Warwick Economics Summer School 2016
Problem set 1
Answers
Question 1
The basic labour supply model assumes that wage income is untaxed. Suppose instead that
a set of marginal tax rates is imposed such that at a wage of £10 per hour, the first 5 hours
of labour is untaxed, the next 10 hours of labor are taxed at a 20% rate, and all labor
thereafter is taxed at a 50% rate. Show on a graph how this would affect the budget line.
How might this affect the number of hours worked?
Question 2
Suppose we are interested in looking at Andrew’s consumption of wine. One way to do this is to
assume Andrew buys wine, and a combination of all other goods: the composite good.
1. How would you adopt this interpretation in writing Andrew’s budget constraint?
a. To adopt this interpretation it is useful to think about the composite good as being
the amount of money (£) left for Andrew to spend on the combination of all the
other goods. Therefore, the price of one unit of the composite good would be set
to 1 (the price of one pound, is one pound) and the budget constraint would be
such as
𝐏𝐰 𝐖 + 𝐂 = 𝐈
Where C represents the composite good (everything else Andrew can spend his money on
that is not wine), W the quantity of wine Andrew purchase, and 𝑷𝒘 the price of wine.
At your high-school's fund-raising picnic, you pay for soft drinks with tickets purchased in advance
-one ticket per bottle of soft drink. Tickets are available in sets of three types:



Small: £3 for 3 tickets
Medium: £4 for 5 tickets
Large: £5 for 8 tickets.
The total amount you have to spend is £12. If fractional sets of tickets cannot be bought (and no
resale of tickets is possible), graph your budget constraint for soft drinks and the composite good.
Short way to answer is to assume \free-disposal" of drinks and money. This means that
if any quantity (x*; y*) is available, then so is (x’; y’) where x*≤ x’ and y*≤y’. Therefore,
the budget line will include all these bundles: (0,12), (3,9), (5,8), (8,7), (11,4), (13,3), (16,2).
And bundles that are in between. This yields a step-shaped budget line as shown in the figure
below.
Longer way to answer is to work out the budget for every quantity:
(1) If no soft drinks are bought then there is £12 remaining to spend on the composite
commodity.
(2) One soft drink can be obtained at lowest cost by buying 3 tickets, 2 can obtained by
buying 3 tickets, 3 can be obtained by buying 3 tickets. Thus, zero to 3 soft drinks cost £3
which leaves £9 to spend on the composite commodity. This is shown in the figure as the line
segment (0,9) (3,9).
(3) Four soft drinks can be obtained at lowest cost by buying 5 tickets at a cost of £4, five soft
drinks cost the same. This is shown in the figure as the line segment (3,8)—(5,8).
(4) And so on...
Composite good (in money terms)
12
9
8
7
4
3
2
Soft drinks (tckts)
3
5
8
11
13
16
Question 3
What is a ‘bad’ good? Given that definition, draw an indifference curve for a consumer making a
choice between Chocolate and Broccoli, where broccoli is assumed to be a ‘bad’.
a. A bad is a commodity that the consumer doesn’t like. There could still be a trade-off between
a good and a bad, in which case, to consume more of the bad, the consumer would want to
be given more of the ‘good’
Broccoli
Chocolate
Question 4
What is a neutral good? Given that definition, draw an indifference curve for a consumer
making a choice between Chocolate and Broccoli, where broccoli is now assumed to be neutral
b. Consumers are indifferent about neutral goods. They do not care about consuming them,
neither do they care about not. In which case, the indifference curve will be vertical at the
chosen amount of the other good, as increasing the consumption of the neutral good does
not change the satisfaction of the consumer.
Broccoli
Chocolate
Question 5
Suppose there are 3 different goods: cups of coffee (x1), ounces of milk (x2) and packets of sugar
(x3). Assume that each of these goods costs 25p and you have an exogenous income of £15.
(a) Suppose that the only way you get enjoyment from a cup of coffee is to have at least one ounce
of milk and one packet of sugar in the coffee, the only way you get enjoyment from an ounce of
milk is to have at least one cup of coffee and one packet of sugar, and the only way you get
enjoyment from a packet of sugar is to have at least one cup of coffee and one ounce of milk.
