ec2104 lecture 2

Quantitative Methods for Economic AnalysisEC2104

Lecture 2August 18, 2015

Outline

Readings: SH, Chapters 6, 7.7-7.12

Notation and Importance of Differentiation Limits Definition and Rules for Differentiation Applications: Growth Rates and Elasticities Continuity Intermediate Value Theorem

Notation for Derivatives

Lagranges notation: Derivative of f (x) is denoted f (x). Also y for y = f (x). Higher order derivatives: f (x), f (x), f (4)(x), . . . , f (n)(x)

Leibnizs notation: Derivative of f (x) is denoted df

dx(x) or

d

dxf (x).

Also dydx

for y = f (x).

Higher order derivatives: d2f

dx2(x),

d3f

dx3(x), . . . ,

dnf

dxn(x)

Newtons notation (only used for time derivatives): Derivative of y = f (t): y Higher order derivative: y

Importance of Differentiation

The derivative gives us a way to find the slopes of functions. This has many many uses both in and out of economics. Rate of Change Marginal Concepts

Graphical Approach: Linear Functions

How do we find slopes? Linear functions: f (x) = ax + b.

We said the slope is a. Why? Slope is 4y4x . If we change x from x1 to x2, f (x) changes from ax1 + b to

ax2 + b. Slope is then:

f (x2) f (x1)x2 x1 =

ax2 + b ax1 bx2 x1 = a

Slope of a line is the same everywhere on the line.

Graphical Approach:Nonlinear Functions

What about for curves? Slope of the curve changes throughout the curve. We want the slope of the curve at a particular point on the

curve.

We are going to find the slope of the line tangent to thecurve at the point.

Nonlinear Functions

-1

0

1

2

3

4

-1 0 1 2 3 4

f(x)

x

a + ha

Nonlinear Functions

Take two points on the function, at x values a and a + h. Draw the line connecting these two points. The slope of this line is:

f (a + h) f (a)h

Gradually reduce h, moving the connecting line. Eventually, as h approaches 0, the connecting line will

approach the tangent line.

This slope then becomes the slope of the tangent line.

Nonlinear Functions

-1

0

1

2

3

4

-1 0 1 2 3 4

f(x)

x

a + ha

Nonlinear Functions

-1

0

1

2

3

4

-1 0 1 2 3 4

f(x)

x

a + ha a + ha

Definition of the Derivative

The derivative of f (x) at x = a, written as f (a), is

f (a) = limh0

f (a + h) f (a)h

Limits

Sometimes, functions are defined around certain points, butnot necessarily defined at those points.

Limits are used to study the behavior of functions aroundthese points.

We say that L is the limit of f (x) as x approaches the value cif, as x gets close to c , f (x) gets close to L.

limxc f (x) = L

This must be true regardless of the direction you approach c . Both one-sided limits must exist and be equal for this limit to

exist.

limxc f (x) = L iff limxc+

f (x) = limxc

f (x) = L

Examples of limits

Let H(t) ={

0, if t < 01, if t 0

limt0

H(t)

t -0.1 -0.01 -0.001 0.001 0.01 0.1H(t) 0 0 0 1 1 1

The left limit is 0 while the right limit is 1. The limit doesnot exist.

limx1

x2 1x 1

x 0.9 0.99 0.999 1.001 1.01 1.1x21x1 1.9 1.99 1.999 2.001 2.01 2.1

As x gets close to 1, x21x1 gets close to 2, but is undefined at

x = 1. The limit is 2.

limt0

H(t)

-1

0

1

2

-1 0 1

H(t)

t

limx1

x2 1x 1

-1

0

1

2

3

-2 -1 0 1 2

f(x)

x

Properties of the Derivative

Power rule: If f (x) = xn, where n is a real number, thenf (x) = nxn1.

Constants: If f (x) = C , where C is a constant, thenf (x) = 0 for all values of x .

Constant multiples: If F (x) = a f (x), then F (x) = a f (x),where a is a constant.

Sums and differences: If F (x) = f (x) + g(x), thenF (x) = f (x) + g (x). Similarly,G (x) = f (x) g(x) G (x) = f (x) g (x).

Products: If F (x) = f (x) g(x), thenF (x) = f (x) g (x) + f (x) g(x).

