UNIVERSITY OF TORONTO SCARBOROUGH

DEPARTMENT OF MANAGEMENT

MGEB02: Price Theory: A Mathematical Approach

(Intermediate Microeconomics I)

Problem Set-2
Pindyck et al.

Chapter-3: 2, 5, 6, 7, 12, 14, 16

Supplemental:

1. Draw a set of indifference curves for the following pairs of goods:

a) Meat and carrots for a vegetarian who neither likes nor dislikes meat.
b) Bread and milk for an individual who always consume them together.
c) Forbes and Fortune for an individual who regard this two magazines as perfect
substitute.

2. Short questions:

a) You and your friend are dining at a local restaurant, which sells only lobsters and
steaks. An economics student working part-time in the restaurant asks the two about their
MRS (Marginal Rate of Substitution) and discovers that your MRS of lobster for steak is
2. You and your friend face the same prices for the two foods. Therefore, we know your
friends MRS of lobster for steak must also equal 2. True/False uncertain, explain

b) TRUE or FALSE: A utility maximizing consumer with U(X, Y ) = 3X + Y will always

choose to spend all income on the commodity with the lower price.

3. (Budget Line) Suppose that John has an income of $500, and has to choose between
consumption of x1 and x2: The price of x1 is $50, and the price of x2 is $20.

a) Write down the equation for the budget line? What is the slope of the budget line?
b) Draw the budget line and note down the intercepts of each axis.
c) Now suppose, John gets 5 units of x1 for free. Draw the new budget line. (He cannot
exchange the free units for x2)
d) The supplier of x2 gives Jack an option of exchanging the free units of units of x1 he
acquired for 2 units of x2. Draw the budget line taking the exchange into consideration.
4. Assume a utility function U(x,y)= xy. Income is equal to 100, the price of x is equal to
20, that one of y is equal to 25.

a) Employ the Lagrange method to calculate the utility maximizing quantities for x and y.
Show your work!
b) Illustrate your results in a graph.
2
5. Zach consumes two goods X and Y, his utility function is U(X,Y )=2XY . Suppose the
price of X is $10, while the price of Y is $15. Zach’s income is $500.

a) Write the expression for indifference curve when Zach gets utility level 40 and along
the indifference curve you found, calculate the numbers of consumption of X when Y=4.
b) Write the expression for Zach’s budget constraint, graph the budget constraint and
determine its slope.
c) Determine the X, Y combination, which maximizes Zach’s’s utility, given his budget
constraint. And figure out what’s marginal rate of substitution (MRS) between two goods
at that maximization point.
d) Suppose now the price of X is changed to 15, calculate the impact on Zach’s optimum
choice. What’s the change to his maximized utility?

6. Mary’s utility function over good X and Y is U(X,Y)=10X+5Y. Her income is 100 and
the price of X is 2 and price of Y is 5.

a) Calculate the marginal rate of substitution (MRS) between X and Y and briefly explain
what it means.

b) How much of X and Y will she buy?

c) What if the utility function was rather U(X,Y) = XY?

7. Ahmed’s preferences over CDs (C) and sandwiches (S) are given by U(S, C) = SC +
10(S + C). Suppose the price of a CD is $9 and the price of a sandwich is $3, and Ahmed
can spend a combined total of $30 each day on these goods,

a) Find Ahmed’s optimal consumption basket.

b) Illustrate your solution in a graph.
8. Considered Julie’s preferences for Gasoline (X) and a composite good (Y). Her utility
function was U(X,Y) = Ln(X)+Ln(Y). Julie has an annual income of 10,000. Suppose the
price of the composite good is $1.

a) The government has introduced a rationing system such that Julie can only consume
2000 liters a year at $.5 a litter. Find Julie’s optimal consumption bundle.

b) The government removes the rationing system and the free market price of gasoline
jumps to $1. Find Julie’s optimal consumption bundle.

c) Is Julie better off with or without the rationing system?

d) Illustrate your solution in a graph.

 Solutions
1.a)
C

U2
U1
U0

M
b)

U2
U1
U0

M
c)
F

U0 U1 U2

2.

a) That is only true if both individuals are characterized by a convex indifference curve.
If either one is characterized by “perfect substitute” or “perfect complement” indifference
curve then the statement will not be true. Therefore, uncertain.

b) FALSE: Since MRS = 3 , the corner solution will be either all X or all Y depending on
whether the relative price is less than 3 or more than 3:
Indifference Curves

Budget Lines
Slope<3
Slope>3
 3.
a) Equation for the budget line is:

I = P1 x1 + p2 x2 => 50 x1 + 20 x2 = 500
p1 50
=− =− = −2.5
p2 20
b)
x2

25

x1
10
c) This will create a kink on the budget line. There will be a new line parallel to the x1
axis for a length of 5 units:
x2