What is the optimal consumption bundle on your budget constraint?
(b) How will your optimal consumption bundle change if the price of coffee rises to 50p?
(c) Return to the original prices of 25p for all goods. Write down the budget constraint.
(d) Write down a utility function that represents the tastes described above.
(e) Suppose that instead your tastes are less extreme and can be represented by the utility function
U (x1, x2, x3) = x1αx2βx3. Calculate optimal consumption of each good when your economic
circumstances are described by the prices p1, p2 and p3 and income is given by I.
At 25p each, 60 units of one good can be bought with £15, assuming nothing else is bought.
The goods are perfect complements, implying with given prices and income x1 = x2 = x3 = 20
and L-shaped indifference curves.
(f) Same quantities of each will still be bought. 0.5x1 + 0.25x2 + 0.25x3 = 15, but x1 = x2 = x3 at any
optimum.
(g) 0.25x1 + 0.25x2 + 0.25x3 = 15
(h) U(x1, x2, x3) = min{x1, x2, x3}
(i) U (x1, x2, x3) = x1αx2βx3. Take logs and maximise utility function. Take FOCs:
α/x1 = λp1
β/x2 = λp2
1/x3 = λp3
P1x1 + p2x2 + p3x3 = I
Solve for x1 = αp3x3/p1 and x2 = βp3x3/p2 by solving equation 3 and substituting into equations
1 and 2. Substitute these into budget constraint and solve for
x1 = αI/(α+β+1)p1 and x2 = βI/(α+β+1)p2 and x3 = I/(α+β+1)p3
Question 6
Suppose you solve a consumer's constrained 2-good optimisation problem for a given economic
environment --- and your answer contains a negative consumption level of good 2. Which of the
following is a valid conclusion on your part?
A. The true optimum has the consumer consume none of good 1.
B. The true optimum has the consumer consume none of good 2.
C. There are multiple “true” optimal consumption bundles.
D. The consumer will sell good 2.
E. None of the above.
Answer is B. FOCs identify tangencies wherever they occur. If consumption is negative for one
good, it means the true optimum has the consumer consume none of that good, with all income
spent on the other good – corner solution.
Question 7
Suppose Joe's utility for lobster (L) and squash (S) can be represented as U = L0.8S0.2. Joe walks into a
restaurant with £90. Lobsters cost £24 each and glasses of squash cost £2 each. How much lobster
and squash will Joe consume if he intends to spend all his money? (There are no taxes and no tips.)
Maximizing Joe's utility subject to his budget constraint yields:
U = L0.8S0.2 + λ(90 - 24L - 2S)
FOCS:
1) dU/dL = 0.8L-0.2S0.2 – λ24 = 0
2) dU/dL = 0.2L0.8S-0.8 – λ2 = 0
3) dU/dλ = 90 - 24L - 2S = 0
From 1) and 2), S/L = 3 or S = 3L. Substituting into 3) yields L = 3. Since S = 3L, S = 9. Thus, Joe will
buy 3 lobsters and wash it all down with 14 glasses of squash.
Question 8
Suppose the utility function for good x and y is given by 𝑈(𝑥, 𝑦) = 𝑥𝑦 + 𝑦.
a. Calculate the uncompensated (Marshallian) demand functions for x and y, describe how the demand
curves for x and y are shifted by changes in income M or the price of the other good.
The Lagrange method yields
𝒚
𝑷𝒙
=
𝒙 + 𝟏 𝑷𝒚
𝑷𝒚 𝒚 = 𝑷𝒙 𝒙 + 𝑷𝒙
By substituting into the budget constraint, we obtain
𝒙=
𝑰 − 𝑷𝒙
𝟐𝑷𝒙
𝒚=
𝑰 + 𝑷𝒙
𝟐𝑷𝒚
Hence, changes in py do not affect x, but changes in px do affect y.
b. Calculate the expenditure function for x and y.
