Download 1. Consumer Theory (Cont.) 1.5- Consumer Choice 1.6

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1. Consumer Theory (Cont.)
1.5- Consumer Choice
1.6- Demand
1.7- Market Demand
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• Mandatory: Varian, H., Intermediate Microeconomics, 5th
edition, Norton, 1999. Chapters 5-6, 15.
1
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After going through the consumer’s budget constraints and
preferences, we now address the issue of optimal choice, which
entails answering the question of how do consumers choose the
best bundle of goods they can afford.
7KH2SWLPDO%XQGOH
• The answer is simple! The consumer chooses the bundle
within the budget set that is on the indifference curve that
provides the highest level of utility.
• If we keep the assumption of monotonic and convex
preferences, we can restrict our attention to the bundles
that lie
RQ
the budget line, because anywhere else in the
budget set is not consistent with optimising consumer
behaviour.
• Thus, to find the RSWLPDO consumption bundle, denoted by
0
*
[
1
5
, [2* , we just start from one extreme of the budget line
and move along until the highest indifference curve is
found.
2
X2
Optimal Choice
X1
• The bundle
0
*
[
1
5
, [2* is optimal because all better bundles
lie beyond the budget line, which means that the consumer
cannot afford them.
• Notice that at the optimal bundle the indifference curve is
WDQJHQW
to the budget line. That must be the case, because
if budget line crossed the indifference curve at the
optimum, it would mean that there would be another point
in the budget line that would be preferred and so that
bundle would not be optimal.
•
5HVXOW
: For well-behaved (monotonic and convex)
preferences, the optimal consumption bundle is found at
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the point where the budget line is tangent to the
indifference curve.
• For well-behaved preferences, WDQJHQF\LVDQHFHVVDU\DQG
VXIILFLHQW FRQGLWLRQ IRU RSWLPDOLW\
. Moreover, for strictly
convex preferences, there is only one point of tangency,
meaning that the optimal bundle is unique.
• This result implies that the slopes of the two lines must be
equal at
0
*
[
1
5
, [2* . This, in turn, means that:
056
2SSRUWXQLW\FRVW
, or
056
=−
S
1
S
2
• The economic interpretation of this implication is that, at
the optimum, the consumer substitutes one good for
another at rate that is identical to the market’s.
Suppose that
056
= −1 2 and the
S
1
S
2
= 1. This
means that the consumer is willing to trade two units of
good 1 for one unit of good 2, whereas in the market the
two goods are traded in a one-to-one basis. Therefore, the
consumer would be willing to save on good 1 to buy more
of good 2, and so the original situation could not be
optimal.
4
([FHSWLRQVWRWKH2SWLPDOLW\5XOH
• In general, the rule may be broken whenever preferences
are not well-behaved. If we keep the assumption of
monotonic preferences all along, then the rule is a
necessary, but not sufficient condition for optimality.
• There many exceptions to the rule, but two in particular
are worth mentioning:
-
.LQN\ 3UHIHUHQFHV
: In this case the optimum occurs
at point where the indifference curve does not have a
tangent.
),*
-
&RUQHU
6ROXWLRQ
: Whenever the slope of the
indifference curve never equals (within the relevant
quadrant) that of the budget line, then the optimal
bundle is found on one of the axis, i.e. the
consumption of one of the goods is zero.
In this case, we have a
FRUQHU
solution, as opposed
to an LQWHULRU solution, to the consumer’s problem.
5
([DPSOHV
•
3HUIHFW
VXEVWLWXWHV
: In this case the slope of the
indifference curve is constant, which means that only by
chance do we get an interior solution:
[
1
%K
=&
K'
P S
1
DQ\ QXPEHU EHWZHHQ
0
<
S =
1
S
1
0 DQG
P
S
1
S
1
>
S
2
S
2
S
2
6
•
&REE'RXJODV 3UHIHUHQFHV
: In this case, what would be
the optimal bundle, given prices and income? To answer
that we will maximise the utility function given the budget
constraint, by the /DJUDQJH¶VPHWKRG.
