Download Problem Set –Chapter 5 Solutions

Econ 3070
Prof. Barham
Problem Set –Chapter 5 Solutions
1. Aunt Joyce purchases two goods, perfume and lipstick. Her preferences are
represented by the utility function
U (P , L ) = PL ,
where P denotes the ounces of perfume used and L denotes the quantity of
lipsticks used. Let PP denote the price of perfume, PL denote the price of lipstick,
and I denote Aunt Joyce’s income.
a. What is Aunt Joyce’s maximization problem?
Max U (P, L) = P.L
L,P
s.t. PL L + Pp P = I
b. What are the endogenous and exogenous variables?
The endogenous variables are: P and L
The exogenous variables are: PL, Pp, I
c. Derive her demand for perfume. Your answer should be an equation that
gives P as a function of PP , PL , and I. Determine this by using calculus and
maximizing the objective function, do not use the tangency condition.
To find the demand for perfume we need to find the optimal amount of perfume, it will
be a function of income and prices.
Step 1: Utility is a function of two variables. Since we don’t know how to maximize
when utility is a function of two variables we need to substitute for one of them.
Since we are trying to find the demand for perfume, we will substitute for lipstick, so
we are utility will be just a function of perfume.
Rewriting the budget constraint:
L=
I − PP P
PL
Now we can substitute this into the utility function
"I −P P%
" I % "P
%
P
Max U(P)=P*$$
'' = P $$ '' − $$ P * P 2 ''
P
# PL &
# PL & # PL
&
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Step 2: Now we need to find the value of P that maximizes utility. We know that we
need the value of P where the slope of the utility curve is zero.
∂U
= gives us the slope of the utility function with respect to P
∂P
∂U
I P
= − P 2P = 0
∂P PL PL
2
PP
I
P=
PL
PL
P=
I PL
PL 2PP
P=
I
2PP
Demand function for perfume
d. Derive her demand for lipstick. Your answer should be an equation that gives
L as a function of PP , PL , and I.
To find the demand function for lipstick, we can repeat a similar exercise that we did in
part A. Or we can substitute our value of P back into the budget constraint. (Or
recognize that the answer has to be symmetric).
! I $
PL L + PP ##
&& = I
" 2PP %
I
PL L = I −
2
I
PL L =
2
I
L=
demand function for lipstick
2PL
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d. Is lipstick a normal good? Draw her demand curve for lipstick when I = 200.
Label the demand curve D1. Draw her demand curve for lipstick when I = 300
and label this demand curve D2.
A normal good is a good that a consumer purchases more of as income rises. Since
Aunt Joyce’s demand for lipstick increases as I increases, lipstick is a normal good.
The two demand curves are depicted in the figure below:
PL
12.25
10
D2
D1
10 12.25
L
e. What can be said about her cross-price elasticity of demand of perfume with
respect to the price of lipstick?
In part a, we found that Aunt Joyce’s demand for perfume is given by
P=
I
.
2PP
Since her demand for perfume does not depend on PL, Aunt Joyce’s cross-price
elasticity of demand of perfume with respect to the price of lipstick is zero. That is,
a 1% change in the price of lipstick generates a 0% change in the demand for
perfume.
2. Ch 5, Problem 5.7
Karl’s preferences over hamburgers (H) and beer (B) are described by the utility
function U(H,B)=min(2H,3B). His monthly income is I dollars, and he only buys
these two goods out of his income. Denote the price of hamburgers by PH and of
beer by PB.
a. Write out the consumer’s maximization problem. Remember this is a case of
perfect complements so the indifference curves would be L shaped.
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Max U (P, L) = min(2H ,3B)
L,P
s.t. PH H + PB B = I
b. Draw a graph in H and B space of where the budget constraint and
indifference curves must be for the utility maximization. Don’t use real
numbers just draw what the basic shape of the curves look like.
Optimum is at the corner, so 2H=3B
IC
BL
c. Derive Karl’s demand curve for beer as a function of the exogenous variables
(hint, you can’t maximize this function normally but you know that to be at
an optimum 2H=3B).
We are maximizing utility at the corner point of the L shaped indifference curve, so at
that point
Equation 1 from Utility function: 2H=3B of H=3/2B (I rewrote it this way as we are trying
to find the demand curve for B.
Equation 2: The budget constraint also has to hold PHH + PBB=I
We have 2 equations and 2 unknowns. We can substitute equation 1 into equation 2.
