Introduction to Economics: Elasticty

8/13/2019 Introduction to Economics: Elasticty

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Econ 100

Lecture 7

The sensivity of quantities demanded and supplied tochanges in price (and other factors):

Elasticity


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In our discussion of the demand and supplyanalysis, we already remarked that demand andsupply curves may have very different shapes

from good to good and from market to market.The shape of a demand or supply curve reflects

most of all the degree by which the quantitydemanded or supplied changes in response to a change in price.

We may focus first on demand curves as we talkabout the responsiveness of Q to P.


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If QD changes by a relatively large amount in response toa relatively small change in P, we would have arelatively flat demand curve.

So a relatively flat demand curve shows that the quantitydemanded is relatively sensitive (or responsive) tochanges in the price.

If, on the other hand, QD changes by a relatively small

amount in response to a relatively large change in P,we would have a relatively steep demand curve.

So a relatively steep demand curve shows that thequantity demanded is relatively insensitive (or

unresponsive) to changes in the price.


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Sometimes it is necessary and/or useful to have aquantitative measure of how sensitive QD is with respectto a change in P.

Example. Suppose the government is worried about theextensive use of cigarettes in the country and consideringthe imposition of a sales tax on the product to cause theprice of cigarettes to increase and quantity demanded andsold to decrease. But what should the amount of the taxper unit (pack) of cigarettes be?

If QD of cigarettes is relatively sensitive to P, a relativelysmall tax per unit may be sufficient to bring about a certaindecrease in the quantity of cigarettes used. But if QD isrelatively insensitive, even a high tax will not be sufficient toachieve the targeted change in Q.


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How to define a measure of the sensitivity (orresponsiveness) of QD with respect to P?


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Consider two demand curves,

one relatively flat,

the other one relatively steep,

that may each be the demand curve for a certain good.

Since each of these may be the demand curve for thesame good, they can be drawn on the same diagram.

Now suppose the price is the same along both curvesand then it changes by a certain amount. Let thechange in price be denoted by P (note that P ispositive if P increases and negative if P decreases).


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If the price decreases (and P is negative),

QD will increase by the amount Q.

Since Q will be larger along the flatter curve than it will

be along the steeper curve,we can also say that (the absolute value of) the ratio

Q/P will be larger along the flatter curve,

or (the absolute value of) the ratio P/Q will be smalleralong the flatter curve.

Does this suggest a measure of sensitivity of Q withrespect to P?


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The ratio P/Q is (approximately) equal to theslope of the D curve at any point along the curve(for small P and Q).

This suggests that we use the slope of the Dcurve at any point as a measure of howresponsive QD is to changes in P.


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But the slope (or P/Q approximately) is not agood or reliable measure of sensitivity of Q withrespect to P

for two reasons:(1) Slope changes as units in which Q is measured are

changed,

(2) we can not compare the sensitivities of QD withrespect to P of two different goods (again becauseunits of Q are different for two different goods).

A few remarks about these follow.


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Consider first the dependence of the numerical value ofthe slope to the choice of units.

Suppose the quantity of a certain good is measured inkgs and

a 1 TL/kg decrease in P causes a 5000 kg increase inQD.

Then

Slope of D curve = 1 (TL/kg) / 5000 kg = 0.0002

The absolute value of this is small and the D curveappears rather flat.


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Now let Q be measured in tons. Since 1 ton = 1000 kgs,

P = 1 TL/kg = 1000 TL/ton

Q = 5000 kgs = 5 tons

P/Q = 1000 TL/ton / 5 tons = 200

The absolute value of this is large and the D curveappears rather steep.

Now which one is the case: Is this D curve flat or steep?Is QD sensitive or insensitive to changes in P?


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Consider also the use of slope in comparing thesensitivities of quantities demanded wrt to changes inprice for two different goods, such as wheat and cars.Quantity of wheat is measured in (say) tons of wheat

and quantity of cars is measured in units of cars.Suppose market studies give the slopes of D curves as

Wheat: 200 TL/ton per ton of wheat

Cars: 500 TL/car per car

Is D for wheat more or less sensitive than D for cars?What would it mean to compare a number in TL/ton perton of wheat with a number in TL/car per car anyway?


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Because of such difficulties, slope of the D curveis not very satisfying as a measure of thesensitivity of QD with respect to changes in P,

and a better measure is needed. And since these difficulties are rooted in the fact

that the numerical value of the slope dependson the choice of units, we may reasonably thinkthat the measure we need should beindependent of the choice of units.

