EC_308_Solutions_to_Problem Set 2_Fall_2023
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Jan 9, 2024
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EC 308
Intermediate Microeconomics
Problem Set 2
Solutions
Problem 1:
Bob’s utility function for soda and pizza is
𝑢(?
1
, ?
2
) =
√
?
1
√
?
2
where
?
1
is the number of
cans of soda and
?
2
is the number of slices of pizza that Bob consumes.
a)
Graph Bob’s indifference curve, labeling at least three points.
b)
If given the choice between 2 slices of pizza and 4 cans of soda or 4 slices of pizza and 2 cans of
soda would Bob prefer the bundle with more pizza, the bundle with more soda, or be indifferent
between the two bundles?
Bob’s utility from bundle (4 sodas, 2 pizza slices) is:
𝒖(?, ?) = √?
√?
= ?√?
.
Bob’s utility from bundle (2 sodas, 4 pizza slices) is
𝒖(?, ?) = √?
√?
= ?√?
.
Since Bob’s utility from both bundles is the same, Bob is indiff
erent between them.
c)
Bob was consuming 2 slices of pizza and 4 cans of soda. Alice offered to give Bob 1 slice of
pizza in exchange for 1 can of soda. Would Bob want to make this trade?
Bob’s utility before the trade is
𝒖(?, ?) = √?
√?
= ?√?
.
If Bob makes the trade, he will consume 3 slices of pizza and 3 cans of soda.
Bob’s utility after the trade would be
𝒖(?, ?) = √?
√?
= ?
.
Since Bob’s utility after the trade is greater than his util
ity before the trade, Bob would
want to make the trade.
d)
Derive the formula for Bob’s marginal rate of substitution bet
ween pizza and soda.
Bob’s marginal rate of substitution (MRS) is given by:
𝑴?? = −
𝑴𝑼
?
𝑴𝑼
?
𝑴𝑼
?
is Bob’s marginal utility from soda. It is the partial derivative of Bob’s utility functi
on
with respect to changes in the consumption of soda (holding Bob’s consumption of pizza
constant). That is, using the ‘power rule’ for derivatives (i.e, the derivative of
?
𝒌
is
𝒌?
𝒌−?
),
Bob’s marginal utility from Soda is::
𝑴𝑼
?
=
?. ??
?
?.?
?
?
?.?
=
?. ?
√
?
?
√
?
?
Similarly,
𝑴𝑼
?
=
?.?
√
?
?
√
?
?
.
Hence, Bob’s MRS is:
𝑴?? = −
𝑴𝑼
?
𝑴𝑼
?
= −
?. ?
√
?
?
√
?
?
?. ?
√
?
?
√
?
?
= − (
?. ?
√
?
?
√
?
?
) (
√
?
?
?. ?
√
?
?
) = −?
?
/?
?
Problem 2:
Alice’s preferences over apples and oranges are represented by utility function:
𝑢(?, ?) =
10?
0.5
?
0.5
where
?
is the number of apples and
?
is the number of oranges she consumes.
a)
What is the relationship between a consumer’s marginal rate of substitution and the consumer’s
indifference curve?
The marginal rate of substitution
is the slope of the consumer’s indifference curve.
b)
Write a general formula for Alice’s marginal rate of substitution (MRS) between apples and
oranges.
Alice’s marginal rate of substitution (MRS) is given by:
𝑴?? = −
𝑴𝑼
?
𝑴𝑼
?
𝑴𝑼
?
=
?√?
√?
, 𝑴𝑼
?
=
?√?
√?
. Hence, Alice
’s MRS is:
𝑴?? = −
𝑴𝑼
?
𝑴𝑼
?
= − (
?√?
√?
) (
√?
?√?
) = −
?
?
.
c)
What is Alice’s marginal rate of substitution (MRS) when she is consuming bundle
(25,5)
?
Her MRS is
−
?
??
= −
?
?
.
d)
What is Alice’s marginal rate of substitution when she is consuming bundle
(25,25)
?
