1 March 2017 Mathematical Foundations of economic analysis John Riley

A.3. Derivatives that economists use

A.3.1 Rules for differentiation 1

A.3.2 The exponential function 6

A.3.3 The natural logarithm 9

A.3.4 Exercises (incomplete) 10

A.3.5 Answers (to be added)

1

A.3.1 Rules for differentiation

The chain rule

Economists often work with functions of variables that are themselves functions of other

variables. For example, consider a monopoly selling in a market with demand price function

( ) 50p q q . Then total revenue is 2( ) 50R q q q and marginal revenue is

( ) ( ) 50 2MR q R q q

The output of the firm depends on the size of the work-force z . Suppose that with z units, the

output is 1/212q z . The output is often called the product of the firm and so the derivative is

the marginal product. Using the rule for differentiating a power function,

1 12 21

2( ) 12( ) 6MP z z z

.

If 4z the marginal product is 126(4) 3

and output is

1212(4) 24 4. Then marginal revenue

is ( ) 50 2(24) 2MR q .

In words, output increases three times as fast as the input and revenue increases twice

as fast as output. Then revenue increases 3 2 6 times as fast as the input.

Let ( ) ( ( ))h z R q z be the function mapping input into revenue. We have just concluded

that

( ) ( ) ( )h z R q q z .

This is called the Chain Rule.

Chain Rule: If ( ) ( ( ))h x g f x , where ( )f x and ( )g y are differentiable functions, then

( ) ( ( ) ( )h z g f x f x

2

Proof (informal1): As long as you find the verbal argument convincing (so that you will

remember the Chain Rule) you need not dwell on the following more formal derivation

Consider the mappings ( )y f x and ( )z g y . For any x define

( ) ( )y f x x f x and ( ) ( )z g y y g y .

( ) ( ) ( ( ) ( ( )) ( ) ( )h x x h x g f x x g f x g y y g y

x x x

( ) ( )g y y g y y

x y

( ) ( )g y y g y y

y x

( ) ( ) ( ) ( )

( )( )g y y g y f x x f x

y x

In the limit the left hand side is the derivative, ( )h x and the right hand side is the

product of the derivatives ( )g y and ( )f x .

QED

Example: We defined the natural logarithm ln( )x to be the antiderivative of 1

x .

Consider ( ) ln ( )h x f x . We define ( ) lng y y . Appealing to the Chain Rule,

1 ( )

( ) ( ) ( ) ( ) ( ) ( )( )

f xh x g y f x f x g y f x

y f x

Product Rule: The derivative of ( ) ( ) ( )h x f x g x is

( ) ( ) ( ) ( ) ( )h x f x g x f x g x

1 To make this a formal proof we would need to work with a formal definition of a limit. However the meaning is clear.

3

Proof: ( ) ( ) ( ) ( ) ( ) ( )h x x h x f x x g x x f x g x

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )f x x g x x f x g x x f x g x x f x g x

( ( ) ( )) ( ) ( )( ( ) ( ))f x x f x g x x f x g x x g x

Therefore

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )h x x h x f x x f x g x x g x

g x x f xx x x

.

Finally we take the limit as 0x

( ) ( ) ( ) ( ) ( )h x f x g x f x g x .

QED

Quotient Rule: The derivative of ( )

( )( )

f xh x

g x is

2

( ) ( ) ( ) ( )( )

( )

f x g x f x g xh x

g x

This can be proved by applying the Product Rule to the functions ( )f x and

11( ) ( )

( )k x g x

g x

Inverse function rule

For monopoly production decision, it was helpful to work not with the mapping from

price to quantity (the demand function ( )q p ) but with the reverse mapping from quantity to

price (the demand price function ( )p q ). This function is known as the inverse mapping and is

sometimes written as follows:

1( ) ( )p q q p .

Suppose, that the demand curve is

2( ) 2400 100q p p p .

4

Then the slope is ( ) 100 2q p p . The demand curve is depicted below with price on the

horizontal axis. At 10p the slope is -120.

If the quantity falls 120 times faster than the price rises, then the price falls a fraction

1/120 of the rate at which the quantity rises. Generalizing, if the rate at which the quantity

changes with price is ( )q p then the rate at which price changes with quantity is 1/ ( )q p .

Inverse function rule: If the mapping ( )x g y is the inverse of the mapping ( )y f x , then

( ) 1/ ( ) 1/ ( ( ))g y f x f g y

Using these rules, it is fairly easy to calculate the derivative of almost any function that

you are likely to see in an economics class. But solving for an antiderivative (or “indefinite

integral”) is a much trickier proposition. First of all, many functions do not have an anti-

derivative. Second, even if someone were to tell you that a function did have an antiderivative,

this does not help in figuring out what it is.

Suppose however that the function to be integrated, ( )h x , is the product of two other

functions ( )f x and ( )g x . Suppose also that you know the anti-derivative of ( )f x . Then the

following result that appeals to the Product Rule can be helpful.

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Integration by parts

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

t t

s s

f x g x dx f t G t f s G s f x G x dx , where ( ) ( )G x g x

Proof: First define ( ) ( ) ( )h x f x G x . Appealing to the product rule,

( ) ( ) ( ) ( ) ( )h x f x G x f x g x .

