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UNIVERSITY COLLEGE LONDON
EXAMI~ATION FOR JNTERNAL STUDENTS
MODULE CODE ECON2601
ASSESSMENT ECON2601D

PATTERN

MODULE NAME Economics 2 (Combined Studies)

DATE· Friday 27 April 2018

TIME 10:00

TIME ALLOWED 3 hrs

This paper is suitable for- candidates who attended classes for this
module in the following academic year(s):
Year
2015/2016, 2016/2017 and 2017/2018
EXAMINATION PAPER CANNOT BE REMOVED FROM THE EXAM HALL. PLACE EXAM
PAPER AND ALL COMPLETED SCRIPTS INSIDE THE EXAMINATION ENVELOPE
2016/17-ECON2601 D-001-EXAM-Statistical Science 114

© 2016 University College London

TURN OVER
SUMMER TERM 2018
ECON2601: Economics II (Combined Studies)
TIME ALLOWANCE: 3 hours
Answer ALL questions from Part A, ONE question from Part B, ana ONE question from Part C.
Correct but unexplained answers will not receive high marks.
Questions in Part A carry five percent of the total mark each, and questions in Parts Band C carry
twenty-five percent of the total mark each.
In cases where a student answers more questions than requested by the examination 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.
PART A
Answer ALL questions from this section.
A.l Your next holiday will be spent at Mayworld, a fantasy-themed amusement park that offers
rides on unicorns to its customers. It has a two-part pricing structure that charges you a fixed
fee of f = 50 (in shillings) to enter and subsequently PI = 1 (in shillings per unit) to go on
each unicorn ride, where Xl denotes the number of unicorn rides you consume. Let X2 be your
expenditUre on all other goods, and you may assume that both Xl and X2 are continuous, non­
negative variables. Suppose that your utility function for unicorn rides and all other expenditure
is U(XI, X2) = Xl + 2JX2. If your exogenous income is m = 100 (in shillings), then how many
unicorn rides are you willing to consume? What is the highest fixed fee that you will pay and still
be willing to consume a strictly positive number of unicorn rides? Call this value the reservation
fee, fR, and find the corresponding number of rides, xr. Note that if you consume zero unicorn
rides, Le., Xl = 0, then you do not pay the fixed fee and have X2 = m.
A.2 The absolute value of the marginal rate of substitution (MRS) between goods i and j is the
rati!J of their marginal utilities, MUi and MUj, respectively. Suppose that a consumer's utility
function for commuting is u(w, t, c) = aw+ ,6t+,c, where w, t, and c represent the walking time
(in hours), total travel time (in hours), and the cost (in £) of commuting, respectively. What
do the constants a, ,6, and, mean in this context? How much is the consumer willing to pay
in order to reduce her trip by one hour? How much of an increase in the total travel time is the
consumer willing to endure in order to reduce her walking time by one hour?
ECON2601 1 TURN OVER
A.3 Suppose e(PI,P2, u) = UVPIP2. Find the Hicksian and Marshallian demand functions for good
1. Verify that the identity hi(PI,P2,U) == Xi(PI,P2,e(PI,P2,U)) is satisfied.
A.4 You have the utility function U(XI' X2) = xi1 xi1 and face prices PI = P2 = 1 with exogenous
income m= 100. Determine the compensating and equivalent variations when there is. a ceteris
paribus increase in PI from 1 to 2. Explain intuitively which income adjustment is greater here.
A.S A gambler with initial wealth Yo > 0 goes to a casino that offers only one type of game: with
probability 0 ::;; P ::;; 1, the gambler ends up with wealth Yo + ZI, and with probability 1 - P,
she ends up with wealth Yo + Z2. Here, ZI > 0 and Z2 < a such that YO + Z2 ::::: O. Find the
gambler's expected terminal wealth from playing a single game at the casino. Next, calculate
her utility of expected wealth, expected utility of wealth, certainty equivalent, and risk premium
if her utility function is u(Y) = .;y.
A.6 Calculate the profit-maximising solution for a monopoly and show how the output price P is
related to the elasticity of demand with respect to that output price, and defined here as c.
Assuming a constant marginal cost, show that dp/dc is positive and sketch a graph of the P
against c. .
A.7 Two countries A and B agree to cooperate by imposing quotas on fishing in shared waters. Their
respective payoffs each year are as follows:
Country B
overfish keep quota
Country A overfish keep quota
(60,60)
(40,110)
(110,40)
(80,80)
where the first number is the payoff to Country A and the second is the payoff to Country B.
Assuming a one-shot simultaneous move game, what is the pure Nash equilibrium? If the game is
repeated infinitely many times and the countries have the same annual discount factor, 8 = 0.7,
check whether an equilibrium in trigger strategies exists and examine if this equilibrium induces
a different outcome in each period compared to the static Nash equilibrium of the game.
A.8 Two firms compete in a duopolistic industry. Firm 1 produces output qi and has a constant
marginal cost k > O. Firm 2 produces output q2 and has cost function C2(q2) = kq2 + f3q~. The
market demand schedule.is given by P = a - (qi + q2) for price per unit P and with a given
positive real parameter a > k. Determine the output levels of each firm (in terms of a, k, and (3)
in a one-shot simultaneous move game where each firm decides, without consultation, the level
of output to produce. Explain briefly how 13 affects the resulting Cournot-Nash equilibrium.
ECON2601 2 CONTINUED
A.9 "A lower rate of population growth over a long period of time will cause a permanent rise
in the rate of growth of output per worker." By considering the Solow Growth Model with
technological progress, explain carefully using a graph and words whether this statement is true,
false, or uncertain.
A.IO During the past twenty years the standard of living (output per head) in Country A has grown
at the same rate on average as the standard of living in Country B. However, the growth rate
. of technological progress has been lower in Country A over the same period. How would you
explain this phenomenon using the Solow Growth Model?
ECON2601 3 TURN OVER
PARTB
Answer ONE question from this section.
2
H.l A consumer has a generic utility function, U(XI, xz), where aau > 0 (for i = 1,2), pau < 0 (for
Xt Xi
i = 1,2), and aa 2 u = O. The two goods have prices PI and pz, and the consumer has anXl aX2
endowment (WI, wz) instead of exogenous income m.
(a) Formulate the Lagrangian for the utility-maximisation problem. Make sure you indicate
the three decision variables. Find the three first-order necessary conditions and indicate
the solutions as Xi(PI,PZ,WI,WZ), X;(PI,PZ,WI,WZ), and >'*(PI,PZ,WI,WZ).
(b) What is the second-order sufficiency condition for this constrained maximisation problem?
Show that it is satisfied here.
(c) Find ~ by substituting the optimal solutions from part (a) into the three first-order nec­
~ .
essary conditions and partially differentiating them with respect to Pl. Can you determine
ax'the sign of -a17Pi
(d) Find ~:~ by substituting the optimal solutions from part (a) into the three first-order
necessary conditions and partially differentiating them with respect to WI. What is the sign
f~?o aWl.
(e) How would the sign of aaxi in part (c) differ if income were exogenous? Explain your answer Pi
intuitively.
ECON2601 4 CONTINUED
B.2 You are the Chief Operating Officer of GlobalHypoMega Corporation which has two U.K. pro­
duction plants, one in Birmingham and one in Manchester. All output from these two plants
belonging to GlobalHypoMega is sold in a perfectly competitive industry, and the cost functions
at the plants are CB(YB) = !Y~ and CM(YlvI) = Yfu., where B and M denote Birmingham and
Manchester, respectively, and Yi is the output of plant i, where i = B, M.
(a) YoUr objective is to produce a total of Y ~ YB + YM units of output from the two U.K.
plants at minimum overall cost. Formulate your constrained cost-minimisation problem.
What is the Lagrangian? Indicate the decision variables clearly.
(b) Find the three first-order necessary conditions to the problem in part (a). Show that the
second-order sufficiency condition, which is that the determinants of all principal minor
submatrices of the bordered Hessian matrix must be negative, is satisfied.
(c) Solve the three first-order necessary conditions in part (b) simultaneously to obtain the
optimal production at each plant as a function of Y, Le., find Y'B(Y) and YNI(Y)' You
should also be able to solve for the Lagrange multiplier, >.*(y). Substitute Y'B(Y) and YNI(Y)
into CB(YB) and CM(YlvI), respectively, and add them to obtain the minimised overall cost
function, c(y).
(d) What is the economic interpretation of the Lagrange multiplier in this context? Prove your
assertion mathematically.
(e) Recall that given the output price, P, a plant's marginal cost curve is also its inverse supply
curve. From this, the supply curves may be found and added to determine the overall U.K.
supply curve for GlobalHypoMega Corporation. Verify that GlobalHypoMega's overall
U.K. supply curve calculated in this way is simply the marginal cost curve from part (c).
ECON2601 5 TURN OVER
PART C

