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25

Fig. XVIII

Fig. XIX

Fig. XX

 i L Decrease in Total Revenueed> 1 i ; 0
 Increase in Total Revenue ed< 1 jC-j D -1-

Constant Total Revenue

ed=1

THE DETERMINANTS OF ELASTICITY

What factors influence the elasticity of demand? First of all, the greater the availability of substitutes, the greater will be the elasdcity. Suppose the price of table salt rises; chances are you will not reduce your consumption much because salt has few good subsdtutes. Hence, wed expect your for salt to be quite low, that is to say, inelastic. But while the elasdcity of demand for salt itself may be quite low, that of a pardcular brand of salt will likely be high because the brands are so similar. Thus, even a small percentage 1 increase in the price of the brand youre currently using is likely to drive you into a competing brand. If M so, your demand for your current brand will be very responsive to changes in its price, which is to say, elastic.

Second, the larger the proportion of a consumers income devoted to a particular product, the greater its elasticity, for example a huge increase in the price of paper clips would little affect your consumpdon, while an equal percentage increase in the price of housing probably would. Wed consequently expect your demand for paper clips to be inelastic and your demand for housing to be elasdc.

Third, time usually increases elasticity. Your demand for gas may be almost perfecdy inelastic o\cv a vei-y short period when youre fift) miles from home and prices sharply rise. Over a longer period of time, you can arrange for car pools or take the bus. - a still longer period your gas guzzler can be replaced by a more fuel efficient car, causing a further decline in quantit) demanded. Hence, your responsiveness to an incrctisc in price vdll likely rise as time goes on, i.e., your demand will become more elastic.

Last of all, necessities tend to have inelastic

demands and luxuries elastic ones. For example, ifthe price of necessities such as salt or drinking water rises, you will not respond much, so that your demand would be inelastic.

MEASURES OF ELASTICITY

Consider the demand schedule segment below, ignoring the column marked TR for now.

Price \$4 \$3

Example One Quantity

IE \$16 \$15

Suppose we wish to calculate the Ed- Its clear that the change in quantity is one unit (5-4). But to express that change as a percent, should we use the old quantity, getting a 25% change (1/4), or the new quantity, getting a 20% change (1/5)? Similarly, ifwe use the old price we get a 25% change in price (1/4), but if we use the new price we have a 33% change (1/3).

One answer to this problem is to use average prices and quantities in calculating these percents. Letting Pi be the old price and be the new price, their average would be V9(Pi+ )- Similarly, the average quantity would be /2(Qi + Qa )• Denoting the change in quantity as AQ and the change in price as AP, where the symbol "A" (delta) always stands for "the change in," we can define the arc elasticity of demand as:

(2) ea

Percentage Change in Quantity Percentage Change in Price

V2(Qi + 02) Vo(Pl + P2 )

Working with our example and remembering to ignore the negative sign in the denominator we get:

(5-4) V2(5 + 4)

(3-4) 7 V2(3 -t 4) 9

Since our elasiicit) of 7/9 is less than 1, we know that demand is inelasdc in this range. However, if we just want to know whether demand is elastic or inelastic, there is a much easier way to fmd out. We only need to see how total revenue (TR) responds to the price change. Inspecting the TR column in Example One above,.we see that TR falls as P falls; consequendy demand must be inelastic in this range. This general relationship is illustrated in Examples Two and Three below.

details, but rather to derive some useful properti which can be established best through the point foJ, mula. Although we shall quiz you on these propenicl none of the problems in this book require you to either the basic mathemadcal defmition in the middlJ of formula (3) above or its rearrangement on the riglJ hand side. Note, however, from the right-hand rearrangement that there are only three terms in ihj point formula: P, Q and the reciprocal of the slope« die demand curve, the run over die rise, AQ/AP. ] the next paragraph we furdier collapse the point ela deity formula into only two terms, thereby making] ideal for deriving certain properdes of demand cur With that as a background consider Fig XXI. Suppose we want die at any arbiuary price! corresponding to a quandty sold of Q. Starting fron Pj, where the subscript z reminds us were interested j the price at which the quantity becomes zero, slope, die rise over die run, (AP/AQ) at any arbit pomt on the demand curve equals (Pj - P)/(Q) we ignore die sign. Therefore flipping over to getl run over the rise (AQ/AP) and subsdtudng into right-hand rearrangement of the point elasticity mula (3) repruited below as (4), we cancel die Qs to