Rearranging terms,

( ) ( ) ( ) ( ) ( )f x g x h x f x G x

Integrating over the interval [ , ]s t

( ) ( ) ( ) ( ) ( )

t t t

s s s

f x g x dx h x dx f x G x dx

( ) ( ) ( ) ( )

t

s

h t h s f x G x dx

( ) ( ) ( ) ( ) ( ) ( )

t

s

f t G t f s G s f x G x dx .

QED

Remark: We use the following short-hand

( ) ( ) ( )t

sh x h t h s

Example: 3

0(1 )

tx

dxx

Define ( )f x x and 3

1( )

(1 )g x

x

. Then 1

2 2

1( )

(1 )G x

x

.

Integrating by parts,

1 12 23 2 2

0 00

1 1

(1 ) (1 ) (1 )

tt tx

dx x dxx x x

6

1 12 22 2

0

1

(1 ) (1 )

tt

dxt x

1 12 22

0

1

(1 ) 1

tt

t x

1 1 12 2 22

1

(1 ) 1

t

t t

A.3.2 The exponential function

The family of functions ( ) xf x a are all defined on the real numbers. The graphs of two

members of this family are depicted in each of the following charts below for positive values of

x .

In the left chart, 1a for both functions so they are both strictly increasing. The right chart

depicts the graphs for two value of 1

1ac

. Since 11

( ) ( )x

c xf x

c these must be

decreasing.

The one very important member of this family is the function with a slope of 1 at 0x .

The slope ( )f x is the limit, as 0h , of

7

( ) ( ) x h xf x h f x a a

h h

Setting 0x , the slope at 0x is the limit of 1ha

h

. We can therefore approximate the

slope by computing this ratio for smaller and smaller values of h . The results from spread-

sheet computations are shown below. Note that the ratio is decreasing so the limiting slope

must be less than 1 when 2a . And for 3a it is pretty clear than the limit must exceed 1.

The next figure shows the computations for 2.71a and 2.72a .

8

Since the sequences are strictly decreasing as h decreases, it follws that the limiting slope must

be less than 1 when 2.71a . We write the value of a for which the slope is 1 as e . In fact, to

three decimal places 2.718e .

Now consider the slope with a e for any x .

0 0 0

( 1) 1( ) lim lim lim

x x h x x h hx

h h h

de e e e e ef x e

dx h h h

Since the limit is 1 it follows that

xxde

edx

.

We conclude with two further useful results.

Proposition: The function ( ) rxh x e has a derivative ( ) rxh x re

Proof: Define ( ) yg y e and ( )f x rx . Then ( ) ( ( ))h x g f x . Appealing to the Chain Rule,

( ) ( ) ( ) ( )rx y rxde h x g y f x e f x e r

dx .

QED

Proposition: The inverse of the mapping xy e is lnx y

Proof: Let ( )x g y be the inverse mapping. Since 0 1e it follows that (1) 0g .

Moreover, the slope of the graph of this function is

xdye y

dx

Then the slope of the inverse function ( )x g y is

9

1

( )dx

g ydy y

.

We have seen this function before. It is called the natural logarithm. Thus the inverse function

is ( ) lng y y

A.3.4 Properties of the natural logarithm

As we shall next show, the function lny x has such two properties that make it one of

the most used functions in economics. Suppose 1 1lny x and 2 2lny x . The inverse of the

natural logarithm is the exponential function. Therefore

1

1

yx e and 2

2

yx e .

Multiplying these,

1 2 1 2

1 2

y y y yx x e e e

The inverse of the exponential is the natural logarithm. Therefore

1 2 1 2lny y x x .

Substituting for 1y and 2y yields the following proposition.

Proposition: The logarithm of the product = the sum of the logs

1 2 1 2ln ln lny y y y

By a similarly short argument we have a second important result.

Proposition: ln lnay a y

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A.3.4 Exercises

Exercise 1: Derivatives

Are all the terms in the second column equal to the derivative of the corresponding function in

the first column.

( )f x ( )f x ?

2xxe 22(1 2 ) xx e

( 2 ) ba x 12( 2 ) ba x

ln(( )( ))a x I px 1 p

a x I px

2 2( ) ( )a x I px 2 2 1

2( ) ( ) ( )p

a x I pxa x I px

Exercise 2: Elasticity of demand

The demand price function is

1

1/

2

( )( ) b

ap q

a q

.

(a) What is the demand function?

(b) What is the derivative of the demand function?

(c) What is the elasticity of demand (i) if 2 0a (ii) if 2 0a ?

Exercise 3: Consumer choice

A consumer has a utility function 1 2( , )U x x . Her income is I . The price of commodity 1 is 1p

and the price of commodity 2 is 2 1p . Substitute for commodity 2 in the utility function.

11

Solve for the utility maximizing choice of commodity 1 if the utility function is (a)

1 2( ) ln(( ) )U x a x x (b) 2 2

1 2( ) ( )U x a x x .

Hint: The Table in Exercise 1 should be helpful.

Exercise 4: Integration

Show that 1 1 1

( )x x a x x a

. Hence solve for the indefinite integral (an antiderivative) of

this function.