To do that, we will first set up the /DJUDQJHDQIXQFWLRQ:
/
=X
0
[ [
1
2
,
5 − λ0
S [
1 1
+
S [
2 2
−P
5
and differentiate to get the 3 first-order conditions:
%K ∂ =
∂
KK ∂
&K∂ =
KK ∂ =
' ∂λ
/
[
1
/
[
D
D[
1
−1
D
[
E
E[ [
1 2
E
2
−1
− λS1 = 0
− λS2 = 0
2
/
S [
1 1
+ S2 [2 − P = 0
7
Solving this system in order to
0
[ [
1
2
,
5 we get the optimal
bundle as:
*
[
1
=
*
2
=
[
D
D
D
P
+E
S
1
E
P
+E
S
2
$SSOLFDWLRQ&KRRVLQJ7D[HV
Now that we know something about how the consumer carries
out his/her optimal choices, we can try to compare the effects of
a
TXDQWLW\ WD[
and an
LQFRPH WD[
that raises the same amount
of revenue and find out which one is less harmful.
The government imposition of a quantity tax on, say good 1,
will probably affect the optimal consumer’s choice. However,
the new optimal bundle
0
*
[
1
5
, [2* still must satisfy the budget
line, so that the following holds:
0
S
1
+W
5
*
[
1
+ S2 [2* = P
It turns out that the revenue generated by the tax is given by:
5
*
= W[1*
The imposition of an income (lump-sum) tax that would raise
5
*
= W[1* would imply the following budget line:
8
S [
1 1
+ S2 [2 = P − W[1*
This means that the budget line would maintain the slope but
would shift inwards in a parallel fashion. Moreover, it would
0
pass through the point
*
[
1
5
, [2* , since this bundle must be
affordable.
However, at this point the
056
=−
0
S
1
+W
5
S
2
, meaning that
the budget line associated with the income tax crosses the
indifference curve at
0
*
[
1
, [2*
5
and so
0
*
[
1
, [2*
5
cannot be
optimal, since there is a bundle positioned at a higher
indifference curve that is still affordable.
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It turns out that the income tax is less harmful than the quantity
tax as it raises the same amount of revenue but leaves the
consumer at a higher indifference curve.
10
'HPDQG
As we have seen, the optimal demand for
[ DQG [
1
2
, depend
on the prices and income. So we use the following notation to
denote the GHPDQGIXQFWLRQ of each of the goods:
[
1
= [1
[
= [2
2
0
0
S
1
, S2 , P
S
1
5
5
, S2 , P
The aim of this section is to analyse how the optimal
consumer’s choice changes as income and price change.
1RUPDODQG,QIHULRU*RRGV
•
1RUPDO *RRGV
: When the optimal consumption of a
particular good increases (decreases) as income rises
(falls), we say that this good is QRUPDO:
∂[1
>0
∂P
•
,QIHULRU *RRG
: If, conversely the consumption of one
good decreases (increases) as income rises (falls), then we
are in the presence of an
LQIHULRU
good. Example: As the
income of a young professional increases he might
consume less sandwiches (and more restaurant meals).
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,QFRPH2IIHU&XUYHVDQG(QJHO&XUYHV
We have seen that increasing the income leads to an outward
shift in the budget line that generates new optimal bundles.
•
,QFRPH 2IIHU &XUYH
: This curve corresponds to the line
that connects the demanded bundles that result from
successive outward shifts of the budget line.
•
(QJHO &XUYH
: This curve corresponds to the graph of the
quantity demanded of one good as a function of income,
with prices held constant.
12
([DPSOHV
: Perfect substitutes, perfect complements and Cobb-
Douglas.
+RPRWKHWLF3UHIHUHQFHV
• Whenever the income offer curve and the Engel curve are
straight lines, the quantity demanded of each good varies
proportionately with income. In this case, preferences are
called KRPRWKHWLF, which implies:
,I
0
[ [
1
2
,
5 0 , 5⇒0
\ \
1
2
W[ W[
1
2
,
5 0
W\ W\
1
2
,
5,
IRU DQ\ W
>0
• If conversely, the demand for one good goes up by a
greater proportion than income, this good is said to be a
13
OX[XU\JRRG
. Also, if the demand for one good goes up by
less than proportionately the increase in income, the good
is said to be a QHFHVVDU\JRRG.
