be invertible, a real valued function f must be strictly
monotonic increasing or strictly monotonic decreasing in
its domain.
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Outline
1. Function as a mapping
2. Inverse function
3. Implicit function and correspondence
4. Microeconomic content
– Production function
– Utility function
– Cost, revenue and profit functions
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Implicit function
• Explicit form: 𝑦 = 𝐹 𝑥1 , … , 𝑥𝑛
– y is an explicit function of the xi’s
• Implicit form: 𝐺 𝑥1 , … , 𝑥𝑛 , 𝑦 = 0
– If the equation determines a corresponding value y
for each (xi , … , xi ), it defines y is an implicit
function of the xi’s
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Correspondence
• A correspondence F from set A to set B is a rule that
maps each x in A into a subset F(x) of B
• A correspondence is different from a function in
that a given domain is mapped into a set (not a single
object as in a function)
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Outline
1. Function as a mapping
2. Inverse function
3. Implicit function and correspondence
4. Microeconomic content
– Production function
– Utility function
– Cost, revenue and profit functions
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Cost function
Given input prices w  R k , and technology T the minimum
cost of producing output y is given by the cost function
C : R k m  R  ,
C (w, y )  min w  x (x, y )  T
x
Note: w, y, and T are exogenously given.
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Revenue function
Given output prices p  R m , and technology T, the
maximum revenue obtainable with inputs x is given by the
revenue function
R : R k m  R  ,
R (x, p)  max p  y (x, y )  T
y
Here x, p, and T are taken as given.
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Profit function
Given input and output prices w  R k , p  R m ,
the maximum profit is given by the profit function
 : R k m  R  ,
 (w, p)  max p  y  w  x (x, y )  T
x,y
Note: prices p and w are exogenously given, quantities y
and x are optimized endogenously.
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Next time – Wed 23 March
• Mathematical programming
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