ECON 102 Tutorial: Week 4
Shane Murphy
www.lancaster.ac.uk/postgrad/murphys4
Maths & Stats Help
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Q1: Chapter 4 Problem 2
You are having lunch at an all-you-can-eat buffet. If you are rational,
what should be your marginal utility from the last morsel of food you
swallow?
Q1: Chapter 4 Problem 2
You are having lunch at an all-you-can-eat buffet. If you are rational,
what should be your marginal utility from the last morsel of food you
swallow?
Since the marginal cost of an additional morsel of food is zero, a
rational person will continue eating until the marginal benefit of the
last morsel (its marginal utility) falls to zero.
Q2: Chapter 4 Problem 3
Martha’s current marginal utility from consuming orange juice is 75 utils per
centilitre and her marginal utility from consuming coffee is 50 utils per centilitre.
If orange juice costs 25 cents per centilitre and coffee costs 20 cents per centilitre, is
Martha maximising her total utility from the two beverages?
If so, explain how you know. If not, how should she rearrange her spending?
Q2: Chapter 4 Problem 3
Martha’s current marginal utility from consuming orange juice is 75 utils per centilitre and
her marginal utility from consuming coffee is 50 utils per centilitre.
If orange juice costs 25 cents per centilitre and coffee costs 20 cents per centilitre, is Martha
maximising her total utility from the two beverages?
If so, explain how you know. If not, how should she rearrange her spending?
We can find Martha’s Marginal Rate of Substitution between orange juice and coffee:
Martha’s MRSO,C :
𝑀𝑅𝑆𝑂,𝐶 =
𝑀𝑈𝑂
𝑀𝑈𝐶
=
75
50
= 1.5
We know the Rational Spending Rule states that utility is maximized if MRSO,C = PO/PC.
We can find the price ratio:
Because 𝑀𝑅𝑆𝑂,𝐶 ≠
𝑃𝑂
𝑃𝐶
𝑃𝑂
𝑃𝐶
=
.25
.20
= 1.25
, Martha is not maximizing her utility.
The Rational Spending Rule states that:
If 𝑀𝑅𝑆𝑂,𝐶 >
𝑃𝑂
𝑃𝐶
, then Martha should spend more on orange juice and less on coffee.
Q3: Chapter 4 Problem 8
Tom has a weekly allowance of €24, all of which he spends on pizza and
movie rentals, whose prices are €6 per slice and €3 per rental, respectively.
If slices of pizza and movie rentals are available only in whole-number
amounts, list all the possible combinations of the two goods that Tom can
purchase each week with his allowance.
Q3: Chapter 4 Problem 8
Tom has a weekly allowance of €24, all of which he spends on pizza and
movie rentals, whose prices are €6 per slice and €3 per rental, respectively.
If slices of pizza and movie rentals are available only in whole-number
amounts, list all the possible combinations of the two goods that Tom can
purchase each week with his allowance.
price:
price:
price:
price:
no.
no.ofofmovies
movies £3/movie
£3/movie slices
slicesofofpizza
pizza £6/slice
£6/slice total
totalcost
cost
0
£0
4
£24
£24
1
£3
3
£18
£21
2
£6
3
£18
£24
3
£9
2
£12
£21
4
£12
2
£12
£24
5
£15
1
£6
£21
6
£18
1
£6
£24
7
£21
0
£0
£21
8
£24
0
£0
£24
Q3: Chapter 4 Problem 8
Tom has a weekly allowance of €24, all of which he spends on pizza and
movie rentals, whose prices are €6 per slice and €3 per rental, respectively.
If slices of pizza and movie rentals are available only in whole-number
amounts, list all the possible combinations of the two goods that Tom can
purchase each week with his allowance.
no. of movies
price:
£3/movie
slices of pizza
price:
£6/slice
total cost
0
£0
4
£24
£24
1
£3
3
£18
£21
2
£6
3
£18
£24
3
£9
2
£12
£21
4
£12
2
£12
£24
5
£15
1
£6
£21
6
£18
1
£6
£24
7
£21
0
£0
£21
8
£24
0
£0
£24
Q4: Chapter 4 Problem 9
Refer to Problem 8. Tom’s total utility is the sum of the utility he derives
from pizza and movie rentals. If these utilities vary with the amounts
consumed as shown in the table below, and pizzas and movie rentals are
again consumable only in whole-number amounts, how many pizzas and
how many movie rentals should Tom consume each week?
pizzas per utils/week movies
week from pizza per week
0
0
0
1
20
1
2
38
2
3
54
3
4
68
4
5
80
5
6
90
6
7
98
7
8
104
8
utils/week
from
movies
0
40
46
50
54
56
57
57
57
Q4: Chapter 4 Problem 9
Refer to Problem 8. Tom’s total utility is the sum of the utility he derives
from pizza and movie rentals. If these utilities vary with the amounts
consumed as shown in the table below, and pizzas and movie rentals are
again consumable only in whole-number amounts, how many pizzas and
how many movie rentals should Tom consume each week?