&KDQJHVLQ3ULFHV
• When prices change we expect consumers to adjust their
optimal choice. If the quantity consumed of one good
increases (decreases) when its price decreases (increases),
the good is said to be RUGLQDU\.
• If the quantity consumed of one good decreases (increases)
when its price increases (decreases), such a good is called
a *LIIHQJRRG.
• If we let p1 change while keeping p2 and m fixed,
geometrically this involves pivoting the budget line, which
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will likely lead to a change in the optimal bundle. If we
connect the optimal bundles associated with different
value of p1 we get the
SULFH RIIHU FXUYH
. If we plot the
optimal level of consumption against its price, we get the
GHPDQGFXUYH
.
15
6XEVWLWXWHVDQG&RPSOHPHQWV
• We have been talking about perfect substitutes and
complements, but often goods turn out to be
LPSHUIHFW
substitutes and complements. Examples of the former are
pens and pencils and of the latter, coffee and sugar.
• Since the demand function for each good depends on its
own price, but also on the price of the remaining goods,
we can use the demand function to ascertain whether two
goods are substitutes or complements. In fact:
If
∂[1
0
S
1
5
, S2 , P
> 0 , good 1 is a substitute for good 2.
∂S2
16
If
∂[1
0
S
1
5
, S2 , P
< 0 , good 1 is a complement to good 2.
∂S2
7KH ,QYHUVH 'HPDQG )XQFWLRQ
This function gives, for each
level of consumption of, say, good 1, the price that would
induce the consumer to choose that quantity of the good. Thus,
this function, which is denoted as
0 5, has the price as a
S [
1
1
function of the quantity demanded of the good, rather than the
quantity as a function of the price. If the demand curve is
downward sloping, so is the inverse demand function.
17
0DUNHW'HPDQG
Until now, we have been analysing the behaviour of consumers
individually considered. To get the total market demand, we
have to aggregate all the individual demand functions.
'HILQLWLRQ
: The market, or aggregate demand for, say good 1, is
the sum of the individual demands over all consumers:
;
1
0
S
1
, S2 , P1 ,..., P = ∑ [ 1
5
=1
0
S
1
5
, S2 , P ,
where
L
indexes consumers.
• Even though the levels of income available to each
consumer can differ, it is convenient to think of a
UHSUHVHQWDWLYHFRQVXPHU
that faces prices p1 and p2 and has
income M (the aggregate level of income). So we can
analyse the market demand using basically the same
concepts and instruments that we used for the consumer
individually considered.
18
• Notice that the above inverse market demand function is
depicted holding the prices of all other goods and the levels
of income constant. Whenever these change the demand
curve shifts.
([DPSOH
: Adding up linear demand curves.
7KHFRQFHSWRI(ODVWLFLW\
One of the main uses of the demand function is to measure how
the demand for a good varies with prices and income. In this
context, elasticity emerges as a very useful tool.
19
• We could think that the responsiveness of the demand of a
good to its price can be given by the slope of the demand
curve, since it gives the magnitude of the variation in the
quantity demanded that follows a price change. However,
that measure is dependent on the units in which quantities
and prices are measured. For example, the response of
demand of chocolate to a change in price is 1000 times
higher when chocolate is measured in grams than when it
is measured in kilograms.
• To get round this limitation, economists have come up
with a unit-free measure of responsiveness called
HODVWLFLW\
.
• The price elasticity of demand is then defined as the
percentage change of quantity demand divided by the
percentage change in the price. Denoting price elasticity
by ε we have:
ε=
S GT
T GS
• Since demand curves are generally downward sloping, the
price elasticity will most of the times be negative. For that
reason, we will often refer to the absolute value of
elasticity.
([DPSOH
: Price elasticity of a linear demand curve. Graph
20
• What information does elasticity convey about the
demand?