PH(3/2B) + PBB=I
Now I can try to solve for B as a function of price and income which give us our demand
curve.
3
B( PH + PB ) = I
2
3
B( PH + PB ) = I
2
I
B=
3
PH + PB
2
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d. You can answer this just by looking at the demand curve. Because it has a larger
coefficient, the price of hamburgers affects the demand for beer more than the price
of beer. A one dollar increase in PH decreases demand for beer more than a one
dollar increase in PB .
3. Uncle Bob purchases two goods, tweed sport coats and bow ties. His preferences
are represented by the utility function
U (B, C ) = B 0.25 C 0.75 ,
where B denotes the number of bow ties purchased and C denotes the number of
sport coats purchased. Let $25 be the price of bow ties and $60 be the price of
sport coats. And finally, let I denote Uncle Bob’s income.
a. Derive Uncle Bob’s Engel curve for bow ties. Your answer should be an
equation that gives B as a function of I.
Uncle Bob’s maximization problem is:
Max U ( B, C ) = B1/ 4C 3/ 4
B ,C
s.t. PB B + PC C = I
I − PB B
Step 1: Rewrite budget constraint. C =
sub into utility function
Pc
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3
" I − P B %4
B
Max U (B) = B * $$
''
B
P
#
&
c
1
4
3
−
" I − P B %4 3 1 " I − P B %
∂U 1
B
B
= B $$
'' + B 4 $$
''
∂B 4
P
4
P
#
&
#
&
c
c
−
3
4
1
4
PB
=0
PC
3
1 " I − PB B % 4
$
'
4 # Pc &
B
3
4
1
3 PB 4
B
4 PC
=
1
" I − P B %4
B
$
'
# Pc &
3
1
1 3
1 " I − PB B % 4 " I − PB B % 4 3 PB 4 4
B B
$$
'' $$
'' =
4 # Pc & # Pc & 4 PC
1 " I − PB B % 3 PB
B
$
'=
4 $# Pc '& 4 PC
P
I PB B
−
=3 B B
Pc
Pc
PC
3
PB
I
B=
Pc
Pc
B=
I Pc
Pc 2PB
B=
I
I
=
4*25 100
Therefore, Uncle Bob’s Engel curve for bow ties is given by
I
.
B=
100
b. Draw Uncle Bob’s Engel curve for bow ties on a graph with B on the
horizontal axis and I on the vertical axis.
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I
Engel Curve
100
1
B
c. Are bow ties a normal good? What can be said about Uncle Bob’s income
elasticity of demand for bow ties?
Bow ties are a normal good because the demand for bow ties increases as income
increases. Since bow ties are a normal good, Uncle Bob’s income elasticity of
demand for bow ties is positive.
We can calculate the income elasticity of demand as follows:
∈B , I =
dB I
⎛ 1 ⎞⎛⎜ I ⎞⎟
= ⎜
= 1.
⎟
dI B
⎝ 100 ⎠⎜⎝ I 100 ⎟⎠
So, Uncle Bob’s income elasticity of demand for bow ties is 1 – a 1% increase in his
income leads to a 1% increase in his demand for bow ties.
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Econ 3070
Prof. Barham
4. Ch 5, Problem 5.9
Rick purchases two goods: food and clothing. He has a diminishing marginal
rate of substitution of food for clothing. Let x denote the amount of food
consumed and y the amount of clothing. Suppose the price of food increases from
Px1 to Px 2 . On a clearly labeled graph, illustrate the income and substitution
effects of the price change on the consumption of food. Do so for each of the
following cases:
a. Case 1: Food is a normal good.
Given the increase in the price of x, we expect to see the following effects:
Substitution Effect
Income Effect
x
↓
↓
y
↑
↓
Because the price of x increased, x became relatively more expensive, and y became
relatively less expensive. As a result, Rick substitutes away from x in favor of y. This
is represented in the table by a down arrow for x and an up arrow for y in the
substitution effect column.
Moreover, the increase in the price of x reduced Rick’s purchasing power. Since x
and y are both normal goods (x being a normal good is given by the problem, y being
a normal good is assumed), the reduction in purchasing power causes Rick to
purchase less of both x and y. This is represented in the table by the down arrows in
the income effect column.
