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Unformatted text preview: Problem Set #1
Name: _____________________
Econ326 Intermediate Micro
Student ID: _____________________
Instructor: Ginger Jin
Section No. _____________________
Due Date: February 14, 2011
TA Name: _____________________
===============================================================
1. Sally consumes two goods, X and Y. Her utility function is given by the expression
The current market price for X is $10, while the market price for Y is $5.
Sally's current income is $600.
a. Sketch two indifference curves for Sally in her consumption of X and Y so that one
curve represents U=240000 and the other curve represents U=480000. Mark two
points on each curve, write down the precise numbers of X and Y for each point.
(Note: you can choose whatever unit of X and Y that you think are most appropriate
in the following graph.) b. Calculate the marginal rate of substitution for each of the four points you have
marked on the above graph.
MRS = MUx /MUy
= Y2/2XY= Y/2X 1 Problem Set #1
Name: _____________________
Econ326 Intermediate Micro
Student ID: _____________________
Instructor: Ginger Jin
Section No. _____________________
Due Date: February 14, 2011
TA Name: _____________________
===============================================================
!" #"
$%&"
'(" )*+,,."
,(("
(" /0+,111"
'(("
,(" ,,+.'0/"
1"
1(" ,(" '/" c. Write the expression for Sally's budget constraint. Graph the budget constraint and
determine its slope.
Sally’s Budget constraint is given by 10X + 5Y = 600 and its slope is 2.This can be seen
by rewriting the budget constraint as Y= 1202X ( Form : y= m*x+c, where ‘m’ is slope
of the line)
d. Determine the X,Y combination which maximizes Sally's utility, given her budget
constraint. Show her optimum point and its corresponding indifference curve on the
graph. Compute the utility level at this optimum point. Note that this utility function belongs to the category Cobb – Douglas Utility functions.
For a utility function of the general form U = A X! Y" , we can write the optimal demands
(X*,Y*) as follows:
X* = [!M] / [(!+ ")Px ]
Y* = ["M] / [(!+ ")Py ]
Where M : Income, Px : Price of good X; Py : Price of good Y
For our problem, ! = 1 and " =2; M= 600;Px=10;Py=5. Therefore,
X* = 20 ; Y*= 80. The utility corresponding to the optimum point is U = 3*20*1600 =
96000
U=3*20*6400=384000 2 Problem Set #1
Name: _____________________
Econ326 Intermediate Micro
Student ID: _____________________
Instructor: Ginger Jin
Section No. _____________________
Due Date: February 14, 2011
TA Name: _____________________
=============================================================== e. Suppose Sally’s utility function changes to U=2lnX + 4lnY. Recalculate the optimal
point that Sally will choose (given the same budget constraint). Is this answer
different from the solution in part d? Why?
If we recalculate the (X*,Y*) using
1. MRS = Px /Py ie. Y/2X = 2
2. Budget constraint : 10x + 5Y = 600
We obtain (X*,Y*) = (20,80)
This is the same as in part d because this utility function is just a MONOTONIC
TRANSFORMATION of the previous utility function and therefore will result in the
same optimal consumption bundle . 3 Problem Set #1
Name: _____________________
Econ326 Intermediate Micro
Student ID: _____________________
Instructor: Ginger Jin
Section No. _____________________
Due Date: February 14, 2011
TA Name: _____________________
===============================================================
2. Antonio buys five new college textbooks during his first year at school at a cost of
$80 each. Used books cost only $50 each. When the bookstore announces that there
will be a 10 percent increase in the price of new books and a 5 percent increase in the
price of used books, Antonio’s father offers him $40 extra.
a. How much did Antonio spend on textbooks before the price hike? With new books on
the vertical axis and used books on the horizontal axis, draw his budget line and his
consumption bundle of textbooks before the price change. 4 Problem Set #1
Name: _____________________
Econ326 Intermediate Micro
Student ID: _____________________
Instructor: Ginger Jin
Section No. _____________________
Due Date: February 14, 2011
TA Name: _____________________
===============================================================
Let N1 be the number of new books and N2 be the number of used books Antonio uses
before the price hike. Let his total budget before the price hike be M1
Before the price hike, his budget constraint could be written as : 80 * N1 + 50*N2 = M1
Using N1 = 5, we can write this as 80 * 5 + 50*N2 = M1
After the price hike, his budget becomes M2 = M1+40, so that he can continue to
consume the same combination of books as earlier.