Example Two P Q IE \$5 4 \$20 \$4 5 \$20 Conclusion: expenditure is constant, so demand is unit elasdc

Example Three

P - Q IE S5 5 \$25 S4 8 \$32 Conclusion: expenditure rises as price falls, so demand is elastic

A check of both conclusions using the arc formula gives Ed = 1.0 in Example Two and = 27/13 in Example Three. The first is unit elastic and the second elastic as predicted bv the total revenue test.

Sometimes, however, were hiterested in calculating Ed for an extremely tinv change in price, so small that, in effect, weve captured the elasticity of demand at one particular price. In such cases we use what is termed the point elasticity of demand. Using the point formula we have:

percentage Change in Quantit) Percentage Change in Price

AQ/Q

AP/P

P AQ

Q AP

Our purpose in introducing the point elasdcit) formula is not to torture vou with arcane mathematical

(4) Ed

P (

Q AP (Q)

(Q) P

(P,-P) (P,-P)

This makes point elasticity calculations for 1 ear (straight line) demand curves really simple. Allj need to know is the price P which makes Q= 0 and I price P were interested in. For example, repict the linear demand curve of Figure XVII as Figure X>

Fig. XXI

1 Iorces

 Fig. XXII Table II P(\$) Calculating witii the Point Formula -- Elastic Demand 0/(6 - 0) = 0 --------ed = 2~ 2/(6 - 2) = 1/2 3/(6 - 3) = 1 • 3 Inelastic Demand 1 ---- 4/(6 - 4) = 2 Ed = / V 1 4 6/(6 - 6) = oo* "" \ = 0 • te approaches <>° IS?approachesS6 1 0 5 10 15 20 25 Q

d nodng that ?;,= S6, we can get e<i at all prices P ng the demand curve as follows in Table 11;

Since Ed can be calculated in two separate ways, Is natural to ask when the methods will yield equiva-it resiUts. Although we shall not bodier to prove it , the smaller the difference in prices over which arc elasdcity is calculated, the smaller becomes the erence benveen the two elasdcity mea.sures. For very price changes it makes \-irtually no difference lether the arc or poiiu formida is u.sed. However, we not wish to belabor the point. The problems in this ik do not require you to make either point elasticity doiis or arc/point disdnctions.

MPARING ELASTICITIES

Consider the two parallel linear demand rves in Figure XXIll below. It follows from formula that the inner curve is more elasdc than the oiuer any single arbitrary price such as P. For example, m formula (4) the elasticit\- along the inner curve at price P is F/(Pz - ?). while the elasdcity along the iter curve at die same price is P/P* - P). Hence, the ter denominator in the outer curxe assures a aller elasticity calculated at P. In such sitiiadons we >osely say that D] is more elastic than Do, meaning it more elasdc onlv when compared at the .same (non-o) price. We will make extensive use of ibis fact in cdon \T concerning monopoly.

You should also verifv using fonnula (4) that en two straight line curves intersect, as in Figure that the flatter is the more ekistic. when com-d at the same non-zero price, P. Again at P on the at curve, Ed = /( - P) and exceeds Ed = P/(P* - P) the steep one because the denominator is smaller

on the flat cur\e. However, note that point A along the less elasdc curve is more elasuc than point B; Ed at A is about 4 while Ed at is about 1/5. Here were comparing at different prices.

Appearances notwithstanding, no general relationship between flatness and elasdcity exists. For example Figure XXV portrays a much flatter demand curve than does Figure XXVI. Nonetheless, the elas-dcities are identical because the demand curves are idenucal. We merely TranslateH" the dataTrorhpounds in Figure into kilos in Figure XXA-l. You can also verify using formula (4) that all curves hi Figure 1 -are equally elastic when reckoned at the same price P, despite the varnng steepness: Pj and P in the formula are idendcal. Moreover, the straight line demand cur\e back in Figure XXll has a constant slope, but dif-

 Fig. XXill \ Outer Cur»e X Is , \ Less Elastic Q 0 • «

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