The following diagram gives us a graphical representation of the information
presented in the table:
y
C
•
B
•
A
•
BL2
BL1
x
The initial consumption bundle is represented by point A, which lies on the initial
budget line BL1. The increase in the price of x causes the budget line to shift
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inwards to BL2. The new consumption bundle is represented by point C. We then
construct point B in order to separate the substitution effect from the income effect.
The movement from A to B represents the substitution effect. Note that as
suggested by the table, the movement from A to B shows x going down and y going
up. The movement from B to C represents the income effect. Once again, note that
as suggested by the table, the movement from B to C shows both x and y going
down.
b. Case 2: The income elasticity of demand for food is zero.
In this case, we expect to see the following effects:
Substitution Effect
Income Effect
x
↓
⎯
y
↑
↓
Once again, because the price of x increased, Rick substitutes away from x in favor
of y. This is represented in the table by a down arrow for x and an up arrow for y in
the substitution effect column.
However, the information in the income effect column has changed. Since the
income elasticity of demand for x is zero, the reduction in Rick’s purchasing power
has no effect on x. This is represented by the horizontal line for x in the income
effect column. (The down arrow for y reflects the fact that we are continuing to
assume that y is a normal good.)
The following diagram gives us a graphical representation of the information
presented in the table:
y
B
•
C
•
BL2
A
•
BL1
x
As before, the initial consumption bundle is represented by point A, and the new
consumption bundle is represented by point C. Point B is constructed in order to
separate the substitution effect from the income effect.
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The movement from A to B represents the substitution effect, and as suggested by
the table, we observe x going down and y going up. The movement from B to C
represents the income effect. As suggested by the table, we observe no change in x
since the income elasticity of demand for x is zero. On the other hand, we do
observe y going down since y is assumed to be a normal good.
c. Case 3: Food is an inferior good, but not a Giffen good.
In this case, we expect to see the following effects:
Substitution Effect
Income Effect
x
↓
↑
y
↑
↓
Once again, because the price of x increased, Rick substitutes away from x in favor
of y. Moreover, the reduction in Rick’s purchasing power reduces his demand for y
(a normal good).
What is new is that x is an inferior good; that is, the reduction in Rick’s purchasing
power causes Rick to purchase more x. This is represented by the up arrow for x in
the income effect column.
The following diagram gives us a graphical representation of the information
presented in the table:
y
B
•
A
•
C
•
BL2
BL1
x
As before, the initial consumption bundle is represented by point A, and the new
consumption bundle is represented by point C. Point B is constructed in order to
separate the substitution effect from the income effect.
The movement from A to B represents the substitution effect, and as suggested by
the table, we observe x going down and y going up. The movement from B to C
represents the income effect. As suggested by the table, we observe y going up (since
y is assumed to be a normal good) and x going down (since x is assumed to be an
inferior good).
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The last thing to take note of is that the diagram indicates that x is not a Giffen good.
The diagram indicates that the income effect is not strong enough to dominate the
substitution effect; that is, the increase in x going from B to C is smaller than the
decrease in x going from A to B.
The last thing to take note of is that the diagram indicates that x is a Giffen good.
The diagram indicates that the income effect dominates the substitution effect; that
is, the increase in x going from B to C is larger than the decrease in x going from A to
B.
5. Ch 5, Problem 5.11 ed. 5.
Ginger’s Utility function is U(x,y)=x2y. She has income I=240 and faces prices Px=$8
and Py=$2.
Part A.
The maximization problem is:
Max U ( x, y ) = x 2 y
x, y
s.t. 8X + 2Y = 240
Part B.
Using the budget constraint we rewrite the maximization problem in terms of one variable.
MaxU ( x) = x 2 (120 − 4 x)
X
∂U
= 240 x − 12 x 2 = 0
∂x
x* = 20
y* = 120 − 4(20) = 40
Her optimal bundle is (x,y)=(20,40) and utility is 16,000
Part C.
To be just as well off as before, her utility must be 16,000. We will use this fact later to help
us. But first we can just set this up as a usual maximization problem like in part A. I’ll start
after I sub in the budget constraint since the steps are just the same as in part A. I will
change the price of y to be $8, and will use Px in the place of the price of X
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Econ 3070
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MaxU ( x) = x 2 (30 −
x
Px x
)
8
3P x 2
∂U
= 60 x − x = 0
∂x
8
8
Px x = 60 = 160
3
Now we can sub PxX into the budget constraint to find Y. 160+8Y=240. So Y* =10.
To find out what X is we can use the fact that utility must equal 16,000
U(x,y)=16,000=x210
X*=40.