What would you suggest?


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Elasticity

Elasticity measures the percentage change in QD thatoccurs as a result of a unit percentage change in P, or

Elasticity = % QD / %P

To be precise, it is called the price elasticity of demand. Suppose the price elasticity of demand for a certain good

is equal to 2.75. What does this mean?

This means that, if the price of the good increases by 1% ,quantity demanded of that good will decrease by2.75% .

Furthermore, we can calculate that, if P increases by 4%,QD will decrease by 4% x 2 .75 = 11%.


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Finally, if P was 20 TL/ton and QD was 500 tons before P(and hence Q) changed, we can calculate the changesin price and quantity as

P = 4% x 20 TL/ton = 0 .80 TL/ton

QD = 11% x 500 tons = 55 tons


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Recall that a percentage changes in Q and P are definedas

%Q = Q / Q

%P = P / P If Q is measured in kgs, for example, the units of the first

of these are (kg / kg) which is unitless, and that of thesecond are (TL/kg) / (TL/kg), also unitless.

Therefore, elasticity itself is a unitless number.


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Therefore, price elasticity of demand will be a numbersuch as 0.5, 1, 2.75, and this number (or thenumerical value of elasticity) will remain unchangedeven if the units of Q are changed.

It is also possible to compare the sensitivity of QD of acertain good with that of QD of another good becausetwo pure numbers can be compared.

Suppose that P elasticity of D for wheat is 0.5 whereasP elasticity of D for cars is 2.75. This will mean thatQD of cars is more sensitive to a change in P of carsthan QD of wheat is to a change in P of wheat.


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Note that, since for a large majority of goods P and QDwill move in opposite directions ,

P and QD that appear in the ratio giving the elasticitywill have opposite signs ,

and, consequently, the price elasticity of demand will bea negative number.

Since it is expected that elasticity will be negative for a

good unless it is explicitly mentioned that the D curveis not negatively sloped, it has become customary todrop the minus sign as price elasticities of demand arereported.

In the rest of this lecture we will follow this practice and...


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give price elasticities of demand as positive numbers.

Therefore, when the price elasticity of demand for a goodis given as 3 instead of 3, it should automatically beunderstood that what is meant is that QD decreases by 3% if P increases by 1% for that good.

Before we begin to talk about other aspects of thismeasure called elasticity, it will be useful to talk aboutwhat factors the price elasticity of demand may bedetermined by.


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Factors affecting the price elasticity ofdemand

The following are among the most important factors thataffect the price elasticity of demand for a specificgood (or service):

Availability of close substitutes Necessities versus luxuries Definition of the market Proportion of income devoted to the product Time horizon

So that...


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(price) elasticity (of demand) tends to behigher...

if close substitutes are available for luxuries than it is for necessities if the good is defined more narrowly if the proportion of income devoted to the

good is larger in the long run than in the short run


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Let us now go back to the definition of (price) elasticity(of demand):

Elasticity = % QD / %P

or, since %Q = Q / Q and %P = P / P, Elasticity = (Q / Q) / (P / P)

or

Elasticity = (Q / Q) x (P / P) which may be rewritten as

Elasticity = (P / Q) x (Q / P)


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One can recognize the ratio Q / P in the lastexpression as the inverse of P / Q which will beapproximately equal to the slope of the demand curveif the changes in P and Q are small.

Therefore, elasticity can be written as

E = (P / Q) x (1 / s)

where E denotes the elasticity and s the slope of the

demand curve.Since not only P or Q but also s will in general change

along a curve, elasticity will not remain constant alonga demand curve except for some special demand

curves.


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This fact creates some difficulties in the use andcalculation of elasticities.

First, about the use of elasticity in calculating the changein Q as a result of a change in P. Suppose you knowthe slope of a demand curve at a point with a given Pand Q. Since

P / Q s or Q (1/s) x P

which can be rewritten asQ / Q (P / Q) x (1/s) x (P / P)

we have

%Q E x %P


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Therefore, using the elasticity calculated by using theformula E = (P / Q) x (1/s) will enable you tocalculate the percentage change in Q onlyapproximately.

And/But the smaller the change in P or the less curvedthe demand curve, the better will the approximation be.

Second, about the calculation of the elasticity itself.

Suppose you find that P and Q will move from a point Aon a demand curve to another point B on the samedemand curve.


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Calculation of elasticity by using A as the initial point willin general give a different result than the calculationthat is done by using B as the initial point.

