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Please type the equations using word processing software (same for graphs and diagrams). Otherwise, draw them and scan them. Do not copy the question text into your answer. Use references as needed. Answer only 2 questions out of 3 in Part A, and only 2 out of 3 in Part B. I don’t know how many pages the assignment will be, we can discuss the price. I need fully explained answers with step-by-step solution

falsication, contract cheating, and falsication of extenuating circumstances
PART A
Answer TWO questions from this section.
A.1 There are two players, called Row and Column, who play a game that can last two periods. In the
rst period they simultaneously choose actions F or C. If they both choose F then the game
continues and they move to the second period. If at least one player chooses C the game ends and
they get the payos in the table below
F C
F continue (2,0)
C (0,2) (0,0)
After playing (F; F) in the rst period, the players again simultaneously choose F or C in the
second period and get the payos below. (The game cannot continue beyond the second period.)
F C
F (-2,-2) (1,-1)
C (-1,1) (-1,-1)
:
(a) Draw the extensive form of this game and list all the row player’s pure strategies.
(b) Find all the Nash equilibria of the game that is played in period 2 (after (F; F) is played in
the rst period).
(c) Find all the subgame perfect equilibria of the entire game.
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(d) Describe the players’ strategies at a Nash equilibrium of this game that is not a subgame
perfect equilibrium.
(e) Now suppose that the game that is played in the second period is the game
F C
F continue (2,0)
C (0,2) (0,0)
And if (F; F) is played in the second period then the strategic form
F C
F (-2,-2) (1,-1)
C (-1,1) (-1,-1)
is played in the third period. Draw the extensive form for this game. Describe how you think
your answers to the questions above will change, but do not calculate all the equilibria. Will
the players benet from this increased number of periods and why?
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A.2 A student has to decide whether to go to a party on the night before an exam. If they go to the
party, then they have the probability of writing a good exam and the probability 1 􀀀 of writing
a bad exam. If they do not go to the party, then they have the probability > of writing a good
exam.
The student has the utility u = x1=2 + H if they go to the party and get the mark x on the exam.
But, if they do not go to the party they have the utility u = x1=2. The professor gets to choose xb
(the mark she gives to bad exams) and xg the mark she gives to good exams. Students also have
the option of going to the party and then quitting the course. This gives them the utility U + H
where U > 0.
(a) What is the student’s expected utility from going to the party and her expected utility from
not going to the party?
(b) Assuming students do not quit the course, what marks (xg; xb) can the professor set to stop
students partying? Draw a picture of this set of marks with yg =
p
xg on one axis and
yb =
p
xb on the other. Explain how this set changes as H and 􀀀 change.
(c) What marks can the professor set so the students prefer partying and taking the exam to
quitting the course? What marks can the professor set so the students prefer not partying and
taking the exam to quitting the course? Show these sets on a new picture.
(d) Suppose that the professor wants to promote equity and thus aims to make the dierence
xg 􀀀 xb as small as possible while still stopping students from going to the party, but she
ignores the possibility that students will quit the course. What marks (xg; xb) should she set
to achieve her objective? Will this result in the outcome she planned?
(e) Suppose now the professor wants to minimise xg 􀀀 xb while still getting students to not party
and attend the course. Plot the contours of her objective function on your (yg; yb) picture.
What marks (xg; xb) should she set to achieve her objective now?
(f) The professor has other variables that she can control: she can make the exam harder, which
decreases and , and she can make the party fail by calling the campus authorities, which
decreases H. How does each of these variables aect her ability to achieve the objectives
described above?
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A.3 A rm can employ two types of workers: A workers and B workers. The A workers have ability
that is uniformly distributed on the interval [0; 2], that is, 0 2. The B workers have ability
that is also uniformly distributed on the interval [0; 2]. All the workers know their own ability.
There are equal numbers of A and B workers. The rm can see whether the workers are type A or
B, but cannot observe the workers’ abilities.
The A workers can earn 22 from setting up their own company whereas the B workers can only
earn 2 from setting up their own company. A rm earns p from every A worker of ability that
they employ and p for every B worker of ability they employ.
(a) If the rm oers the wage v for A workers and w for B workers, then what workers come to
the rm?
(b) What is the rm’s expected prot from the wages (v;w) and how should it choose wages to
maximise prots? Or to make zero prots in perfectly competitive labour markets?
(c) Does the rm employ more A workers or B workers? Explain why this is so.
(d) A piece of anti-discrimination legislation is published. The rm is no longer able to pay
dierent wages to A and B workers, but must pay them all the same wage u. What is the
rm’s prot now? How should it choose u to maximise prots? Or to make zero prots?
