Economics 2301

Lecture 15

Differential Calculus

Difference Quotient

argument. its

in change a respect to with changes ,function,

a of value thehow shows , quotient, difference The

f(x)yx

y

Example excise tax revenue

Qt

Qt

t

R

Qt

QttQtQtRRRQtR jijiji

Quotient Difference

demand inelasticPerfectly :1 Case

Example excise tax revenue

tPttP

CP

t

R

tPttP

tPC

tPttP

tttPtttttPPtC

tPttP

tCttPCtttP

tP

tC

ttP

CttR

CtPtP

tCR

2

22

constant. and ratetax price,supply where

Demand Elasticity Unitary :2 Case

Excise Tax Example Continued

Tax Rate

R=30t R=100t/(2+t)

R R/t R R/t

1 30 33.33

30 17.67

2 60 50

30 10

3 90 60

General Form of Difference Quotient

8333.41495

1/545/5415

8333.46/546

1 and 5for /54

generalin that Note

generalIn

ffΔxfxf

fΔxxf

Δxxxxf

Example

ΔxfxfΔxxf

Δx

xfΔxxf

Δx

Δy

Difference quotient for linear and quadratic functions

xccxbx

cxbxaxcxcxcxxbbxa

x

cxbxaxxcxxba

x

y

cxbxay

bx

xb

x

bxaxxba

x

y

bxay

2

2

:function Quadratic

:functionLinear

222

22

2

Change at the margin and the derivative

In general, the difference quotient of a function y=f(x) depends upon the parameters of the original function, as well as the argument of the function, x, and the change in the argument of the function, x. We can, in a sense, standardize the difference quotient by specifying a common x to be used in the calculation of all difference quotients. In calculus, we consider the limit of the difference quotient as x approaches zero. We call the difference quotient as x approaches zero the derivative.

Derivative

exists.limit theprovided

is , as

written is which ,function theof derivative The

lim0 x

xfxxfdxdy

dxdy

f(x)y

x

Derivative Linear function

bb

x

xfxxfdxdy

bx

xfxxf

x

y

bxa

x

x

lim

lim

0

0

thatyfunction linear our for Recall

Derivate of Quadratic function

cxbxccxbdxdy

xccxb

x

22

therefore,2 asquotient w difference the

cxbxayfunction quadraticour for that Recall

lim0

2

Derivative of our tax function for constant elasticity demand

220

2

lim

asfunction w revenue tax for thequotient difference the

function, demand elasticityconstant our for that Recall

tP

CP

tPttP

CPdtdR

hencetPttP

CP

t

R

t

Derivative and Economics

Since the derivative is defined for very small changes in x, it is some times referred to as the instantaneous rate of change of the function.

The mathematical concept of the derivative has a direct correspondence to the economic concept of looking at relationships “at the margin.”

Economic concepts such as marginal revenue, marginal productivity and marginal propensity to consume.

Geometry of Derivatives

The difference quotient R/t was also referred to as the slop of the secant line.

The derivative of a function at a given value represents the slope of a line tangent to that function at that value.

Figure 6.2 Secant Lines and a Tangent Line

Derivative as a function

In many cases, the derivate dy/dx is itself a function of x. i.e. dy/dx=f’(x).

Our constant elasticity tax revenue function

2

asfunction w theof derivative The

asfunction w

demand elasticityconstant afor function revenueOur tax

tP

PCdtdR

tP

tCR

Plot of Tax Revenue and Marginal Tax Revenue

Tax Revenue and Marginal Tax Revenue

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

90.00

0 2 4 6 8 10 12

Tax Rate

Rev

enu

e

R

dR/dt

Geometry of Derivatives

point.

inflectionan is which at zero equals derivative The 4.

. and , See point. extreme global

or localany at zero equalsfunction theof derivative The 3.

zero. is tangent to

line a of slope when thezero equals derivative The 2.

. to range

in the as negative, is tanget toline a of slope when the

negative is ). to range seepositive.(or negative is

itself whether of regardless positive is tangent to

line a of slope when thepositive is derivative The .1

)g(x

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

g(x)

g'(x)

x x

g(x)

g'(x)xx

g(x)g(x)

g'(x)

d

cba

cb

ba

Figure 6.4 A Function and Its Derivative