Thus his budget constraint becomes
88* 5 + 52.5*N2 = M2
Using M2= M1+40 =80 * 5 + 50*N2
We get
88* 5 + 52.5* N2= 80*5 + 50N2 +40
Solving which gives us that N2 = 0
Thus M1= 400. i.e Anonio spent $400 before the price hike.
Therefore his consumption bundle before the price change was (X,Y) = (0,5). 5 Problem Set #1
Name: _____________________
Econ326 Intermediate Micro
Student ID: _____________________
Instructor: Ginger Jin
Section No. _____________________
Due Date: February 14, 2011
TA Name: _____________________
===============================================================
b. Let X=# of used books, Y=# of new books. Suppose Antonio’s utility is
U=100X+165Y. Draw the indifference curve that Antonio is on before the price
change. Calculate the marginal rate of substitution at this point. Has Antonio made an
optimal choice of new versus used textbooks before the price change? Explain why.
MRS = 100/165 = .606
Before the price change, Antonio consumes (0,5) so his utility U = 100*0+5*165 = 825.
Yes, Antonio has made an optimal choice, because new books and old books are perfect
substitutes and MRS =.606 < .625 = (Px/Py). Therefore, Indifference curves are flatter
than the budget line and Antonio is better off consuming only good Y i.e new books.
c. What happens to Antonio’s budget line after the price change (including both price
changes and the extra money from his father)? Illustrate the new budget line in the
above graph.
Slope of the Budget Line is given by Px/Py
The slope of the budget line changes from  .625 (i.e 50/ 80) to .59 (i.e 52.5/88) d. How many new and used textbooks will Antonio buy after the price hike? Is Antonio
worse or better off after the price change? Explain.
Since new books and old books are perfect substitutes and MRS =.606 > .5966
= (P’x/P’y). Therefore, Indifference curves are steeper than the budget line and Antonio
is better off consuming only good X i.e used books. Substituting N1 ‘=0 in budget
constraint, we get
0+ 52.5 *N2’ = 440
Ie. N2 ‘ = 8.3810 # 8
Antonio’s utility is now U’ = 100*8 + 0 = 800. He is worse off than before as a result of
the price change. 6 QUESTION 3
Elmer’s utility function is U (x, y ) = min{x, y 2 }. a. If Elmer consumes 4 units of x and 3 units of y, his utility is ___4___.
If Elmer consumes 4 units of x and 2 units of y, his utility is ___4___.
If Elmer consumers 5 units of x and 2 units of y, his utility is ___4___.
Answer:
U (4, 3) = min{4, 9} = 4
U (4, 2) = min{4, 4} = 4
U (5, 2) = min{5, 4} = 4
Hint: Since the above 3 points gives the same utility level, connecting the 3 points will give
you an idea of the shape of the indiﬀerence curve, which is right angled. And preference is this
form is called perfect complementary.
On the graph below, draw the indiﬀerence curves that go through (4,2), (1,1) and (16,5)
respectively. By now, you may notice that Elmer’s indiﬀerence curves have kinks. If you connect
the kinks in one curve, what is the equation for this curve? (16,5) (1,1) (4,2) Hint: The steps to draw the indiﬀerence curves is to ﬁrst identify which indiﬀerence curves
do these points lie. And secondly, draw these indiﬀerence curves.
