SUMMER ASSESSMENT 2022
ECON0013: MICROECONOMICS
Answer the question in Part A, and ONE question from Part B.I and ONE question from Part B.II.
This assessment accounts for 60 per cent of the marks for the course. Each question carries an equal
percentage of the total mark.
In cases where a student answers more questions than requested by the assessment rubric, the policy of
the Economics Department is that the student’s first set of answers up to the required number will be the
ones that count (not the best answers). All remaining answers will be ignored. No credit will be given for
reproducing parts of the course notes. The answer to each part of each question should be on at most one
page (for example A.1 has 6 parts and there should be at most 6 pages of answers to this). Any part of
any answer that violates this will be given zero marks.
ECON0013 1 TURN OVER
PART A
You must answer the question in this section.
A.1 Individuals in a market each have a total budget y to spend on two goods q1 and q2 at the prices p1
and p2 respectively. Their preferences are described by the utility function
u (q1, q2) =
{
Bq1 − 12 (A− q2)2 , if q2 ≤ A;
Bq1, if q2 > A;
(1)
where A, B > 0 are preference parameters.
(a) Show that the quantity demanded of good 2 is
f2 (y, p1, p2) =

y
p2
, if p2p1 <
A
B − yBp2 ;
A−B p2p1 , if AB −
y
Bp2
≤ p2p1 ≤ AB ;
0, if AB <
p2
p1
.
Assume for parts (b) and (c) that prices are such that individuals consume positive quantities of both
goods.
(b) Find the expenditure function corresponding to these preferences.
(c) Suppose that the price of good 1 is fixed at 1 but the price of good 2 rises from p02 to p
1
2. Show
that compensating and equivalent variation are equal to each other and equal to consumer
surplus for each individual. Comment.
Henceforth, suppose there are N consumers each with the utility described in equation (1) above,
that p1 = 1, and that y >
A2
4B .
(d) Suppose that the good q2 is supplied by a monopolist with a cost function cQ
2 where Q is
its total output of good 2. Draw this monopolist’s demand, MR and MC curve. Calculate
the monopolist’s profit-maximizing prices and quantity and compare them with their perfectly
competitive equivalent values.
(e) Explain why this monopolist does not increase its output without limit as N , the size of its
market, increases. Can regulation of the monopolist solve this problem?
(f) How much would supply increase (and price fall) if another firm entered this market and the
competition was in quantities? (Assume the entrant has the same costs as the incumbent.)
ECON0013 2 CONTINUED
PART B.I
Answer ONE question from this section.
B.I.1 A firm has two types of customer. High-value customers have utility 3 ln(1 + x) − p if they buy x
units of the good for the price p. Low-value customers have utility 2 ln(1+x)− p if they buy x units
of the good for the price p. There are 10 customers of each type. The firm has the marginal cost
c < 1 for producing each unit of x. If customers do not buy anything they get zero utility
(a) The firm chooses to offer x units of the good in a pack for the price p. What conditions must
(x, p) satisfy if: (i) Only the high value customers buy it? (ii) Both types of customer buy it?
(iii) Both types of customer are willing to buy one pack but are not willing to buy two packs.
Explain why this is can occur only if the pack size is large.
(b) What is the maximum profit from selling one pack only to high-value customers? What is the
maximum profit from selling to both low-value and high-value customers if the customers are
restricted to only buy one pack?
(c) Show that when costs are high it is better for the firm to only sell to the high types.
(d) The firm chooses to offer two different packages: an A pack (xA, pA) and a B pack (xB, pB).
List the conditions (xA, pA) and (xB, pB) must satisfy if the high-value customers choose to
buy the A pack and the low-value customers choose to buy the B pack.
(e) Explain what constraints the packs (xA, pA) and (xB, pB) must satisfy if the firm is choosing
these packs to maximise it profits. State your answer in the simplest possible form.
(f) Explain what economic interpretations can be placed on the above conditions.
ECON0013 3 TURN OVER
B.I.2 A worker produces a product that is subject to two separate tests. If the worker puts in effort there
is: probability p2 that the product passes both tests, probability 2p(1 − p) that it passes one test,
and probability (1−p)2 that it passes no tests. If the worker does not put in effort these probabilities
are: q2, 2q(1− q), (1− q)2; where q < p. The manager decides to pay the worker the wage u if both
tests are passed, v if only one test, and w if no tests are passed. The worker has the utility
√
x− c
if she receives the wage x ∈ {w, v, u} and puts in effort. If she does not put in effort she has the
utility
√
x. The worker can earn the utility U from working elsewhere.
(a) What conditions must the wage contract (u, v, w) satisfy if the worker is willing to work at the
firm?
