Methods of Economic Analysis

8/11/2019 Methods of Economic Analysis

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EC115 - Methods of Economic AnalysisSpring Term - Lecture 1

Elasticity

Renshaw - Chapter 9

University of Essex - Department of Economics

Week 16

Domenico Tabasso (University of Essex - Department of Economics)

Lecture 1 - Spring Term Week 16 1 / 26
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Hello!

Domenico Tabasso

Room 5.408 (5th floor, Economics building)

Office Hour: Wednesday 2-3pm

E-mail: [email protected]

Reminder: Mid-Term Test, Thursday 19 February 2009


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Main Topics for this Term

1 Elasticity (last topic for functions of one variable)

2 Functions of two variables:

Definition Differentiation Maximisation Economic Examples

3 Constrained Optimization


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Introduction

Our starting point is a simple function y= f(x).

The elasticityof a function is a way to measure the relation

between a proportionate change in xand the resultingproportionate change in y.

The word proportionateis extremely important.

Domenico Tabasso (University of Essex - Department of Economics)Lecture 1 - Spring Term Week 16 4 / 26
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Absolute, proportionate and percentage change - 1

Our initial weekly income is Y0= 200. Then the incomegoes up to Y1= 240. We can then define three types ofchanges:

1 Absolute Change:Y Y1 Y0= 240 200 = 40 per week;

2 Proportionate Change:YY0

= 40200

= 15 (= 0.2);

3 Percentage Change:Prop. Change100 = Y

Y0 100 = 1

5 100

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Absolute, proportionate and percentage change - 2

In general, the proportionate change is more informativethan the absolute one.

Important note: the proportionate change is pure number,

it does not depend on the units in which the variableexperiencing the change is measured.

Rule: If any variable

ychanges by a finite amount

y, theproportionate change in y is measured by y

y0wherey0 is

the initial value ofy.

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The elasticity of supply

Consider a supply function of the form: q= f(p). Thequestion that we would like to answer is:How is the quantity supplied by the producers going tochange when there is a change in the price they receive?

The elasticity will answer this question in terms ofresponsivenessof the supplied quantity to the change inprice.We will discuss two different definitions of elasticity:

Arc elasticity;

Point elasticity.

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The elasticity of supply - The arc elasticity - 1

The arc elasticity of supply (denoted ass) is defined as:

s = percentage change in quantity suppliedpercentage change in price

(1)

Playing a bit with this equation we can get:

s = proportionate change in quantity supplied 100

proportionate change in price 100(2)

Since the 100 in the numerator and in the denominatorcancel out we can write the previous equation as:

s =q/q0p/p0

(3)

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Since:

s =q/q0

p/p0(4)

is the ratio of two proportionate changes we call it theratio ofproportionate changeofyand measures the proportionate change inqper unit of proportionate change in p.

Example: the price of beer is p0 = 2and the suppliers sell 500millions pints. The price then rises to p1 = 2.5. After this changethere is an increase in production of 50 millions pints. Can wecalculate the elasticity of the supply (i.e. the rate of proportionatechange in q)?

s =q/q0p/p0

=50, 000, 000/500, 000, 000

0.5/2 =

0.1

0.25= 0.4 (5)

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The elasticity of supply - The arc elasticity - 3 - A graph

0

D

Ap0

C

q0 q1q

p1

p

E

F

GH

The arc elasticity of supply is

B

FGAB

ADGH

pp

qqS=

0

0

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In eq. 4we defined the elasticity of supply as

s =q/q0p/p0

.

This equation can be rewritten as :

s =q

q0

p0

p. (6)

This formulation is more useful when we try to compare thearc elasticity with the point elasticity.

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The elasticity of supply - The point elasticity

In general the arc elasticity is not a very precise method formeasuring proportionate changes (unless the function weredealing with is linear). Hence in economics we tend to relymore on the point elasticity, that we define as:

s = limp0

q

q0

p0

p

dq

dp

p0

q0(7)

which can also be expressed as:

s dq/dp

q0/p0

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Relations between arc and point elasticity - A graph

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Relations between arc and point elasticity - 2

Summarising:

1 Arc elasticity: s = qq0

p0p

.