Assuming that Tom spends his entire weekly allowance, £24, the affordable combinations
and their corresponding utilities are as listed in the table below:
Combinations of pizza and rentals that cost €24 per week
Total Utility
0 pizzas, 8 rentals
0 + 57 = 57
1 pizza, 6 rentals
20 + 57 = 77
2 pizzas, 4 rentals
38 + 54 = 92
3 pizzas, 2 rentals
54 + 46 = 100
4 pizzas, 0 rentals
68 + 0 = 68
From this table, we can see that Tom gets the most utility from 3 pizzas per week and 2
movie rentals. This is his optimal combination.
Question 5(a)
Ann buys only orange juice and yogurt.
In 2014, Ann earns £20,000, orange juice is priced at £2 a carton, and yogurt is
priced at £4 a tub.
Draw Ann’s budget constraint.
(For each part of this question, put the quantity of orange juice on the y-axis, and the quantity of yogurt on the x-axis)
Question 5(a)
Ann buys only orange juice and yogurt.
In 2014, Ann earns £20,000, orange juice is priced at £2 a carton, and yogurt is
priced at £4 a tub.
Draw Ann’s budget constraint.
(For each part of this question, put the quantity of orange juice on the y-axis, and the quantity of yogurt on the x-axis)
XOJ
10,000
5,000
XY
The diagram above show’s Ann’s budget constraint.
If Ann buys only orange juice, she can afford 10,000 cartons.
If she buys nothing buy yogurt she can afford 5,000 tubs.
Note that we can calculate the slope of this line by: -10,000/5,000 = -2.
Question 5(b)
Suppose that all prices increase by 10 percent in 2015, and Ann’s salary remains
constant. Draw Ann’s new budget constraint.
We know that in 2014: Ann earned £20,000; Poj = £2 a carton; Py = £4 a tub.
Question 5(b)
Suppose that all prices increase by 10 percent in 2015, and Ann’s salary remains constant. Draw Ann’s
new budget constraint.
We know that in 2014: Ann earned £20,000; Poj = £2 a carton; Py = £4 a tub.
So then, in 2015: No change to earnings; Poj = £2.20 a carton; Py = £4.40 a tub.
Ann’s new budget line is depicted in red.
XOJ
Since both prices have increased by 60%, the relative price of yogurt to
orange juice did not change.
This means that the slope of the budget line is the same as in 2014. 9,090
However, since both prices have increased, Ann is not able to afford the
same bundles as before.
After the price increase, the price of orange juice is £2.2, and the price
of yogurt is £4.4.
So, if Ann buys nothing but orange juice, she can afford 20,000/2.2 =
9,090.90 cartons. However, since we can’t buy .9 of a carton of juice,
Anne can afford only 9,090 cartons.
If Ann buys only yogurt, she can afford 20,000/4.4 = 4,545.45 tubs.
Again, because if we can’t buy .45 of a tub of yogurt, Anne can afford
4,545 tubs of yogurt.
Notice that a proportional increase in all prices (leaving income
unchanged) has the same effect as a decrease in income (leaving all
prices unchanged).
4,545
XY
Question 5(c)
Now suppose that all prices increase by 10 percent in 2015 and that Ann’s salary
increases by 10 percent as well.
Draw Ann’s new budget constraint.
How will Ann’s optimal combination of orange juice and yogurt in 2015 compare to
her optimal combination in 2014?
We know that in 2014: Ann earned £20,000; Poj = £2 a carton; Py = £4 a tub.
Question 5(c)
Now suppose that all prices increase by 10 percent in 2015 and that Ann’s salary
increases by 10 percent as well.
Draw Ann’s new budget constraint.
How will Ann’s optimal combination of orange juice and yogurt in 2015 compare to
her optimal combination in 2014?
We know that in 2014: Ann earned £20,000; Poj = £2 a carton; Py = £4 a tub.
So then, in 2015: Ann earns £22,000;
Poj = £2.2 a carton; Py = £4.4 a tub.
XOJ
10,000
Ann’s new budget constraint is exactly the same as in part a.
When all prices increase by 10 percent, and her income
increases by 10 percent, she is able to afford exactly the
same bundles as in 2014.
Since her budget constraint is unchanged (and nothing about
her preferences has changed), her optimal combination of
goods is exactly the same as in 2014.
5,000
XY
Question 6
Tim’s marginal rate of substitution between goods 1 and 2 is given by:
2𝑥1
𝑀𝑅𝑆1,2 =
3𝑥2
The price of good 1 is $3/unit, and the price of good 2 is $1/unit.
How many units of each good will Tim purchase if he has $20 to spend?