(b) What conditions must the wage contract (u, v, w) satisfy if the worker prefers high effort to
low effort? Write the condition you find in the simplest possible form and explain why the firm
would not want v = w in this case?
(c) The firm decides that it is content with low effort from the worker. Write down and solve a
constrained optimization that describes the cheapest way for the firm to achieve this. Interpret
what you find.
(d) The firm decides it wants high effort from the worker, but ignores the possibility that the worker
might leave the firm and go elsewhere. The firm also decides to pay a wage of zero if both tests
are failed. Write down the constrained optimization that describes the firm’s cost minimization
in this case.
(e) Solve this constrained optimization and explain what economic effects it captures.
ECON0013 4 CONTINUED
PART B.II
Answer ONE question from this section.
B.II.1 Suppose individuals consume two goods, baked beans q1 and apples q2. Total budget is y and prices
are 0 ≤ p1 < y/A and p2 ≥ 0. Preferences are represented by the indirect utility function
v (y, p1, p2) = ln (y − p1A)− 1
1− β ln
[
αp1−β1 + (1− α)p1−β2
]
where
• A ≥ 0 is a parameter reflecting baked bean needs,
• 0 < α < 1 is a parameter reflecting taste for baked beans and apples, and
• β ≥ 0 is a parameter reflecting substitutability between baked beans and apples.
(a) Find the uncompensated demand functions for the two goods.
(b) Which, if either, good is a necessity?
Suppose that in a base period p1 = p2 = 1 and in a later period p1 = 1 + δ > 1 and p2 = 1.
(c) Explain what a true (or Konu¨s) cost of living index is and show that a true (or Konu¨s) cost of
living index at base-period utility can be expressed for an individual with total expenditure y as[
α(1 + δ)1−β + (1− α)
]1/(1−β)
+
(
1 + δ −
[
α(1 + δ)1−β + (1− α)
]1/(1−β)) A
y
.
(d) Find an expression for the Laspeyres cost of living index as a function of preference parameters,
y and δ.
ECON0013 5 TURN OVER
Figure 1: Laspeyres and true cost of living indices
(e) Figure 1 plots Laspeyres and Konu¨s cost of living indices against total expenditure for different
values of β and for specific values A = 0.5, α = 0.5, δ = 0.1. Discuss and explain, in the
context of this Figure,
• the difference between the values of the Laspeyres and Konu¨s indices,
• the way in which each index varies with total expenditure, and
• the way in which indices vary with β.
“The current inflation basket measures the changing price of over 700 specific items that
are widely bought, but what happens if the prices of individual items change? If one variety
of apple goes up in price while another falls, do some people switch varieties to avoid a
price rise? And given that people of different means undoubtedly buy different varieties of
products, what happens to the price of own brand versus branded baked beans?”
Measuring the changing prices and costs faced by households, Mike Hardie, Head of
Inflation Statistics at the Office for National Statistics, January 26, 2022
(f) To what extent can modelling of the sort above help in understanding and addressing the
considerations raised in this quotation?
ECON0013 6 CONTINUED
B.II.2 An optimising consumer has total budget y and consumes two goods, cheese q1 and bread q2,
purchased at the prices p1 and p2. Suppose that her Marshallian demand for cheese is
f1 (y, p1, p2) = max
{
y
p1
−A, 0
}
, p1 > 0.
where A > 0 is a preference parameter.
(a) Find the consumer’s Marshallian demand for bread. Draw diagrams to illustrate her income
expansion paths and the Engel curves for the two goods. What class of preferences do these
demands describe?
Suppose that the cheese supplier reduces the price of cheese to p1 − δ on purchases above E units
(where δ < p1). In other words, buying q1 units of cheese costs p1q1 if q1 < E and p1q1− δ(q1−E)
if q1 ≥ E.
(b) Describe the budget set and discuss difficulties in modelling demand for budget sets of this type.
(c) Suppose y > p1(A + E). Explain carefully what bundle of cheese and bread she will choose.
How do you know, solely from the information given so far, that she prefers a bundle with
q1 ≥ E to any bundle with q1 < E?
(d) Suppose her preferences are given by utility function u (q1, q2) = q1 +A ln q2. Show that this
is compatible with the Marshallian demand curves described above.
(e) Show that she consumes q1 > E if
y ≥ p1E + p1A
(p1
δ
− 1
)
(ln p1 − ln (p1 − δ)) .
Hence draw a diagram showing the path followed by the bundles consumed by the consumer as
y increases under the budget constraint with the discount. Comment.
(f) How would your analysis change if the cheese supplier made the cheese discount available only
to customers who pay a flat fee F to join a club for cheese lovers?
ECON0013 7 END OF PAPER