Download E3 Elasticity

Elasticity
John Eckalbar
For many practical problems, it matters a lot just how responsive buyers are to changes in P.
Examples from class: effect of tuition increase on enrollment, effect of oil supply interruption on
price of oil.
When you look at the graphs of these cases, it looks like slope D is critical, but it’s not quite that
simple. Look at the egg example given in class.
What we use to measure responsiveness is “elasticity.” The elasticity of demand, ,d is
where Zb is the “base” value of Z.
Note: ,d is greater than or equal to zero. And large values of ,d mean the buyers are
responsive to price, or demand is elastic.
It is important to note that generally speaking, entire demand curves are neither elastic nor
inelastic. As a rule, elasticity varies along a demand curve. For straight-line demand curves,
there are two polar exceptions to this statement.
A vertical demand curve is everywhere “perfectly inelastic,” meaning ,d = 0.
0 < ,d < 1....we say D is “inelastic.”
,d = 1 ......................D is “unit elastic.”
1 < ,d < 4, we say D is “elastic.”
,d 6 4...D is “perfectly elastic.” In this case D is horizontal.
Real world practicing economists compute elasticities all the time. Your book reports on some
numerical values, and I’ll give more in class. For example, the elasticity of demand for
cigarettes is estimated to be 0.4... So D is “inelastic.” This means that if you use a tax to jack
up the price of cigarettes by 10%, Qd will drop by 4%. That can be important to know.
Elasticity can be estimated a variety of ways. Here’s one.
Suppose we have the following D.
We need to compute the %)Qd, and %)P, and then divide %)Qd by %)P.
Look at the data and compute %)Qd. If Q went from 200 to 280, we would call that a 40%
increase, since (80/200) x100 = 40. In this case we are using Q = 200 as our base quantity. But
if Q went from 280 to 200, we would call that a 28.57% drop in Q, since (80/280) x100 = 28.57.
But on a demand curve, it is not as if “first we were here, then we were there.” There is no
natural base point. So what we do if we are using data from two points on D is use the average P
or Q as the base. We don’t use X and we don’t use Y, we use the average of the two. In effect
we are computing elasticity with point A as our base. Here’s the computation:
So in the neighborhood of point A, demand is elastic.
Elasticity has an important relation to total revenue, TR = P x Q.
If demand is elastic, when we cut P a little, Q rises a lot, and that increases TR.
If demand is inelastic, when we cut P a lot, Q rises a little, and TR falls.
It would be nice if we could use graphs rather than math, and if we do a little math now, we can
then use graphs when we talk about elasticity. That seems worthwhile to me.
The connection between ,d and the slope of D.
To begin, the slope of D is )P/)Q. Notice that this is not the same as ,d, but it is contained in
the formula. Here’s how:
Notice that ,d is the product of 1/Slope D and Pb/Qb. This has two important implications:
Fact 1: If we compute the elasticity at two different base points on a linear demand curve, the
elasticity will be different at the two points, and elasticity will be higher at the point with the
higher price. For example, if we compute the elasticity in the neighborhood of point 1, we will
get a higher elasticity than we would get if you computed elasticity in the neighborhood of point
2. This is why linear demand curves generally
don’t have a single elasticity. Instead,
elasticity varies along the linear demand
curve.
(Proof: The slope is the same at points 1 and
2, but Pb/Qb is higher at point 1.)
Fact 2: If two linear demand curves intersect, then in the neighborhood of the point of
intersection, the flatter one is more
elastic. (Proof: At the point where the
two curves meet, Pb/Qb is the same,
but the flatter curves has a lower
absolute value for Slope D, so it has
a higher absolute value for 1/Slope
D, and therefore a higher elasticity.
The gas consumption example:
country
gas/person/year
P
USA
484
.95
Sweden
221
2.81
Italy
146
3.96
Japan
133
3.47
Denmark
172
3.47
Britain
175
2.52
Draw a D curve through the data.
Its slope is -.007168, so the
elasticity near the US point is
(1/.007168)(.95/484) = 1.4. This
is very crude, but it give you some
idea of how the numbers are
found.
What determines elasticity?
Availability of substitutes
size of expenditure
time: cigarettes are estimated to have elasticity of .4 in short run and .75 in long run
You should read about elasticity of supply, income elasticity, and cross elasticity on your own.