Economists often study how demand for a certain commodity such as coffee reacts to price changes. We can ask by how many units such as kilograms the quantity demanded will change per dollar increase in price. In this way, we obtain a concrete number, measured in units of the commodity per unit of money. There are, however, several unsatisfactory aspects to this way of measuring the sensitivity of demand to price changes. For instance, a $1 increase in the price of a kilo of coffee may be considerable, whereas a $1 increase in the price of a car is insignificant.
This problem arises because the sensitivity of demand to price changes is being measured in the same arbitrary units as those used to measure both quantity demanded and price. The difficulties are eliminated if we use relative changes instead. We ask by what percentage the quantity demanded changes when the price increases by 1%. The number we obtain in
S E C T I O N 7 . 7 / W H Y E C O N O M I S T S U S E E L A S T I C I T I E S 229 this way will be independent of the units in which both quantities and prices are measured.
This number is called theprice elasticity of demand, measured at a given price.
In 1960, the price elasticity of butter in a certain country was estimated to be−1. This means that an increase of 1% in the price would lead to a decrease of 1% in the demand, if all the other factors that influence the demand remained constant. The price elasticity for potatoes was estimated to be−0.2. What is the interpretation? Why do you think the absolute value of this elasticity is so much less than that for butter?
Assume now that the demand for a commodity can be described by the function x=D(p)
of the pricep. When the price changes fromptop+p, the quantity demanded,x, also changes. The absolute change in x isx = D(p+p)−D(p), and the relative(or proportional) change is
x
x = D(p+p)−D(p) D(p)
The ratio between the relative change in the quantity demanded and the relative change in the price is
x x
p p = p
x x
p = p
D(p)
D(p+p)−D(p)
p (∗)
Whenp =p/100 so thatpincreases by 1%, then(∗)becomes(x/x)ã100, which is the percentage change in the quantity demanded. We call the proportion in(∗)the average elasticity ofxin the interval[p, p+p]. Observe that the number defined in(∗)depends both on the price changepand on the pricep, but is unit-free. Thus, it makes no difference whether the quantity is measured in tons, kilograms, or pounds, or whether the price is measured in dollars, pounds, or euros.
We would like to define the elasticity ofDatpso that it does not depend on the size of the increase inp. We can do this ifDis a differentiable function ofp. For then it is natural to define the elasticity ofDw.r.t.pas the limit of the ratio in(∗)asptends to 0. Because the Newton quotient [D(p+p)−D(p)]/ptends toD(p)asptends to 0, we obtain:
The elasticity ofD(p)with respect topis p D(p)
dD(p) dp
Usually, we get a good approximation to the elasticity by lettingp/p=1/100=1% and computingp x/(x p).
The General Definition of Elasticity
The above definition of elasticity concerned a function determining quantity demanded as a function of price. Economists, however, also consider income elasticities of demand, when demand is regarded as a function of income. They also consider elasticities of supply, elasticities of substitution, and several other kinds of elasticity. It is therefore helpful to see
how elasticity can be defined for a general differentiable function. Iff is differentiable atx andf (x)=0, we define the elasticity off w.r.t.xas
Elxf (x)= x
f (x)f(x) (elasticityoff w.r.t.x) (1) E X A M P L E 1 Find the elasticity off (x)=Axb(Aandbare constants, withA=0).
Solution: In this case,f(x)=Abxb−1. Hence, ElxAxb =(x/Axb)Abxb−1=b, so
f (x)=Axb⇒Elxf (x)=b (2)
The elasticity of the power functionAxbw.r.t.xis simply the exponentb. So this function has constant elasticity. In fact, it is the only type of function which has constant elasticity.
This is shown in Problem 9.9.6.
E X A M P L E 2 Assume that the quantity demanded of a particular commodity is given by D(p)=8000p−1.5
Compute the elasticity ofD(p)and find the exact percentage change in quantity demanded when the price increases by 1% fromp=4.
Solution: Using (2) we find that ElpD(p)= −1.5, so that an increase in the price of 1%
causes quantity demanded to decrease by about 1.5%.
In this case we can compute the decrease in demand exactly. When the price is 4, the quantity demanded isD(4)=8000ã4−1.5 =1000. If the pricep=4 is increased by 1%, the new price will be 4+4/100=4.04, so that thechangein demand is
D(4.04)−D(4)=8000ã4.04−1.5−1000= −14.81
The percentage change in demand fromD(4)is−(14.81/1000)ã100= −1.481%.
E X A M P L E 3 LetD(P )denote the demand function for a product. By sellingD(P )units at priceP, the producer earns revenueR(P )given byR(P )=P D(P ). The elasticity ofR(P )w.r.t.