Self Checklist: Some Examples of Derivatives

f1(x) = 5x f2(x) =

x + 2

f3(x) = x2 + 2x 3 f4(x) = (x + 3)(2x 1) f5(x) = a0 + a1x + a2x2 + . . . + anxn

Quotient Rule

If F (x) = f (x)g(x) , then F (x) =g(x) f (x) f (x) g (x)

(g(x))2.

Example: f6(x) = x55x2+2

Application: Relationship between Average Cost andMarginal Cost

Show that MC exceeds Average Cost iff AC increases asoutput increases

Hint: Differentiate Average Cost wrt output

file:///C|/Users/ECSCYHW/Desktop/mc_ac.gif[16/1/2015 12:07:31 PM]

Chain Rule

If F (x) = (f g)(x) = f (g(x)), then F (x) = f (g(x)) g (x).

Differentiate this: f7(x) = (5x2 + 1)3

Exponential Function

If f (x) = ax where a is a constant, then f (x) = ax ln a. A special case: If f (x) = ex , then f (x) = ex .

Differentiate this: f8(x) = 5ex

Logarithm Function

If f (x) = loga x where a is a constant, then f (x) = 1x ln a . A special case: If f (x) = ln x , then f (x) = 1x .

Differentiate this: f9(x) = ln(x)x

Application: Growth Rates

By definition, Growth Rate of N = g = NN

=dN/dt

N

Now, prove that g = ddt

lnN(t)

Application: Elasticities

In economics, we frequently deal with how certain variableschange in response to changes in other variables.

This is exactly how the derivative works. So, if a $1 increase in the price of a kilogram of coffee reduces

demand for coffee by 500000 kg, we could have found this bytaking the demand function for coffee and differentiating:

dQ

dp=

dD(p)

dp= 500000 kg/$

However, this is sensitive to the units used as well as the typeof good. We cannot compare this to other goods meaningfully.

So, we use relative changes instead: by what percentage doesquantity demanded change when price increases by 1%?

This is known as the price elasticity of demand:

D =%4 in Q%4 in p =

4QQ4pp

=dQ

dp

p

Q

Elasticities

Using this, we can compare the price responsiveness ofdifferent goods to each other.

More generally, we can have elasticities of anything. Theelasticity of f with respect to x is:

ELx f (x) =df (x)

dx

x

f (x)=

d ln f (x)

d ln x

Let us prove that now !

Example: Elasticities

Assume that demand for a product is given by

D(p) = ln (p + 1) + 5

Find the elasticity of D(p) when p = 100?

Continuity

A function f (x) is continuous at x = a if all of the following aretrue:

f (a) exists lim

xa f (x) exists

f (a) = limxa f (x)

Continuity

-1

0

1

2

3

-2 -1 0 1 2

f(x)

x

Differentiability

A function f (x) is differentiable at x = a as long as the limitin the definition of the derivative exists:

limh0

f (a + h) f (a)h

Essentially, the slope of the function from the right and fromthe left must both exist and be equal.

If a function is differentiable at x = a, then it is alsocontinuous at x = a (Differentiability Continuity).

This means, logically, that (Not Continuous NotDifferentiable)

However, the reverse is not true. (Continuity 6Differentiability).

Interpretations: Continuity - graph connects. Differentiability- slopes connect.

Example: Continuity and Differentiability

Determine whether f (x) = |x | is continuous and/or differentiableat x = 0.

f (x) = |x |

-1

0

1

-1 0 1

f(x)

x

Question

Draw a graph of the function f (x) = min(3x+2, 9 4x)where x is a real number

Is the function differentiable? Is the function continuous?

Increasing/Decreasing Functions

We know the derivative gives the slope of a function at anypoint.

Derivative positive at a point function upward sloping function is increasing

Derivative negative at a point function downward sloping function is decreasing

Graph of 13x3 + 2x2 3x + 1

-1

0

1

2

3

4

-1 0 1 2 3 4

f(x)

x

Extension of Limits: LHopital Rule

If f(a)=g(a)=0, and g(a) is non zero:

limxa

f (x)

g(x)=

f (x)g (x)

Example:limx0

ex 1x

Intermediate Value Theorem

Let f be a function continuous in the closed interval [a,b] If f(a) and f(b) have different signs, then there is at least 1 c

in (a,b) such that f(c) = 0.

If f(a) not equal to f(b), then for every intermediate value y inthe open interval between f(a) and f(b), there is at least one cin (a,b) such that f(c) = y.

ec2104 lecture 2

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