Transcript Intermediate Microeconomics

Intermediate Microeconomics
Math Review
1
Functions and Graphs

Functions are used to describe the relationship between two
variables.

Ex: Suppose y = f(x), where f(x) = 10 - 2x
 This means
 if x is 2, y must be 10 – 2(2) = 6
 if x is 4, y must be 10 – 2(4) = 2
* This relationship can also be described via a graph.
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Rate-of-Change and Slope

We are often interested in rate-of-change of one variable
relative to the other.

For example, how do profits (y) change as a firm increases
quantity supplied (x)?

This is captured by the slope of a graph.
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Rate-of-Change and Slope

For linear functions, this is constant and equal to “rise/run” or
Δy/Δx.
y
y
2
6
-4
1
4
-2
2
2
2
4
x
slope = rise/run
3
4
x
slope = rise/run = -2/1 = -2
=(change in y)/(change in x)
= -4/2 = -2
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Non-linear Relationships

Things gets slightly more complicated when relationships are “non-linear”.

Consider the functional relationship y = f(x), where f(x) = 3x2 + 1
y
y
slope = 9/1 = 9
slope = 24/2 = 12
28
24
13
9
4
1

4
2
3
x
1
1
2
x
For non-linear relationships, rise/run is just an approximation of the slope at any
given point.

This approximation is better, the smaller the change in x we consider.
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Non-linear Relationships

Analytically:

Consider again the relationship y = f(x), where f(x) = 3x2 + 1

Starting at x = 1, if we increase x by 2 what will be the
corresponding change in y?
f (1  2)  f (2) (3(3) 2  1)  (3(1) 2  1) 28  4


 12
2
2
2

Similarly, starting at x = 1, if we increase x by 1 what will be the
corresponding change in y?
f (1  1)  f (1) (3(2) 2  1)  (3(1) 2  1) 13  4


9
1
1
1

So this functional relationship between x and y means that how
much y changes due to a change in x depends on how big of a
change in x and where you evaluate this ratio.
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The Derivative

As discussed before, we get a better approximation to the relative
rate-of-change the smaller the change in x we consider.

In particular, given a relationship between x and y such that y = f(x) for
some function f(x), we have been considering the question of “if x
increases by Δx, what will be the relative change in y?”, or
y f ( x  x)  f ( x)

x
x

The derivative is just the limit of this expression as Δx goes to zero, or
f ( x)
f ( x  x)  f ( x)
 lim x0
x
x

We will also sometimes express the derivative of f(x) as f’(x)
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The Derivative

Given y = f(x), where f(x) = 3x2 + 1,

f’(x) = (2)3x2-1 = 6x
y

So at x = 1, f’(1) = 6(1) = 6

This means the slope of f(x) = 3x2 +
1 at x = 1 equals 6.

28
slope = 18
Equivalently, this means that at x =
1, y is increasing at a rate 6 times
faster than x.
slope = 6
4

Alternatively, at x = 3, f’(3) = 6(3) = 18

This means the slope of f(x) = 3x2 +
1 at x = 3 equals 18.

1
3
x
Equivalently, this means that at x =
3, y is increasing at a rate 18 times
faster than x.
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The Derivative


This shows that rise/run method
with non-linear functions will
give an approximation of the
slope at any given x*, where
this approximation is essentially
the average slope between x*
and x* + Δx.
Obviously, the smaller the Δx,
the better the approximation, or
the closer the rise/run
calculation will be to the
derivative.
y
28
slope = 12
slope = 9
13
slope = 6
4
1
2 3
x
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Rules for Taking Derivatives

Basic functions:


f’(x) = a/x
f(x) = g(x)h(x)
f’(x) = g’(x)h(x) + g(x)h’(x)
Quotient Rule:


f(x) = a log x
Product Rule:


f’(x) = (b)axb-1
Log functions:


f(x) = axb + c
f(x) = g(x)/h(x)
f’(x) = [g’(x)h(x) – g(x)h’(x)]/h(x)2
Chain Rule:

f(x) = g(h(x))
f’(x) = g’(h(x))h(x)
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Second Derivatives

A Second Derivative is just taking the derivative of the derivative.


Going back to y = f(x), where f(x) = 3x2 + 1,
 f ’(x) = 6x > 0
 f ”(x) = 6 > 0
Intuitively, if the first derivative give you the slope of a function at a
given point, the second derivative gives you the slope of the slope of
a function at a given point.

For y = f(x), where f(x) = 3x2 + 1,
 The positive first derivative tells us that y increases as x increases,
 The positive second derivative tells us that the slope of f(x) increases
as x increases, meaning y increases more quickly as x increases.
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Thinking about Derivatives Graphically
y
f(x) = 3x2 + 1
x
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Thinking about Derivatives Graphically
y
f(x) = 3x2 + 1
f’(x) = 6x
x
x
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Thinking about Derivatives Graphically
y
f(x) = 3x2 + 1
f’(x) = 6x
f”(x) = 6
x
x
x
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Thinking about Derivatives Graphically
Alternatively, consider y = f(x), where f(x) = 10 – 2x0.5.
y
f(x) = 10 – 2x0.5
x
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Thinking about Derivatives Graphically
Alternatively, consider y = f(x), where f(x) = 10 – 2x0.5.
y
x
f(x) = 10 – 2x0.5
f’(x) = -x-0.5
x
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Thinking about Derivatives Graphically
Alternatively, consider y = f(x), where f(x) = 10 – 2x0.5.
y
x
f(x) = 10 – 2x0.5
f’(x) = -x-0.5
f”(x) = x-1.5
x
x
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Partial Derivatives

Often we will want to consider functions of more than one variable.

For example: y = f(x, z), where f(x, z) = 5x2z + 2

We will often want to consider how the value of such function changes
when only one of its arguments changes.
This is called a Partial derivative.

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Partial Derivatives

The Partial derivative of f(x, z) with respect to x, is simply the
derivative of f(x, z) taken with respect to x, treating z as just a
constant.

Examples:
 What is the partial derivative of f(x, z) = 5x2z3 + 2 with respect to x?
With respect to z?

What is the partial derivative of f(x, z) = 5x2z3 + 2z with respect to
x? With respect to z?
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