Answer ONE question from this section.

C.l A monopolist produces a single good x and sells it to two consumer types, A and B, whose
preferences are given by the utility functions
where x is the quantity consumed of the monopolist's good and y is the consumer's disposable
income. There is an equal number (say N) of both types of consumer in the population and the
monopolist has a constant marginal cost equal to 4.
(a) Determine each consumer type's demand function for the monopolist's good based on utility
maximisation.
(b) Suppose that the monopoly can accurately identify consumer type and wishes to design
specific "take it or leave it" contracts to perfectly price discriminate and maximise
profits. These contracts take the form of a specific quantity qi sold for an overall price of ri
for each consumer type i = A, B. (i) Write down the profit-maximisation problem and the
two constraints faced by the monopolist. (ii) Determine the specific quantity and overall
price for each contract, (qA,rA) and (qB,rB).
(c) Suppose now that the monopoly cannot accurately identify consumer type and so the
contracts must be tailored in such a way that the consumers of each type have the incentive
to self select a given contract. (i) Write down the willingness-to-pay and self-selection con­
straints for the monopolist's profit-maximisation problem and highlight which constraints
are slack and which constraints are binding. (ii) Graphically or otherwise determine the
revised contracts (qA, rA) and (qB' rB) for this second-degree price discrimination problem
that maximise the monopolist's profit.
ECON2601 6 CONTINUED
•

C.2 Consider an exchange economy consisting of two agents, A and B, and two goods. Agent A has
preferences UA(:XA,yA) = x A + alnyA and starts with an endowment consisting of a units of
good x and 1 unit of good y. Agent B has preferences UB(xB, yB) = xB + 2ln yB and starts
with an endowment consisting of 2 units of good x and 2 units of good y.
(a) Define the Walrasian equilibrium of an exchange economy.
(b) In an exchange economy, can one agent be made strictly better off if we are already at a
Pareto-efficient allocation? If we are not at a Pareto-efficient allocation, can we make all
agents strictly better off by trading to a Pareto-efficient allocation? Explain your answers.
(c) Determine agent A's and agent B's demand functions for each good.
(d) Find the Walrasian-equilibrium price and allocation.
(e) What would be the Walrasian-equilibrium price and allocation if agent B only has one unit
of good x and zero units of good y?
(f) Are the Walrasian-equilbrium allocations obtained above Pareto efficient? Explain your
answer.
ECON2601 7 END OF PAPER