%K
&K
'
,I
,I
,I
ε >1
ε <1
ε =1
HODVWLF GHPDQG
LQHODVWLF GHPDQG
XQLW
− HODVWLF
GHPDQG
• In general, the elasticity of demand for a good depends
largely on the availability of substitutes. If a particular
good has many close substitutes then, as the price of that
good increases, the consumer will shift way from that
good and towards its substitutes, so that the demand
suffers a big drop.
(ODVWLFLW\DQG5HYHQXH
For any firm, it is interesting to analyse what happens to its
revenue if it changes the price of the good it produces.
• Revenue is defined as:
5
=
ST Since an increase in the
quantity leads to a reduction in the price, the revenue can
either go up or down. Specifically, the direction of the
variation in revenue depends on the responsiveness of
demand to a given change in price, i.e. on the price
demand elasticity.
21
• To establish the relation between revenue and elasticity,
let’s define revenue in a bit more rigorous way:
0 5 = 0 5. Differentiating revenue with respect to
5 S
S
ST S
, we get:
G5
GS
G5
GS
=
S
=
S
GT
GS
GT
GS
+T
GS
GS
,
RU
+T
• If the revenue variation triggered by a price change is
positive, then we must have:
G5
GS
S
=
GT
GS
S
GT
GS
+ T > 0,
+T>0⇔
RU
S GT
T GS
,
UH
− DUUDQJLQJ
= ε > −1
• Referring to the absolute value of elasticity, the expression
above amounts to:
ε < 1, that is, the revenue variation that follows a price
increase will be positive if demand is inelastic, will be
negative if demand is elastic and will remain unaltered
if demand is unit-elastic.
22
(ODVWLFLW\DQG0DUJLQDO5HYHQXH
A firm knows that if it increases the quantity of a good, the price
must come down in order to induce consumers to buy the extra
amount of good put in the market. Since the quantity increases
but the price decreases, the firm to would be interested in
knowing what would happen to its total revenue.
• To establish the relation between PDUJLQDOUHYHQXH05
and elasticity, let’s define revenue in a slightly different
(but still rigorous) way:
0 5 = 0 5 . Differentiating revenue with respect to
5 T
S T T
T
,
we get:
G5
GT
G5
GT
=
S
=
S
GT
GT
+T
+T
GS
GT
GS
GT
,
,
RU
RU
G5
GT
=
1 + = 1 + 1
ε
S
TGS
SGT
S
• If the marginal revenue associated by a change in quantity
is to be positive, then we must have:
G5
GT
=
S
1 − 1 > 0,
ε
RU WKDW
1
>1⇔ ε >1
ε
23
So, marginal revenue is positive if demand is elastic, will
be negative if demand is inelastic and will be zero if
demand is unit-elastic. Naturally, if demand is little
responsive to changes in prices than the firm must reduce
the price a lot to be able to sell the extra amount of the
good and that decreases total revenue.
0DUJLQDO5HYHQXH&XUYHV
We’ve seen that marginal revenue can be written as:
G5
GT
=
S
+T
GS
GT
From that expression, one can see that the
T
05
=
S
when
= 0 . After that, 05 < S , since as quantity increases the price
must decrease. It turns out that, by increasing the quantity the
firm receives less for all the other units of good it was
supplying, meaning that MR will be equal to the price minus the
revenue it loses from having to sell all the production at a lower
price.
([DPSOH
: Linear (Inverse) Demand Curves
Consider the following linear inverse demand function:
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0 5=
S T
D
− ET
The slope of the associated curve is simply − E , so the MR
curve is given by:
G5
GT
=
0 5+
S T
T
GS
GT
=
0 5−
S T
ET
= D − 2ET
So, this MR curve has the same vertical intercept as the demand
curve, but twice the slope.
,QFRPH(ODVWLFLW\
This concept is used to measure the responsiveness of the
demand for a good to changes in income; is defined as the ratio
of the percent change in the quantity demanded and the percent
change in income; and is given by:
,QFRPH HODVWLFLW\
=
GT P
GP T
When this elasticity is negative, we are in the presence of a
LQIHULRU JRRG
. If the elasticity is greater than one, we are in the
presence of a OX[XU\JRRG.
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