Answer: Step1 :
U (4, 2) = min{4, 4} = 4, thus point (4, 2) is on indiﬀerence curve that yields the utility level of 4. 1 U (1, 1) = min{1, 1} = 1, thus point (1, 1) is on indiﬀerence curve that yields the utility level of 1. U (16, 5) = min{16, 25} = 16, thus point (16, 5) is on indiﬀerence curve that yields the utility level of 16. Step2 :To graph indiﬀerence curve U (x, y ) = 4, the key is to ﬁnd the kink point, where we
ﬁnd by equating the two component of the min function,
x = y 2 = 4 ⇒ (x, y ) = (4, 2) To graph indiﬀerence curve U (x, y ) = 1, again we ﬁnd the kink point by equating the two
component of the min function,
x = y 2 = 1 ⇒ (x, y ) = (1, 1) To graph indiﬀerence curve U (x, y ) = 16, again we ﬁnd the kink point by equating the two
component of the min function,
x = y 2 = 16 ⇒ (x, y ) = (16, 4) The kinks lies on the locus x = y 2 or y = √ x. b. On the same graph, draw Elmer’s budget line where the price of x is 1, the price of y is
3, and his income is 4. Write down the budget constraint equation.
Answer: The budget constraint equation is
x + 3y = 4 c. What bundle does Elmer choose in this situation? Write down your steps below.
Hint: The optimal bundle with perfect complementary goods is the intersection between
the locus and the budget line.
Answer : Solving the system of the following two equations
x + 3y = 4
x = y2
Plug the second equation into the ﬁrst one, we get y 2 + 3y = 4
y 2 + 3y − 4 = 0
(y + 4)(y − 1) = 0
y=1 2 and x = 1. Thus the optimal bundle Elmer chooses is (x, y ) = (1, 1).
d. Suppose that Elmer’s income increases to 8 while both prices remain unchanged. What
bundle does Elmer choose now? Does doubling the income lead to doubling consumption in x
and y? Explain.
Answer : Again solving the system of the following two equations with the new income
level,
x + 3y = 8
x = y2
Plug the second equation into the ﬁrst one, we get
y 2 + 3y = 8
y 2 + 3y − 8 = 0
y 1. 7
x 2. 9
The new optimal bundle Elmer chooses is (x, y ) = (1.7, 2.9). Thus the consumption is not
doubling the original bundle after the income doubled. It is because we have nonlinear locus,
and the direction of indiﬀerence curve increasing does not follow a straight line. Mathematically,
when utility is not homogeneous of degree one in two goods, we lose the property of doubling
the income doubles the optimal consumption in two goods. 3 QUESTION 4.
Gentle Charlie consumes only apples and bananas. Let the number of apples be xA , and
the number of bananas be xB . His utility function is U (xA , xB ) = xA ∗ xB . The price of apples is $2, the price of bananas is $8, and Charlie’s income is $160 a day. The price of bananas
suddenly falls to $2.
a. Before the price change, how many apples and bananas do Charlie consume? With these
consumptions, how much utility can he achieve? Write down your steps below.
Answer: Let’s focus on the the market before the price change. Denote the optimal consumption bundle before the price change as (x0 , x0 ), where the superscript 0 stands for initial
AB
optimal choices. Denote original price pair as (pA , pB ). We have two conditions to solve for
( x 0 , x0 ) ,
AB
M RSAB = pA
pB which can be written as
M RSAB = M UA
xB
pA
1
=
=
=
M UB
xA
pB
4 and hence we have relationship x0 = 4x0 .
A
B
Also we have the condition that the optimal bundle (x0 , x0 ) is on the budget constraint, or
AB
pA x 0 + pB x 0 = I
A
B
2x0 + 8x0 = 160
A
B
Plug x0 = 4x0 into the budget constraint, we have
A
B
8x0 + 8x0 = 160
B
B
x0 = 10
B
and x0 = 4x0 = 40. Thus we have solve for the optimal bundle is that
A
B
(x0 , x0 ) = (40, 10)
AB
which corresponds to point A in the graph. And the utility level achieved by this bundle is
U = 40 ∗ 10 = 400
On the graph, Charlie’s original budget line is in blue and A is his original chosen consumption.
4 b. If, after the price change, Charlie’s income had changed so that he stays on the same
indiﬀerence curve as before, how much would his new income be and how many apples and
bananas will he consumer with this income and the new prices? Write down your steps below.