We know PxX=160 and X* is 40, so Px=4 if Ginger is just as well off as before the price
change.
6. Ch 5, Problem 5.18
The demand function for Kendamas is given by D(P)=16-2P (note that D(P) is just a
way of saying writing the demand function where the quantity demanded is a
function of P which you are used to seeing as QD. Compute the change in consumer
surplus when the price of a widget increases from $1 to $3. First show your results
graphically.
First lets graph this demand curve it is linear, so the slope is -2
If P=0, Q=16
If Q=0 then P= ? just write the inverse demand curve P=(16-QD)/2 so P=8
If P=1 then QD or D(1) =14
If P=3 then QD or D(1) =10
For price of a widget equal to $1 consumer surplus is
CS$1 = ½ · (8 – 1) · D(1) = ½ · 7 · 14 = 49.
When price is equal to $3 consumer surplus is
CS$3 = ½ · (8 – 3) · D(3) = ½ · 5 · 10 = 25.
So the change in consumer surplus is 49-25=24 or Area EBDC
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P
$8
A
Area of ABE triangle
CS when P = $3 is 25
D(P) = 16 – 2P
$3
$1
E
D
B
Area of ACD triangle
CS when P = $1 is 49
C
D(P)
7.
Ch 5, Problem 5.26 ed. 5.
Suppose that Bart and Homer are the only people in Springfield who drink
7-UP. Moreover, their inverse demand curve for 7-UP are:
Bart: P=10-4QB
Homer: P=25-2QB
Neither one can consume a negative amount.
Write down the market demand curve for 7-UP in Springfield, as a function of
all possible prices.
Bart will only consume when the price is less than 10. To see this, see what the price has
10 − P
to be if QB is zero. Therefore his demand curve for 7-UP is QB =
, when P<10
4
and zero otherwise.
Homer will only consume if the price is less than 25 so his demand curve is
25 − P
QH =
, when P < 25 and zero otherwise.
2
Therefore the market demand curve for 7-UP as a function of all possible values of price
is:
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Econ 3070
Prof. Barham
Q M = 0, if P > 25
25 − P
QM =
, if 10 < P < 25
2
60 − 3P
QM =
, if P < 10
4
Ch, Problem 5 5.20
8.
Lou’s preferences over (x) and other goods (y) are given by U(x, y) = xy.
His income is $120. You can use your calculations from HW 3 Problem 1
where you found the demand function for this type of utility function:
I
X=
demand function for X
2PX
Y=
I
2PY
demand function for Y
a. Calculate his optimal basket when Px = 4 and Py = 1.(Note you the
demand function given or or you can practice optimizing again).
Plugging in the values into the demand functions :
120
demand function for X
2(4)
120
Y=
demand function for Y
2(1)
X=
(X*, Y*)=(15, 60)
b. What is Lou’s utility if he consumes the optimal basket determined in a?
U(X,Y)= X*Y*=(15*60) = 900
c. Graph the budget line and indifference curve and mark the optimal point,
call this point A. Call the budget line BL1 and indifference curve U1. You
can just approximate the indifference curve but get the shape right.
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Econ 3070
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Y
B
A
U2
C
U1
BL2
BL1
BL’2
X
D
15 X X2
d. Now the price of pizza falls to $3. On the Graphs put on a new budget
line and indifference curve for the new optimal bundle, and call the bundle
B. Call the budget line BL2 and indifference curve U2. Don’t
worry about calculating the exact bundle. Mark the quantity of X
consumed in this bundle as X2 on the X axis.
e. The decomposition bundle is (17.3,51.9). Show on the graph how you
would calculate this decomposition bundle. What indifference curve and
budget line are tangent to find this point? Mark this tangency point as C on
the graph. Mark the quantity of X consumed by XD on the X axis.
You need to find point C, you need to find a tangency where the original utility
U1, and a budget line with the new prices price, BL’2 , are tangent. This is the
decomposition bundle. I’ve given it to you as it is a little tricky to figure out as
you can tell by the decimal points.
f. Calculate the compensating variation of the price change.
The compensating variation is the amount of income Lou would be willing to give
up after the price change to maintain the level of utility he had before the price
change. This equals the difference between the consumer’s actual income, $120,
and the income needed to buy the decomposition basket at the new prices. This
latter income equals: 3*17.3 + 1*51.9 = 103.8. The compensating variation thus
equals 120 – 103.8 = $16.2.
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