The indirect utility function is
(𝑰 + 𝑷𝒙 )𝟐
𝑽=
𝟒𝑷𝒙 𝑷𝒚
which yields an expenditure function of
𝑬 = √𝑽𝟒𝒑𝒙 𝑷𝒚 − 𝑷𝒙
c. Use the expenditure function calculated in (b) to compute the compensated demand functions for
goods x and y. Describe how the compensated demand curves for x and y are shifted by changes in
income or by the changes in the price of the other good.
Clearly the compensated demand function for x depends on py whereas the uncompensated
function doesn’t. By Shepherd’s Lemma:
𝒙𝒄 =
𝝏𝑬
= 𝑽𝟎.𝟓 𝒑−𝟎.𝟓
𝒑𝟎.𝟓
𝒙
𝒚 −𝟏
𝝏𝑷𝒙
Question 9
A college student who loves chocolate has a budget of £10 a day, and out of that income she
purchases chocolate x and a composite good y. The price of the composite good is £1. The student’s
preferences are represented by the quasi-linear utility function 𝑈(𝑥, 𝑦) = 2√𝑥 + 𝑦.
a. Find the marginal utilities
𝑴𝑼𝒙 =
𝟏
√𝒙
; 𝑴𝑼𝒚 = 𝟏
b. Suppose the price of chocolate is initially £0.50 per ounce. How many ounces of chocolate and how
many units of the composite good are in the student’s optimal consumption basket?
𝑴𝑼𝒙 𝑴𝑼𝒚
𝟏
=
→
= 𝑷𝒙
𝑷𝒙
𝑷𝒚
√𝒙
The student’s demand curve is therefore given by 𝒙 = 𝟏⁄
. At a price of £0.50, the student
(𝑷𝒙 )𝟐
buys 𝟏⁄(𝟎. 𝟐 = 𝟒 ounces of chocolate. From the budget constraint, we can obtain the amount of
𝟓)
the composite good the student consumes:
𝑷𝒙 𝒙 + 𝑷𝒚 𝒚 = 𝑰 → 𝟎. 𝟓(𝟒) + (𝟏)𝒚 = 𝟏𝟎 → 𝒚 = 𝟖
c. Suppose the price of chocolate drops to £0.20 per ounce. How many ounces of chocolate and how
many units of the composite good are in the new optimal consumption basket?
𝑴𝑼𝒙 𝑴𝑼𝒚
𝟏
=
→
= 𝑷𝒙
𝑷𝒙
𝑷𝒚
√𝒙
The student’s demand curve is therefore given by 𝒙 = 𝟏⁄(𝑷 )𝟐 . At a price of £0.20, the student
𝒙
buys 𝟏⁄(𝟎. 𝟐 = 𝟐𝟓 ounces of chocolate. From the budget constraint, we can obtain the amount
𝟐)
of the composite good the student consumes:
𝑷𝒙 𝒙 + 𝑷𝒚 𝒚 = 𝑰 → 𝟎. 𝟓(𝟐𝟓) + (𝟏)𝒚 = 𝟏𝟎 → 𝒚 = 𝟓
d. What are the substitution and income effects that result from the decline in the price of chocolate?
Illustrate these effects on a graph.
Initially the consumer consumes 𝒙 = 𝟒 and 𝒚 = 𝟖 and obtains a utility level 𝑼𝟏 = 𝟐√𝟒 + 𝟖 = 𝟏𝟐.
At the compensated basket, it must therefore be true that 𝑼(𝒙, 𝒚) = 𝟐√𝒙 + 𝒚 = 𝟏𝟐. We also
know that
𝑴𝑼𝒙
𝑷𝒙
=
𝑴𝑼𝒚
𝑷𝒚
must be such that
𝟏
√𝒙
= 𝟎. 𝟐𝟎. Solving for x and y from those two equations
with two unknowns we get 𝒙 = 𝟐𝟓 and 𝒚 = 𝟐.
The substitution effect is therefore 𝟐𝟓 − 𝟒 = 𝟐𝟏 ounces of chocolate and the income effect is the
rest of the effect, that is, zero.
