Description

1
Fun tions of One Variable
Notes 10
1.1 Introduction
ã Functions arise routinely in the study of economic problems.
ã For example, a demand function relates quantity demanded to price.
ã Need to study properties of such functions, such as slope
ã The slope and elasticity measure responsiveness of demand to price variations.
ã A function, f , from a set X to a set Y is a rule associating with each
element of a set X some element in the set Y .
ã This is written f : X → Y .
ã In words, to each

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1
Notes 10
1.1 Introduction
ã Functions arise routinely in the study of economic problems.ã For example, a demand function relates quantity demanded to price.ã Need to study properties of such functions, such as slopeã The slope and elasticity measure responsiveness of demand to price vari-ations.ã A
function
,
f
, from a set
X
to a set
Y
is a rule associating with eachelement of a set
X
some element in the set
Y
.ã This is written
f
:
X
→
Y
.ã In words, to each point in
X
,
f
associates a point,
y
=
f
(
x
) in
Y
.
1
E255-Matrices E255-Matrices
ã The set
X
is called the domain and the set
Y
is called the range.ã The inverse demand function, relates price to quantity demanded,
p
(
Q
).
1.2 Some Examples.
DEMANDã write,
D
(
p
), for the aggregate demand as a function of own price.ã Writing
Q
for the quantity demanded,
Q
=
D
(
p
).ãã With this function, we can consider the impact of price changes on quan-tity demanded.ã Consider a price
p
′
=
p
with corresponding demand
Q
′
=
D
(
p
′
)ã the variation in quantity resulting from the price variation is
Q
′
−
Q
=
D
(
p
′
)
−
D
(
p
)ã The change in aggregate quantity demanded per unit change in price is:
Q
′
−
Q p
′
−
p
=
D
(
p
′
)
−
D
(
p
)
p
′
−
p
ã This gives the slope of the demand function.
***
2
E255-Matrices E255-Matrices
ã Similarly, the percentage change in quantity resulting from a percentagechange in price is given by:
Q
′
−
QQ p
′
−
p p
=
D
(
p
′
)
−
D
(
p
)
D
(
p
)
p
′
−
p
)
p
=
p D
(
p
)(
D
(
p
′
)
−
D
(
p
)(
p
′
−
p
)The negative of this term gives the elasticity of demand.ã Note that the elasticity of demand is computed at a point on the demandcurve: at different points, the value of the elasticity will vary.ã Thus, we may write:
ǫ
(
p
)
=−
p D
(
p
)
(
D
(
p
′
)
−
D
(
p
))(
p
′
−
p
)
***
3
E255-Matrices E255-Matrices
P
RESENT
V
ALUE
ã The present value of a cash ﬂow may be viewed as a function of the in-terest rate.ã A bond is characterized by a face value,
P
, an interest rate,
ρ
, a maturitydate
T
,ã It pays the holder a ﬁxed payment,
f
=
ρ
P
, each time period (such asevery six months or every year), and the face value
P
at the end of the
T
th period.ã Thus, the cash ﬂow generated by the bond is:(
f
,
f
,
f
,...,
f
,
P
)where
f
is paid at the end of periods 1, 2, up to
T
−
1.ã What is the value of this cash ﬂow?ã If the market interest rate is
r
,
x
dollars invested today for one period isworth
y
=
x
(1
+
r
) at the end of the period.ã To have
y
dollars at the end of the period,
x
=
y
(1
+
r
)
dollars should beinvested now.ã How much should one pay for a payoff of
y
dollars in one period fromnow?
***
4

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