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Unformatted text preview: Elasticities In this handout, we look at an important concept in economics, the concept
of elasticies. FormallyI it is deﬁned as follower Given a function 2: : ﬂash ...,mn)  a
The elasticity of z with respect to m.» = evaluated at the point of interest.
a
(1)
In economics, particuiarly in consumer theory, we usually deal with the following
elasticities of demand function: ' ' Given a demand‘function of a: : a: = f(pm,py, m) (2)
where m = income
13, = price of'good i
 Own elasticity
Pa: 8f
= — — 3
a Cross elasticity
= .131 . 2f.  (4)
m 5;)“
to Income elasticity
m 6
 :5 ' as (53
where =absainte value of 2:2 i.e., = as when :1: 2 0, : *1 when
m < 0.
Examples We now consider some examples. First refer back to Handout #9 where we
deal with the consumer problem when the utility function has the CobbDouglas (0D) form.
maximize may?“ subject to pma: + my: in 55 0:920
0 <1 a<1. We have derived the demand functions for both a: and y, restated below: “la
Pu Pa: Now, computing all three elasticities yield: glam; m: I Own elasticity = 2'23; = ohm :1
:r 6P3: I (Pay;
0 Cross elasticity
=e £120
:1: pg, 0 Income elasticity 3.8m _m pm Note that the results from CD utility function are extremely and deceptively
simple. The demand function is unit elastic and it doesn‘t depend on the price
of the other good, 1;. Do not take this as a general result. This holds for this speciﬁc utility function.
Let's now consider a. different utility function and see how different the result Imight be. Consider maximize min{a:,y}
subject to page: +pyy = m
56' 2 Dry 2 0 To solve this, note that we want as" = y" always. If that is so, we go to
budget constraint and write pzm+pym=m and solve to get: * m
:1: =  . . 7
Pm'i‘py ()
* m l
y = . 8
P1: +Pv () One can verify that
 Own elasticity = weevil = We?“  (oz—ﬁnd = )2 new .( —m j: ._.2.u._
ITI o e 'c' = E“ m i! _
Cross last: my I ((—W) (p,+p., P=+Pv
‘ ‘ — E —]'s——. t
7 0 Income elast1c1t3’ — 3 “1+1,”
t. :1 I‘m. 4”" , A.
’ r l a? '13. an . “an: .t or" ...
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 '06
 MASSON
 Microeconomics, Income Elasticity, Supply And Demand, Cross Elasticity

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