Starting from A: E = (PA / QA) x (Q/P)

Starting from B: E = (PB / QB) x (Q/P)

For this reason, some researchers prefer to use anotherelasticity formula which is called the midpoint

elasticity (or arc elasticity): E = (Pm / Qm) x (Q/P)

where Pm is the arithmetic average of PA and PB andQm is the arithmetic average of QA and QB so that....


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the point at which P = Pm and Q = Qm is the midpointbetween points A and B (that is, the midpoint on astraight line connecting A and B).

Midpoint elasticity is thought to be useful or reliablebecause it gives you the same elasticity value betweentwo given points on a demand curve regardless of thedirection of the change along the curve.


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Classifying demand curves according totheir elasticity

We will now see different demand curves and comparethem with respect to their elasticity.

There will be five demand curves. Let us first considertwo extreme ones:

QD extremely unresponsive to P QD extremely responsive to P


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One extreme: Totally unresponsive Q

When QD is extremely unresponsive to P, it will notchange no matter how large P is.

This means that

Q = 0 or %Q = 0 and therefore E = 0

The graph of demand will be a vertical line.

Such a demand curve is designated as perfectlyinelastic .

This is the limit case for steeper and steeper demandcurves.


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The other extreme: Infinitely responsive Q

When QD is extremely responsive to P, it will change bya very large amount no matter how small P is.

This means that

Q = or %Q = and therefore E =

The graph of demand will be a horizon tal line.

Such a demand curve is designated as perfectlyelastic .

This is the limit case for flatter and flatter demand curves.


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Cases in between

All the other demand curves will be in between these twoextremes.

Those that are relatively steep and close to a vertical linewith zero elasticity will have small elasticities and willbe called inelastic,

And those that are relatively flat and close to a horizontalline with infinite elasticity will have large elasticities andwill be called elastic.

Then there must be a certain demand curve that is juston the border that separates inelastic curves fromelastic ones. What is the elasticity of that curve?


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We take E = 1 to be the demand curve thatseparates elastic from inelastic demand curves.So,

E < 1 corresponds to % Q < % P, inelastic D, E > 1 corresponds to % Q > % P, elastic D. And a D curve with E = 1 is called unit -elastic.

But why should a demand curve with unit elasticitybe the one between elastic and inelasticcurves? Why not take E = 5 for example?


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The answer is as follows. Consider first a market inequilibrium, so QD = QS. Therefore,

P x QD = P x QS

orTotal expenditure = Total revenue

(paid by buyers) (received by sellers)

Therefore we can takeTotal revenue = P x QD.

Now... if E = 1, then...


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QD will decrease (or increase) by the same percentageas P increases (or decreases), and, consequently, totalrevenue will remain approximately unchanged.

Example. Suppose E = 1 and P increases by 4%. ThenQD will decrease also by 4%. This means that P willrise to 1 .04 times its initial value and QD will fall to 0 .96times its initial value, and, consequently, total revenuewill change to 1 .04 x 0 .96 = 0 .9984 times its initial

value, which means that total revenue will remainunchanged because 0 .9984 1.

In other words, if E = 1 for a demand curve, PxQ will beconstant along that curve.


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What about the total revenue along other demandcurves? (Let us denote total revenue by R.)

If the demand curve is inelastic, P and R will move in thesame direction. In other words,

An increase in P will cause R to increase

and a decrease in P will cause R to decrease.

Think in terms of the extreme case: If D curve isperfectly inelastic and E = 0, Q is constant, and as Pdoubles, R will also double.


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If, on the other hand, the demand curve is elastic, P andR will move in opposite directions. In other words,

An increase in P (hence, a decrease in Q) will cause Rto decrease

and a decrease in P (hence, an increase in Q) willcause R to increase.

You can think of this case as Q and R moving in the

same direction. In terms of an extreme case: If Dcurve is highly elastic so that E =100, as P decreasesby 1% it (stays roughly constant), Q will increase by100% (it will double). As P is (roughly) constant and Qdoubles, R will also (roughly) double.


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Example. Consider farmers producing agriculturalproducts, say, wheat. Suppose one year the weathergoes unusually well in the entire country and thequantity produced of wheat is unusually large. This

sounds like good news for farmers, but is it really so?Quantity produced being large at the end of a season means that

supply of wheat increases and, as a result, equilibrium price ofwheat decreases (along the demand curve). This could still begood news for the farmers because their total revenue couldincrease if the demand for wheat were elastic. But wheat, likeother necessities, has an inelastic demand and consequentlythe total revenue decreases when production of wheatincreases. Good weather may be bad news for farmers.