2 Point elasticity: s = dqdp

p0q0

Note that in the case of linear equations, the two formulasare identical since q

p and dq

dp coincide.

The point elasticity should only be used in case of small

changes in p. Nonetheless, in economics the point elasticityis the most commonly used measure of elasticity, becuaseits generally easier to calculate that the arc elasticity.

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An example

Given the supply function: q= 10p3 10001

find the arc elasticity when pincreases from 10 to 11;2 find the point elasticity when p= 10

Since p0 = 10 from the equation we obtain:q

0= 10, 000 1, 000 = 9, 000.

Furthermore, p1 = 11, hence q1= 12310 and so: p= 11 10 = 1andq= 12310 9000 = 3310.

We have all the elements for calculating the arc elasticity:

s =q

q0

p0

p=

3310

9000

10

1 = 3.678

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An example - 2

How about the point elasticity?

Givenq= 10p3 1000 we havedqdp= 3 10 p31 = 30p2.

So, following the definition, when p= 10 we have:

s = dqdp p0

q0= (30 (10)2)

dq/dp

p0/q0

109000 = 3.333

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The elasticity of demand - The arc elasticity

Given a generic demand function q= g(p), the elasticity of

demand measure the responsiveness of the quantity of gooddemanded by buyers to a change in the price they have topay. Just as for the supply function we define the arcelasticityof demand as:

d =q

q0

p0

p=

proportionate change in quantity demanded

proportionate change in price(8)

or equivalently we can write:

d =q

p

p0

q0


h l f d d h l h
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The elasticity of demand - The arc elasticity - A graph


Th l i i f d d Th l i i 2
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The elasticity of demand - The arc elasticity - 2

Note: The elasticity of demand is usually negative (if theprice of a good goes up, the quantity demanded should godown). Very often economist prefer to express the elasticityas a positive number so that they redefine it either with anextra minus sign in front or expressing it in absolute values.

Value of arc Implied absolute Terminologyelasticity, d elasticity,

dd < 1

d>1 Demand is elastic

d > 1d


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The elasticity of demand - The point elasticity

Just as for the case of the supply function we can also

define the point elasticity of the demand function, which isof course given by:

s =dq

dp

p0

q0(9)

or alternatively:

s =dq/dp

q0/p0

The point elasticity is more commonly used than the arcelasticity and it is the definition we will refer to when wetalk about elasticity.


Th l i i f d d A h
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The elasticity of demand - A graph


Th l ti it f d d A i t t l ifi ti
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The elasticity of demand - An important clarification

It is tempting to assume that the elasticity of demand of alinear function is constant. This is wrong!

||=

Elastic20 >1

Elastic

15||=1

10

||


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The relationship between the elasticity of demand andmarginal revenue

Rule:

Given a certain demand function p= f(q)the followingrelation between marginal revenue and the elasticity ofdemand function can be established:

MR= p1 + 1D

(10)Furthermore it can be proved that:

1 If marginal revenue is positive, than the demand iselastic

2 If the demand is elastic, then the marginal revenue ispositive


The relationship between the elasticity of demand and
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The relationship between the elasticity of demand andmarginal revenue - A graph


The elasticity of demand under perfect competition
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The elasticity of demand under perfect competition

So far we have assumed that the demand is sloped downward. In caseof perfect competition, though, the demand is horizontal at the levelof the market price, p. In this case the slope of the function isdp

dq= 0, and dq

dpdoes not exist, so calculating the elasticity seems to

be impossible. Nonetheless, using the limits we can still say that asdpdq 0 then dqdp andD. Note that this implies that

in perfect competition:

MR= p

1 +

1

D

= p

1 +

1

= p (1 0) = p

which is the standard result we always obtain in perfect competition


Elasticity Generalisations
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Elasticity - Generalisations

We have focused on the elasticity of demand and supply.Nonetheless the concept of elasticity can be applied to anyfunction. In general the rule is:For any function y= f(x)the elasticity ofywith respect

to x is: y,x=dy

dx

x

y

.

So for example the elasticity of the total cost function TC= f(q)willbe: TC = dTC

dq

q

TC

Furthermore it is useful to note (but we dont prove it!)that:

y,x=dy

dx

x

y=

dln(y)

dln(y)

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Methods of Economic Analysis