(e) Does the rm now employ more A workers or B workers? Explain why this is so.
(f) The government also decides the rm must employ equal numbers of A and B workers. What
workers would the rm in part (d) like to sack to achieve this additional objective?
(g) What can we learn about the economics of discrimination from the answers you have given to
the above?
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PART B
Answer TWO questions from this section.
B.1 An individual consumes two goods, food q1 and clothing q2. Their total budget is y, the prices of
food and clothing are p1 and p2 and their utility is . Suppose their preferences are represented by
expenditure function
c(; p1; p2) =
p
p1p2 ep1=p2 :
Assume that prices and utilities take values such that this function has all the required properties of
an expenditure function, including concavity in prices, and that consumers demand positive
quantities of food and clothing. Note that is allowed to be positive or negative.
(a) Use Shephard’s Lemma to show that compensated budget shares of food and clothing are
1
2 + p1=p2 and 1
2 􀀀 p1=p2.
Suppose that prices in a base period are p1 = p2 = 1. In a later period they have changed to
p1 = 2; p2 = 1=2. You are asked to consider whether the overall cost of living should be regarded
as having risen or fallen.
(b) Explain what a true or Konus cost of living index is and show that a Konus cost of living index
evaluated at initial utility 0 for these preferences and these prices has the form
K(0) = e30
(c) Explain what a Laspeyres cost of living index is and show that a Laspeyres cost of living index
for these preferences and these prices has the form
L(0) =
5
4
+
3
2
0
(d) Discuss the facts that
i. according to either index, whether the cost of living has risen of fallen depends on the
initial utility 0 at which it is evaluated
ii. the cost of living has risen according to L(0) but fallen according to K(0) if initial
utility is such that 0 > 0 > 􀀀1
6 .
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B.2 Suppose a consumer consumes two goods, nutmegs q1 and raisins q2. Preferences are represented
by indirect utility function
v (y; p1; p2) =
p
y 􀀀
p
p1
p
p2
where y is total budget and p1 and p2 are prices of nutmegs and raisins. You should assume that
y > p1.
(a) Find the consumer’s Marshallian demand for nutmegs. Hence show that if the individual has
an endowment !1 of nutmegs and !2 of raisins, no other source of income and the relative
price of nutmegs is P = p1=p2 then they will demand
p
!1 + !2=P nutmegs. (Assume that
!1 > 1 and therefore that !1 + !2=P > 1 for all P.)
Eufemia is an exchange economy in which H individuals consume these two goods and all
individuals have preferences as described above.
(b) i. Suppose that all consumers have the same endowments of each good, !h
1 = 1 and
!h
2 = 2, h = 1; : : : ;H where 1 > 1. Show that the unique equilibrium relative price of
nutmegs is
P =
2
1 (1 􀀀 1)
and that the equilibrium allocation is such that each individual consumes their
endowment.
ii. Suppose instead that half of consumers have all of the endowments of raisins.
Specically, assume !h
1 = 1 and !h
2 = 0 for h = 1; : : : ;H=2 whereas !h
1 = 1 and
!h
2 = 22 for h = H=2 + 1; : : : ;H. (Assume H is even and, as before, 1 > 1.) Find the
unique equilibrium relative price of nutmegs. Discuss how the change in initial
endowments has aected the equilibrium outcome.
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B.3 Marie has preferences over two goods, cake q1 and bread q2. She chooses quantities to consume so
as to best satisfy these preferences subject to the budget constraint p1q1 + p2q2 y where p1 and
p2 are prices and y is total budget.
Suppose that Marie’s preferences are represented by utility function u(q1; q2) = q1 + ln (bq1 + q2)
where b 0 is a preference parameter. Assume that p1=bp2 y= (p1 􀀀 bp2) 1.
(a) Show that Marie’s indierence curves are downward sloping and that her weakly preferred sets
are convex for all possible values of b.
(b) Show that her Marshallian demand for cake q1 is
f1 (y; p1; p2) =
y
p1 􀀀 bp2
􀀀 1
and nd her Marshallian demand for bread, f2 (y; p1; p2). Discuss the shape of Engel curves
for the two goods.
(c) Explain why Marshallian demand curves for normal goods slope down. Are there any values of
b for which either cake or bread could be a Gien good for Marie? Discuss.
(d) Find the form of the indirect utility function and expenditure function and hence establish
expressions for the Hicksian demands for the two goods.
ECON0013 7 END OF PAPER