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Now recall two of the things we have seen so far:(a) In general, elasticity changes along a demand curve

(so E may be < 1 over some portion of a demand curvewhereas it is > 1 over some other portion of the samecurve), and

(b) As P rises, total revenue rises also if E < 1 but it fallsif E > 1.

If we put these two together, we arrive at the conclusionthat as P rises, total revenue may increase for somerange of P and it may decrease for some other rangeof P. Let us see an example...


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A linear demand curve

Elasticity varies between zero and infinity over a lineardemand curve.

At the Q intercept (when P = 0), E = 0

At the midpoint, E = 1 At the P intercept (when Q = 0), E =

As P changes, E will change continously so that Between Q intercept and midpoint, 0 < E < 1

(inelastic) Between P intercept and midpoint, 1 > E < (elastic)


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It follows that, as P falls starting from the P-intercept ofthe D curve (as Q rises starting from 0), R will first rise(until the midpoint) and then fall (after the midpoint).

If we combine this with the fact that R = 0 both

at the Q intercept (because P = 0) and

at the P intercept (because Q = 0),

we can draw a sketch of the graph of how the revenuechanges as P and Q vary.


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We can now change the variable in the denominator ofthe ratio giving the elasticity and define otherelasticities of demand, for example:

Income elasticity of demand

E = % QD / %I

where I denotes income, Cross-price elasticity of demand

E = %QD of good A / %P of good B

where B is one of the other goods and can be either asubstitute or a complement.


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Income elasticity will be Positive for normal goods Negative for inferior goods

and Smaller for necessities Larger for luxuries

Cross-price elasticity will be Positive for substitutes Negative for complements

l f l


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Elasticities of supply

We can also change the variable in the numerator of theratio giving the elasticity to quantity supplied of a goodand define elasticities of supply. For example, if wekeep identifying the variable in the denominator with

the price of the good, we obtainPrice elasticity of supply

E = % QS / %P

or the percentage increase in QS per unit percentageincrease in P.


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Price elasticity of supply measures the responsiveness ofquantity supplied of a good by producers or sellers orfirms to changes in the price of the good.

The easier it is to increase the amount of the goodproduced or available for sale, the higher will be thisresponsiveness.

Hence an important factor affecting the price elasticity ofsupply is the time horizon because as time passes it iseasier for existing producers to acquire and allocatemore resources (land, machines, factory buildings) tothe production of this good and for more producers toenter the market.

Classifying supply curves according to their


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Classifying supply curves according to theirelasticity

We will now see different supply curves and comparethem with respect to their elasticity.

There will be five supply curves. Let us first consider twoextreme ones:

QS extremely unresponsive to P

QS extremely responsive to P

O t T t ll i Q


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One extreme: Totally unresponsive Q

When QS is extremely unresponsive to P, it will notchange no matter how large P is.

This means that

Q = 0 or %Q = 0 and therefore

E = 0

The graph of supply will be a vertical line.

Such a supply curve is designated as perfectlyinelastic .

This is the limit case for steeper and steeper supplycurves.

Th th t I fi it l i Q


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The other extreme: Infinitely responsive Q

When QS is extremely responsive to P, it will change bya very large amount no matter how small P is.

This means that

Q = or %Q = and therefore

E =

The graph of supply will be a horizon tal line.

Such a supply curve is designated as perfectly elastic .

This is the limit case for flatter and flatter supply curves.


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We take E = 1 to be the supply curve thatseparates elastic from inelastic supply curves.So,

E < 1 corresponds to % Q < % P, inelastic S,E > 1 corresponds to % Q > % P, elastic S.

And a S curve with E = 1 is called unit -elastic.


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One can verify that a linear supply curve with E =1 will pass exactly through the origin (it will haveneither a positive Q-intercept nor a positive P-intercept), regardless of its slope.

A linear supply curve with E < 1 will have apositive Q-intercept, and

A linear supply curve with E > 1 will have apositive P-intercept.


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In general, elasticity will change along a supplycurve. One example is a supply curve which isrelatively flat (so, E > 1) for relatively smallvalues of Q and relatively steep (E < 1) forrelatively large values of Q.

Such a supply curve reflects the fact that it isrelatively easy to increase production when the

production is well below capacity but it becomesharder and harder to increase it when thecapacity limit is reached.

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Introduction to Economics: Elasticty