Rational Spending Rule
If consumption bundle (x1*, x2*) maximises the
utility of an individual, given her budget set,
then it satisfies the condition
MRS1,2
benefit of
1 additional
unit of good 1
(expressed in
units of good 2)
p1
=
p2
cost of
1 additional
unit of good 1
(in units of good 2)
• If MRS1,2 > P1/P2 then buy more of good 1 and less of good 2.
• If MRS1,2 < P1/P2 then buy more of good 2 and less of good 1.
Question 6
Tim’s marginal rate of substitution between goods 1 and 2 is given by:
2𝑥1
𝑀𝑅𝑆1,2 =
3𝑥2
The price of good 1 is $3/unit, and the price of good 2 is $1/unit.
How many units of each good will Tim purchase if he has $20 to spend?
Question 6
Tim’s marginal rate of substitution between goods 1 and 2 is given by: 𝑀𝑅𝑆1,2 =
3𝑥2
2𝑥1
The price of good 1 is $3/unit, and the price of good 2 is $1/unit.
How many units of each good will Tim purchase if he has $20 to spend?
We know that Tim is rational and wants to maximize utility. To maximize utility, the rational spending rule tells us that
MRS1,2 should equal the price ratio, p1/p2.
So, we set MRS1,2 = p1/p2, which gives:
𝑀𝑅𝑆1,2 =
2𝑥1
3𝑥2
=
𝑝1
𝑝2
3
1
9𝑥2 = 2𝑥1
We can cross multiply to get:
9
2
𝑥1∗ = 𝑥2∗
This can be simplified to:
The optimal bundle must exhaust Tim’s income. So the budget constraint is: 𝑝1 𝑥1∗ + 𝑝2 𝑥2∗ = 𝑚, where m is the
$20 he has available to spend.
Plugging in the prices, we can re-write this as:
3𝑥1∗ + 𝑥2∗ = 20
Now, we have a system of two linear equations, with two unknowns.
9
We can substituting 𝑥1∗ = 𝑥2∗ from t he rational spending rule into the second equation, Tim’s budget
2
constraint:
9
2
3( 𝑥2∗ ) + 2𝑥2∗ = 20
Solving this equation gives x*2 =
We can then plug x*2 =
40
29
40
29
.
in to either of our first two equations to find x*1 =
180
.
29
Question 7: Ch. 5 Problem 1(a)
Zoe is trying to decide how to divide her time between her job as a wedding
photographer, which pays €27 per hour for as many hours as she chooses to
work, and as a fossil collector, in which her pay depends both on the price of
fossils and the number of them she finds. Earnings aside, Zoe is indifferent
between the two tasks, and the number of fossils she can find depends on
the number of hours a day she searches, as shown in the table below.
Hours per day
Total fossils per day
(1)
(2)
1
5
2
9
3
12
4
14
5
15
Derive a table with price in euro increments from €0
to €30 in column (1) and the quantity of fossils Zoe is
willing to supply per day at that price in column (2).
Question 7: Ch. 5 Problem 1(a)
Derive a table with price in euro increments from €0 to €30
in column (1) and the quantity of fossils Zoe is willing to
supply per day at that price in column (2).
1. If the price of a fossil is less than €6, Zoe should devote all her
time to photography because when the price is, say, €5 per fossil, an
hour spent looking for fossils will give her 5(€5) = €25, or €2 less
than she’d earn doing photography. If the price of fossils is 6, Zoe
should spend one hour searching, will supply 5 fossils, and will get
€30 revenue, which is €3 more than she’d earn from photography.
However, an additional hour would yield only 4 additional fossils or
€24 additional revenue, so she should not spend any further time
looking for fossils. If the price of fossils rises to €7, however, the
additional hour gathering fossils would yield an additional €28, so
gathering fossils during that hour would then be the best choice,
and Zoe would therefore supply 9 fossils per day. Using this
reasoning, we can derive a price-quantity supplied relationship for
fossils as follows:
Price of fossils (€)
0-5
6
7, 8
9-13
14-26
27+
Hours per day
Total fossils per day
(1)
(2)
1
5
2
9
3
12
4
14
5
15
Number of fossils supplied per day
0
5
9
12
14
15
Question 7: Ch. 5 Problem 1(b)
Using the table you found in part (a), plot these points in a
graph with price on the vertical axis and quantity per day
on the horizontal. What is this curve called?
If we plot these points, we get Zoe’s daily supply curve for
fossils:
Price of
fossils (€)
0-5
6
7, 8
9-13
14-26
27+
Number of fossils
supplied per day
0
5
9
12
14
15
Next Week
 Multiple Choice Exam in Week 6 on Friday
 Moset questions from Prof. Rietzke (economics questions)
 Covering material up until the Tutorial 6 worksheet
 Some questions from Prof. Peel (maths questions)
 Exactly like his maths practice questions at the end of lecture notes.
 Check your timetable for exam time and location.
 Study & Review lecture slides, tutorial questions, and maths
questions.