P is
ElPR(P )= P P D(P )
d
dP[P D(P )]= 1
D(P )[D(P )+P D(P )]=1+ElPD(P ) Observe that if ElPD(P )= −1, then ElPR(P )=0. Thus, when the price elasticity of the demand at a point is equal to−1, a small price change will have (almost) no influence on the revenue. More generally, the marginal revenuedR/dP generated by a price change is positive if the price elasticity of demand is greater than−1, and negative if the elasticity is less than−1.
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NOTE 1
• If|Elxf (x)|>1, thenf is elastic atx.
• If|Elxf (x)| =1, thenf is unit elastic atx.
• If|Elxf (x)|<1, thenf is inelastic atx.
• If|Elxf (x)| =0, thenf is perfectly inelastic atx.
• If|Elxf (x)| = ∞, thenf is perfectly elastic atx.
NOTE 2 Ify=f (x)has an inverse functionx=g(y), then Theorem 7.3.1 implies that Ely(g(y))= y
g(y)g(y)= f (x) x
1
f(x)= 1
Elxf (x) (3)
A formula that corresponds to (7.3.5) is
Elyx= 1
Elxy (4)
There are some rules for elasticities of sums, products, quotients, and composite functions that are occasionally useful. You might like to derive these rules by solving Problem 9.
Elasticities as Logarithmic Derivatives
Suppose that two variablesxandyare related by the equation
y=Axb (x,y, andAare positive) (5)
as in Example 1. Taking the natural logarithm of each side of (5) while applying the rules for logarithms, we find that (5) is equivalent to the equation
lny =lnA+blnx (6)
From (6), we see that lny is a linear function of lnx, and so we say that (6) is alog-linear relation betweenx andy. The transformation from (5) to (6) is often seen in economic models, sometimes using logarithms to a base other thane.
For the function defined by (5), we know from Example 1 that Elxy =b. So from (6) we see that Elxyis equal to the (double) logarithmic derivativedlny/dlnx, which is the constant slope of this log-linear relationship.
This example illustrates the general rule that elasticities are equal to such logarithmic derivatives. In fact, wheneverx andy are both positive variables, withy a differentiable function ofx, a proof based on repeatedly applying the chain rule shows that
Elxy =x y
dy
dx = dlny
dlnx (7)
P R O B L E M S F O R S E C T I O N 7 . 7
1. Find the elasticities of the functions given by the following formulas:
(a) 3x−3 (b) −100x100 (c) √
x (d) A/x√
x (Aconstant) 2. A study in transport economics uses the relationT =0.4K1.06, whereKis expenditure on
building roads, andT is a measure of traffic volume. Find the elasticity ofT w.r.t.K. In this model, if expenditure increases by 1%, by what percentage (approximately) does traffic volume increase?
3. (a) A study of Norway’s State Railways revealed that, for rides up to 60 km, the price elasticity of the volume of passenger demand was approximately−0.4. According to this study, what is the consequence of a 10% increase in fares?
(b) The corresponding elasticity for journeys over 300 km was calculated to be approximately
−0.9. Can you think of a reason why this elasticity is larger in absolute value than the previous one ?
4. Use definition (1) to find Elxyfor the following (aandpare constants):
(a) y=eax (b) y=lnx (c) y=xpeax (d) y=xplnx 5. Prove that Elx(f (x))p=pElxf (x)(pis a constant).
6. The demandDfor apples in the US as a function of incomerfor the period 1927 to 1941 was estimated asD=Ar1.23, whereAis a constant. Find and interpret the elasticity ofDw.r.t.r. (This elasticity is called the income elasticity of demand, or theEngel elasticity.)
7. Voorhees and colleagues studied the transportation systems in 37 American cities and estimated the average travel time to work,m(in minutes), as a function of the number of inhabitants,N. They found thatm=e−0.02N0.19. Write the relation in log-linear form. What is the value ofm whenN=480 000?
8. Show that Elx
Af (x)
=Elxf (x) (multiplicative constants vanish) Elx
A+f (x)
= f (x)Elxf (x)
A+f (x) (additive constants remain)
HARDER PROBLEMS
⊂SM⊃9. Prove that iff andgare positive-valued differentiable functions ofxandAis a constant, then the following rules hold (where we write, for instance, Elxf instead of Elxf (x)).
(a) ElxA=0 (b) Elx(f g)=Elxf +Elxg (c) Elx
f g
=Elxf −Elxg (d) Elx(f +g)=fElxf +gElxg f +g (e) Elx(f −g)=fElxf −gElxg
f −g (f) Elxf
g(x)
=Eluf (u)Elxu (u=g(x))
S E C T I O N 7 . 8 / C O N T I N U I T Y 233
10. Use the rules of Problem 9 to evaluate the following:
(a) Elx(−10x−5) (b) Elx(x+x2) (c) Elx(x3+1)10 (d) Elx(Elx5x2) (e) Elx(1+x2) (f) Elx
x−1 x5+1