Her MRS is
−
??
??
= −?
.
Problem 3:
Suppose the price of good 1 is
𝑝
1
= $5
, the price of good 2 is
𝑝
2
= $2
, and four consumers,
Alice, Bob, Charlotte, and David, each have
𝑚 = $100
available to spend on these two goods.
a)
Alice has utility function
𝑢(?
1
, ?
2
) =
√
?
1
√
?
2
. Find Alice’s demand
functions for her optimal
quantities to consume of goods 1 and 2 (i.e., find formulas for her optimal values of
?
1
and
?
2
).
Given the prices and the money she has available, what is Alice’s optim
al consumption of goods
1 and 2?
We solve the utility maximization problem for utility function U(x
1,
x
2
) =
√
?
?
√
?
?
= (x
1
1/2
)(x
2
1/2
).
Step 1: Write down the budget constraint: p
1
x
1
+ p
2
x
2
= m.
Step 2: Write down the optimality condition: MRS = - MU
1
/MU
2
= - p
1
/p
2
.
Step 3: Find MU
1
and MU
2
which are the partial derivatives of U(x
1,
x
2
) with respect to x
1
and x
2
:
MU
1
= 0.5x
1
-1/2
x
2
1/2
MU
2
= 0.5x
1
1/2
x
2
-1/2
Recall from the first day of class that the rules of exponents imply:
?
?
?
?
= ?
?−?
.
So MU
1
/MU
2
=
𝑴𝑼
?
𝑴𝑼
?
=
?.??
?
−?.?
?
?
?.?
?.??
?
?.?
?
?
−?.?
= (
?.?
?.?
) (?
?
−?.?−?.?
)(?
?
?.?−(−?.?)
) = ?
?
−?
?
?
?.?+?.?
=
?
?
?
?
.
So the optimality condition is: x
2
/x
1
= p
1
/p
2
. So x
2
= x
1
p
1
/p
2
= x
1
p
1
/p
2
Step 4: Substitute the above formula for x
2
into the budget constraint:
p
1
x
1
+ p
2
x
2
= p
1
x
1
+ p
2
(x
1
p
1
/p
2
) = p
1
x
1
+ p
1
x
1
= 2p
1
x
1
= m.
Step 5: Solve for x
1
: x
1
= m/2p
1.
Step 6: Substitute the solution for x
1
into the formula for x
2
and solve for x
2
:
?
?
=
?
?
𝒑
?
𝒑
?
= (
𝒎
?𝒑
?
) (
𝒑
?
𝒑
?
) =
𝒎
?𝒑
?
Step 7: Substitute the given values into the formulas for x
1
and x
2
:
p
1
= $5, p
2
= $2, m = $100. So x
1
= m/2p
1
= 100/10 = 10 and x
2
= m/2p
2
= 100/4 = 25.
The optimal bundle is (x
1
,x
2
) = (10, 25).
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Related Questions
True or false with reasoning:
1) _______When we claim that utility can be ordinally measured, we assume that the consumer is able to measure the total and marginal utility received when one extra unit of a commodity is consumed.
2)_______If MRS between two goods is constant, then having more of one good without having more of the other does not increase utility.
3)_______Marginal Utility increases until total utility is at a maximum and then marginal utility decreases.
arrow_forward
RM2
1. Assume Good X and Good Y are perfect complements and both are normal goods. Suppose the price of X increases while the price of Y remains unchanged. On a preference map, a. demonstrate the change in the optimal consumption bundle. b. On a separate graph, decompose the substitution effect and the income effect.
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10) Suppose that the utility function of an individual can be described as U(X,Y) = 4X +2Y. For this utility function the MRSA) is always X*YB) is always constantC) is always X/YD) is always X+YE) is always X-Y.