Hint: The key to this answer is that when Charlie “stays on the same indiﬀerence curve
as before ”, he is still making his optimal consumption choice, but under the new price ratio,
so we have condition M RSAB = pA /p . And graphically, we have the budget line with the
B
slope representing the new price ratio tangent with the original indiﬀerence curve. So the steps
part(b) takes is to ﬁrst solve for the optimal consumption bundle, we denote as (x1 , x1 ) that
AB
makes Charlie stays on the same indiﬀerence curve under the new price ratio. Secondly, we can
solve for the amount of money Charlie needs to be able to aﬀord bundle (x1 , x1 ), which is his
AB
new income, we denote as I .
Answer : Denote the new price of banana as p . (x1 , x1 ) satisﬁes the condition that it
B
AB
makes Charlie stays on “the same indiﬀerence curve as before”, which the one we solved in
part(a), xA xB = 400, thus (x1 , x1 ) must satisfy equation
AB
x1 x1 = 400
AB
Also (x1 , x1 ) is an optimal choice for Charlie, we have M RSAB =
AB
M RSAB = M UA
xB
pA
=
= =1
M UB
xA
pB 5 pA
pB , or Thus we have the second equation x1 /x1 = 1 or x1 = x1 . Plug it into x1 x1 = 400, we
A
B
A
B
AB
have
x1 x1 = 400
AA
So we solve for x1 = x1 = 20, which is the consumption bundle charlie chose under the
A
B
new price if he is to keep himself on the same indiﬀerence curve as before. It is indicated in the
graph as B.
The income that Charlie needs to spend on (x1 , x1 ) is
AB
pA x1 + p x1 = 2 ∗ 20 + 2 ∗ 20 = 80
A
BB
In the graph, the budget line corresponding to this income is in light blue line. The bundle
that Charlie would choose at this income is labelled at B.
c. After the price change (and income remains at $160 a day), how many apples and bananas
will Charlie actually buy? Write your steps below.
Answer: Denote the consumption bundle after the price change as (x2 , x2 ), and the new
AB
price pair as (pA , p ). We have two conditions to solve for (x2 , x2 ),
B
AB
M RSAB = pA
pB which can be written as
M RSAB = M UA
xB
pA
=
=
=1
M UB
xA
pB and hence we have relationship x2 = x2 .
A
B
Also we have the condition that the optimal bundle (x2 , x2 ) is on the budget constraint, or
AB
pA x 2 + pB x 2 = I
A
B
2x0 + 2x2 = 160
A
B
Plug x2 = x2 into the budget constraint, we have
A
B
2x2 + 2x2 = 160
B
B
x2 = 40
B
and x2 = x2 = 40. Thus we have solve for the optimal bundle is that
A
B
(x2 , x2 ) = (40, 40)
AB
d. With points A, B, C on the graph, from which point to which point is the substitution
6 eﬀect of the price fall of bananas? From which point to which point is the income eﬀect? How
would the consumption of apples and bananas change because of the substitution eﬀect? How
would the consumption of apples and bananas change because of the income eﬀect?
Answer : Substitution eﬀect is the change in consumption bundle due to price change and
keeps the consumer at the original indiﬀerence curve. And hence, from A to B is the substitution eﬀect of the price fall of bananas. From B point to C is the income eﬀect. Because of the
substitution eﬀect, consumption of apples drops from 40 to 20 and bananas increases from 10 to
20. Because of income eﬀect, the consumption of apples increases from 20 to 40, and bananas
increases from 20 to 40.
e. Is banana a Giﬀen good? Is apple an inferior good? Explain.
Answer : A Giﬀen good is one that the consumption of it change in the same direction as
its price changes. Consumption of Banana increases when its price decreases, thus Banana is
not Giﬀen good.
An inferior good is one that the consumption of it decreases when the income increases, and
consumption of it increases when income decreases (keeping prices unchanged). Consumption
of apple increases when income increases, so apple is not an inferior good. 7 ...
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This note was uploaded on 09/07/2011 for the course ECON 326 taught by Professor Hulten during the Spring '08 term at Maryland.
 Spring '08
 Hulten
 Utility

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