25
x2

x1
5 15
d)
x2
27
25
x2

x1
5 15
 4. Lagrange Method

Utility Function U (x,y) = xy

Budget Constraint I = pxx + pyy with I=100, px=20, py=25
20x + 25y=100
a) Transform Budget Constraint
0 = 20x + 25y -100

b) Build the Lagrangian

L(x,y,λ) = xy - λ(20x + 25y -100)

c) Calculate partial derivatives for x, y, and λ and set them equal to zero

1. ∂L
= y − 20λ = 0
∂x
2. ∂L
= x − 25λ = 0
∂y
3. ∂L
= − ( 20 x + 25 y − 100 ) = 20 x + 25 y − 100 = 0
∂λ
d) Solve for x and y
From (1) & (2) y 20 4
= => y = x
x 25 5
plug into 3: 4
20 x + 25( ) x − 100 = 0
5
|Solve for x and y => x = 2.5, y = 2
U = 2 × 2.5 = 5

U=5
4

x
2.5 5
5.

a)
U = 40 = 2 XY 2
Y = 4 => 40 = 2 X 4 2 => X = 40 / 32 = 1.25

1.25, 4

U=40
X

1.25
1.25

b)
500 10
10 X + 15Y = 500 => Y = − X
15 15
PX 10 2
Slope : − =− =−
PY 15 3

33.3

X
50
c)
MU X PX
MRS = =
MU Y PY
2Y 2 10 Y
=> = =
4 XY 15 2 X
=> X = 0.75Y
10 X + 15Y = 500 = 10(0.75Y ) + 15Y = 500
=> Y * = 22.22
=> X * = 16.67
=> U * = 2 ×16.67 × 22.22 2 = 16460.91

d)

2Y 2 15 Y
=> = =
4 XY 15 2 X
=> X = 0.5Y
15 X + 15Y = 500 = 15(0.5Y ) + 15Y = 500
=> Y * = 22.22
=> X * = 11.11
=> U * = 10973.94

His utility declines.

Y Old Budget Line

New Budget Line

22.22

X
11.1 16.67 50
6.

MU X 10
a) MRS = = =2
MU Y 5

The MRS is constant which means that Mary is willing to give up one unit of “good X”
for every two more units of “good Y”, while keeping her overall utility (satisfaction)
constant. In other words, she is willing to exchange one unit of X for two units of Y no
matter how much X and Y she has.

b) There will be a corner solution because:

MU X 10
MRS = = =2
MU Y 5
PX 2
=
PY 5
PX
MRS >
PY

Utility is Maximized U = 500

20

50

c) No corner solution here:

MU X Y P 2
MRS = = = X =
MU Y X PY 5
=> X = 2.5Y
2 X + 5Y = 100 = 2( 2.5Y ) + 5Y = 100
=> Y = 10, X = 25
 7.

a) The fact that Ahmed’s indifference curves touch the axes should immediately make
you want to check for a corner point solution. But let us check the tangency condition
first as we always do:

S + 10
MRS = =3
C + 10
=> S = 3C + 20
Budget Line : 9C + 3S = 30
− 30
=> C =
18
This implies a negative number for CD. Since it’s impossible to purchase a negative
amount of something, our assumption that there was an interior solution must be false –
there has to be a corner solution. Ahmed spends all his money on sandwiches: S = 10 and
nothing on CD : C=0.

b) The corner optimum is reflected in the fact that the slope of the budget line is steeper
than that of the indifference curve, even when C = 0. Specifically, note that at (C, S) =
(0, 10) we have PC / PS = 3 > MRSc,s = 2. Thus, even at the corner point, the marginal
utility per dollar spent on CDs is lower than on sandwiches. However, since Ahmed is
already at a corner point with C = 0, he cannot give up any more CDs. Therefore the best
Ahmed can do is to spend all her income on sandwiches: (C, S) = (0, 10). [Note: At the
other corner with S = 0 and C = 3.3, PC / PS = 3 > MRSc,s = 0.75. Thus, Ahmed would
prefer to buy more sandwiches and less CDs, which is of course entirely feasible at this
corner point. Thus the S = 0 corner cannot be an optimum.]

S
10

C
3.33
8.

a) If there was no restriction:

1
Y P 0.5
MRS = X = = X =
1 X PY 1
Y
=> X = 2Y
0.5 X + Y = 10,000 => Y = 5,000, X = 10,000
U = Ln(10,000 ) + Ln(5000 ) = 17.73

But she cannot consume more than 2,000 liter at this price therefore:

X = 2000
0.5 X + Y = 10,000 => Y = 9,000, X = 2,000
U 0 = Ln( 2000 ) + Ln(9000 ) = 16.706

b)

1
Y P 1
MRS = X = = X =
1 X PY 1
Y
=> X = Y
X + Y = 10,000 => Y = 5,000, X = 5,000

c) Better off without the rationing because:

With
U with = Ln( 2,000 ) + Ln(9,000 ) = 16.706
Without
U without = Ln(5,000 ) + Ln(5,000 ) = 17.034
U without > U with
 d)

U0

10,000

U1

5,000

X
2,000 5,000 10,000