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4
Assume that a person's utility over two goods is given by U(g1; g2) = g1 + ln g2. The price of good g1 is equal to p1 and the price of good g2 is p2. The total income of the individual is given by I. The marginal rate of substitution between g1 and g2 is given by 1/(1/g2). Then, the expressions for this person's (1) budget constraint, (2) budget line's slope (assume that, graphically, g1 is on the horizontal axis and g2 on the vertical axis), and (3) the person's demand function for g2 (that is, g2 as a function of price ratio) are respectively:
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7)
Suppose that upon solving for optimal bundles of hamburgers and french fries, we determine that when the price of hamburgers is $5 and the price of fries is $2, that we consume 2 hamburgers and 5 orders of fries. Suppose further that, upon changing the price of fries to $4, we only consume 3 orders of fries.
Draw out a demand curve for fries, labeling two points. This is not a trick question, this one is supposed to be easy.
arrow_forward
27. If you have Marginal utility of 2
with a Total utility equals to 56,
then when marginal utility is
reduce to zero while Total utility
remains the same, what does it
mean?
a. at 56 utils of MU equals zero, utility
begins to gain momentum
b. at 56 utils of MU equals zero, utility
begins to diminish
c. at 56 utils of MU equals zero. utility
begins to increase
d. at 56 utils of MU equals zero, utility
begins to upswing
arrow_forward
#10. Assume that Thomas can afford to buy as many candy bars and ice cream cones as he wants. He would continue to consume both candy bars and ice cream until the a. marginal utility of each decreases. b. marginal utility of each becomes negative. c. total utility of each becomes negative. d. marginal utility of candy bars and ice cream bars is equal. e. total utility of candy bars and ice cream bars is equal.
arrow_forward
A consumer finds only three products, X, Y, and Z, are for sale. The amount of utility which their consumption will yield is shown in the table below. Assume that the prices of X, Y, and Z are $10, $2, and $8, respectively, and that the consumer has an income of $74 to spend.
Product X
Product Y
Product Z
Quantity
Utility
Marginal
Utility
per $
Quantity
Utility
Marginal
Utility
per $
Quantity
Utility
Marginal
Utility
per $
1
42
NA
1
14
NA
1
32
NA
2
82
4
2
26
6
2
60
3.5
3
118
3.6
3
36
5
3
84
3
4
148
3
4
44
4
4
100
2
5
170
2.2
5
50
3
5
110
1.25
6
182
1.2
6
54
2
6
116
0.75
7
182
_0
7
56.4
_1.2
7
120
_0.5_
Why would the consumer not be maximizing utility by purchasing 2 units of X, 4 units of Y, and 1 unit of Z?
arrow_forward
A consumer finds only three products, X, Y, and Z, are for sale. The amount of utility which their consumption will yield is shown in the table below. Assume that the prices of X, Y, and Z are $10, $2, and $8, respectively, and that the consumer has an income of $74 to spend.
Product X
Product Y
Product Z
Quantity
Utility
Marginal
Utility
per $
Quantity
Utility
Marginal
Utility
per $
Quantity
Utility
Marginal
Utility
per $
1
42
NA
1
14
NA
1
32
___NA__
2
82
4
2
26
6
2
60
__3.5_
3
118
3.6
3
36
5
3
84
__3___
4
148
3
4
44
4
4
100
__2___
5
170
2.2
5
50
3
5
110
_1.25___
6
182
1.2
6
54
2
6
116
_0.75__
7
182
0
7
56.4
1.2
7
120
_0.5_
How many units of X, Y, and Z will the consumer buy when maximizing utility and spending all…
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6. MRS and utility maximization
Suppose your classmate Bob loves to eat dessert—so much so that he allocates his entire weekly budget to apple crisp and pie. The price of one bowl of apple crisp is $1.25, and the price of a piece of blueberry pie is $3.75. At his current level of consumption, Bob's marginal rate of substitution (MRS) of apple crisp for pie is 2. In other words, Bob is willing to sacrifice two bowls of apple crisp for one piece of pie per week.
Does Bob's current consumption bundle maximize his utility? That is, does it make him as well off as possible? If not, how should he change it to maximize his utility?
Bob could increase his utility by buying more apple crisp and less pie per week.
