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Intermediate Microeconomics

Math Review

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Functions and Graphs

Functions are used to describe the relationship between two variables. Ex: Suppose y = f(x), where f(x) = 2x + 4

This means if x is 1, y must be 2(1) + 4 = 6 if x is 2, y must be 2(2) + 4 = 8

* 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 does the amount of output (y) change as a firm increases the quantity of an input (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.

-4

2

y

6

2

2 4 x

slope = rise/run

=(change in y)/(change in x)

= -4/2 = -2

y

4

2

3 4 x

1-2

slope = rise/run = -2/1 = -2

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Slope

Positive Slope means that the relationship between the two variables is such that as one goes up so does the other, and vice versa.

Negative Slope means that the relationship between the two variables is such that as one goes up the other goes down, and vice versa.

y y

x x

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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

For non-linear relationships, rise/run is a discrete approximation of the slope at any given point. This approximation is better the smaller the change in x we consider.

y

28

4

1 3

y

13

4

1 2

2

24

slope = 24/2 = 12

9

1

slope = 9/1 = 9

x x

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Non-linear Relationships

We can also approximate the slope 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?

Similarly, starting at x = 1, if we increase x by 1 what will be the corresponding change in y?

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.

122

428

2

)1)1(3()1)3(3(

2

)2()21( 22

ff

91

413

1

)1)1(3()1)2(3(

1

)1()11( 22

ff

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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

The derivative is just the limit of this expression as Δx goes to zero, or

We will also sometimes express the derivative of f(x) as f’(x)

x

xfxxf

x

y

)()(

x

xfxxf

dx

xdfx

)()(lim

)(0

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The Derivative

Given y = f(x), where f(x) = 3x2 + 1, what is expression for derivative?

So what is slope of f(x) = 3x2 + 1 at x = 1?

What is slope of f(x) = 3x2 + 1 at x = 3?

How do we interpret these slopes?

y

28

4

1 3 x

slope = ?

slope = ?

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Derivatives

Rules for calculating derivatives - See “Math Review”

Second Derivative - the derivative of the derivative. Intuitively, if the first derivative gives 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.

In other words, second derivative is rate-of-change of the slope.

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Derivatives

y

x

y

x

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Finding maxima and minima

Often calculus methods are used for finding what value maximizes or minimizes a function. A necessary (but not sufficient) condition for an “interior” maximum

or minimum is where the first derivative equals zero.

x* x

y

f(x)

f(x)

x* x

y

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Finding maxima and minima

This means that when trying to find where a function reaches its maximum or minimum, we will often take the first derivative and set it equal to zero. Often referred to as “First Order Condition”

f(x) = 10x – x2

F.O.C.: 10 – 2x = 0

x* = 5

How do we know if this is a maximum or a minimum?

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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.

For example, output is function of labor and capital. How does output change as we increase labor but hold capital fixed?

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?

A partial derivatives of the function f with respect to x is denoted

z

zxf

),(