Other goods
A
C
𝑈2 = 15
B
C
𝐵𝐿2
𝑈1 = 12
𝐵𝐿1
Chocolate
Question 10
Morgane purchases two goods, food and clothing. She has the utility function 𝑈(𝑋, 𝑌) = 𝑥𝑦, where
denotes the amount of food consume and y the amount of clothing. Denote the price of 𝑥 as 𝑃𝑥 and
the price of 𝑦 as 𝑃𝑦 .
(a) Find the marginal utilities.
𝑴𝑼𝒙 = 𝒚 ; 𝑴𝑼𝒚 = 𝒙
(b) In terms of marginal utilities found above, write down the consumer’s optimum condition and
rearrange to find an expression for x in terms of y.
𝑴𝑼𝒙 𝑴𝑼𝒚
=
𝑷𝒙
𝑷𝒚
𝑷𝒚
𝒚
𝒙
=
→ 𝒙 = 𝒚( )
𝑷𝒙 𝑷𝒚
𝑷𝒙
(c) Write the budget constraint, substitute for x from the expression found in (b).
𝑷𝒙 𝒙 + 𝑷𝒚 𝒚 = 𝑰
𝑷𝒙 𝒚 (
𝑷𝒚
) + 𝑷𝒚 𝒚 = 𝑰
𝑷𝒙
𝟐𝑷𝒚 𝒚 = 𝑰
𝒚=
𝑰
𝟐𝑷𝒚
(d) What can you say about the expression found above?
It is the equation for Morgane’s demand curve.
(e) Is clothing a normal good?
Yes, it is a normal good. As her income increases, holding the price of y (clothing) constant, the
quantity demanded by Morgane of clothing increases.
Suppose now that Morgane has an income of £72 per week, that the price of clothing is 𝑃𝑦 = £1.
Suppose also that the price of food is initially 𝑃𝑥1 = £9 per unit, and that the price falls to 𝑃𝑥1 = £4.
a. Find the initial consumption basket when the price of food is £9, by using the two equilibrium
conditions (budget constraint and tangency condition).
𝑷𝒙 𝒙 + 𝑷𝒚 𝒚 = 𝑰 → 𝟗𝒙 + 𝒚 = 𝟕𝟐
𝑴𝑼𝒙 𝑴𝑼𝒚
=
→ 𝒚 = 𝟗𝒙
𝑷𝒙
𝑷𝒚
Solving for x and y we get: 𝒙 = 𝟒 and 𝒚 = 𝟑𝟔
b. Repeat this step to find the consumption basket when is price of food is £4
𝑷𝒙 𝒙 + 𝑷𝒚 𝒚 = 𝑰 → 𝟒𝒙 + 𝒚 = 𝟕𝟐
𝑴𝑼𝒙 𝑴𝑼𝒚
=
→ 𝒚 = 𝟒𝒙
𝑷𝒙
𝑷𝒚
Solving for x and y we get: 𝒙 = 𝟗 and 𝒚 = 𝟑𝟔
c. Find the utility level corresponding to the initial consumption basket.
𝑼(𝒙, 𝒚) = 𝒙𝒚 = 𝟒(𝟑𝟔) = 𝟏𝟒𝟒
d. Find the compensated basket, and therefore the substitution effect. Remember that at that point,
Morgane faces the new budget constraint, but is compensated to obtain the same level of utility as
before the change in price.
𝒙𝒚 = 𝟏𝟒𝟒
𝑴𝑼𝒙 𝑴𝑼𝒚
=
→ 𝒚 = 𝟒𝒙
𝑷𝒙
𝑷𝒚
Solving for x and y we get: 𝒙 = 𝟔 and 𝒚 = 𝟐𝟒
The substitution effect is therefore 𝟔 − 𝟒 = 𝟐, or 2 units of food.
e. Using your answer in b. and d., find the income effect.
The income effect is the remaining change in quantity to obtain the new consumption basket,
or 𝟗 − 𝟔 = 𝟑, that is 3 units of food.
f.
Graph your results.
Clothing, y
𝑈2 = 324
36
24
𝑈1 = 144
𝐵𝐿𝑐𝑜𝑚𝑝
𝐵𝐿1
4
6
9
𝐵𝐿2
Food, x