Bob could increase his utility by buying less apple crisp and more pie per week.
Bob's current bundle maximizes his utility, and he should keep it unchanged.
arrow_forward
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arrow_forward
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- 27. If you have Marginal utility of 2 with a Total utility equals to 56, then when marginal utility is reduce to zero while Total utility remains the same, what does it mean? a. at 56 utils of MU equals zero, utility begins to gain momentum b. at 56 utils of MU equals zero, utility begins to diminish c. at 56 utils of MU equals zero. utility begins to increase d. at 56 utils of MU equals zero, utility begins to upswingarrow_forward#10. Assume that Thomas can afford to buy as many candy bars and ice cream cones as he wants. He would continue to consume both candy bars and ice cream until the a. marginal utility of each decreases. b. marginal utility of each becomes negative. c. total utility of each becomes negative. d. marginal utility of candy bars and ice cream bars is equal. e. total utility of candy bars and ice cream bars is equal.arrow_forwardA consumer finds only three products, X, Y, and Z, are for sale. The amount of utility which their consumption will yield is shown in the table below. Assume that the prices of X, Y, and Z are $10, $2, and $8, respectively, and that the consumer has an income of $74 to spend. Product X Product Y Product Z Quantity Utility Marginal Utility per $ Quantity Utility Marginal Utility per $ Quantity Utility Marginal Utility per $ 1 42 NA 1 14 NA 1 32 NA 2 82 4 2 26 6 2 60 3.5 3 118 3.6 3 36 5 3 84 3 4 148 3 4 44 4 4 100 2 5 170 2.2 5 50 3 5 110 1.25 6 182 1.2 6 54 2 6 116 0.75 7 182 _0 7 56.4 _1.2 7 120 _0.5_ Why would the consumer not be maximizing utility by purchasing 2 units of X, 4 units of Y, and 1 unit of Z?arrow_forward
- A consumer finds only three products, X, Y, and Z, are for sale. The amount of utility which their consumption will yield is shown in the table below. Assume that the prices of X, Y, and Z are $10, $2, and $8, respectively, and that the consumer has an income of $74 to spend. Product X Product Y Product Z Quantity Utility Marginal Utility per $ Quantity Utility Marginal Utility per $ Quantity Utility Marginal Utility per $ 1 42 NA 1 14 NA 1 32 ___NA__ 2 82 4 2 26 6 2 60 __3.5_ 3 118 3.6 3 36 5 3 84 __3___ 4 148 3 4 44 4 4 100 __2___ 5 170 2.2 5 50 3 5 110 _1.25___ 6 182 1.2 6 54 2 6 116 _0.75__ 7 182 0 7 56.4 1.2 7 120 _0.5_ How many units of X, Y, and Z will the consumer buy when maximizing utility and spending all…arrow_forward6. MRS and utility maximization Suppose your classmate Bob loves to eat dessert—so much so that he allocates his entire weekly budget to apple crisp and pie. The price of one bowl of apple crisp is $1.25, and the price of a piece of blueberry pie is $3.75. At his current level of consumption, Bob's marginal rate of substitution (MRS) of apple crisp for pie is 2. In other words, Bob is willing to sacrifice two bowls of apple crisp for one piece of pie per week. Does Bob's current consumption bundle maximize his utility? That is, does it make him as well off as possible? If not, how should he change it to maximize his utility? Bob could increase his utility by buying more apple crisp and less pie per week. Bob could increase his utility by buying less apple crisp and more pie per week. Bob's current bundle maximizes his utility, and he should keep it unchanged.arrow_forward2. Tom spends all his $100 weekly income on two goods, apples and bananas. His utility function is given by U (A, B) = AB, where A and B stand for the quantity of apples and bananas consumed by Tom. If PA = $4 and PB= $10, how many apples and bananas will he consume? Make sure you write out the utility maximization problem explicitly, including the decision variable(s). What if his utility function is given by U (A, B) = A0.5B 0.5?arrow_forward
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