Assignment 4
ECON30010 2021
Due: Sunday 16 May by 11:59 pm
(+ 4 hour grace period with no penalty)
Problem 1 [35pt]
In this problem, we will look at self-driving cars, which make driving (on highway) less
costly.
Suppose that there are two options for travel from A to B: local roads l and highway
h. The travel time on local roads is tl = 25 min and the travel time on highway is
th = 20 + 20h, where h is the mass of drivers travelling on the highway (I know, it is not
a very good highway).
(a) What is the equilibrium number of drivers on the highway and travel times on local
roads and the highway?
Suppose that there is a third option: public transport. Unlike driving, travel time on
public transport differs among people. Let us index people by i (there are infinitely many
people; there is a person with index i = 0, with i = 0.11, with i = 0.93, and with i = 1).
The person with index i has a travel time on public transport given by 10+40i. You may
think of this specification as saying that they live at different distances from a train stop.
For example, a person with index i = 0 would spend 10 + 40 × 0 = 10 min on public
transport, 25 min on local roads and at least 20 min on the highway; that person would
definitely choose to travel on public transport. A person with index i = 1 would spend
10 + 40 = 50 min on public transport; local roads would take 25 min, so that person will
not use public transport.
As a warm-up, let us suppose, for (b) and (c) only, that highway is closed and people
can only travel on local roads or by public transport.
(b) What are the indices of people who will travel on public transport? What are the
indices of people who will travel on local roads? (Assume that a person is taking public
transport when indifferent.)
If we want to talk about travel time for a group of people, it is reasonable to take the
average of all their times. If, for example, we consider all people with i ≤ 3/8, their
average travel time is 10+(10+40×3/8)
2
= 10+25
2
= 17.5. In general, if we consider people with
i ≤ p, then their average travel time is 10+(10+40p)
2
= 10 + 20p.1 If people with i ≤ p are
taking public transport and the rest are driving on local roads, then the average travel
time for the population is p× (10 + 20p) + (1− p)× 25.
(c) What is the average travel time of the population if they behave according to (b)?
We will now return to our full-fledged problem, with highway, local roads and public
transport as three options. Suppose that h mass of people travel on highway, l mass travel
on local roads and p travel on public transport, with h + l + p = 1.
(d) Find one equilibrium, which you can express by finding values for h, l, and p.
1Wednesday session will have a problem explaining why, but you can take it as given.
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(e) What is the average travel time in this equilibrium?
Suppose now that all people own self-driving cars; or maybe not completely self-driving,
but at least Teslas. This means that when they drive on highway, people can do something
pleasant. Suppose they continue to spend 3/4 of the time driving and 1/4 of the time
doing pleasant things, which means that their effective driving time is 3/4 of the actual
driving time. That is, if there are h drivers on the highway, the effective driving time is
3/4× (20 + 20h).
(f) Find new equilibrium values for h, l, p (they are not necessarily all positive).
(g) What is the total travel time in the new equilibrium?
Problem 2 [30pt]
Consider a congestion game similar to the one in class, with infinitely many drivers choos-
ing which route from A to B to take, on the following road network:
A B
L
2 min
x
l
3 min
1− x
H
{
2 if y ≤ 0.5
1.5 + y if y > 0.5
min
y
h{
1 if 1− y ≤ 0.7
0.3 + (1− y) if 1− y > 0.7 min
1− y
C
D
c min s
Numbers on the edges show the travel time (in minutes) on each road; x, 1−x, y, 1−y are
fractions of drivers on each road. Unlike the model in class, the travel time on highways
h and H increases only after a certain threshold is reached (when more than 0.5 mass of
drivers travel on H and more than 0.7 mass of drivers travel on h).
In parts (a)–(b), let c = 1.
(a) Find one pure-strategy Nash equilibrium of this game.
(b) Find all pure-strategy Nash equilibria. Explain why no other pure-strategy Nash
equilibria exist.
(c) Let c = 0.3. Find one pure-strategy Nash equilibrium of the new game.
Problem 3 [19pt]
Consider the panic buying model with correlated signals (Topic 11). I have assumed (Slide
13) that every consumer who searches for toilet paper has an equal chance to buy it. In
this problem, we will change this assumption: we will assume that consumers who have
0 days of toilet paper left (it0) purchase ahead of any consumers in other groups. Within
the group, they have equal chance to purchase toilet paper, and if they do purchase, they
can buy as many rolls as they want (there are no restrictions on the number of rolls).
Consumers who have 1 day left (it1) purchase next, if there is any remaining stock; they
are followed by it2, i
t
3, etc. Everything else remains the same as in Topic 11; in particular,
i10 = i
1
1 = i
1
2 = 1/3 and S = 1/3.
2
(a) Suppose that, as in lectures, i10 attempt to buy two rolls, and i
1
1, i
1
2 attempt to buy
one roll. What would be the masses of i20, i
2
1, and i
2
2 in period 2?
(b) Suppose that each consumer who was not able to buy toilet paper in period 1 try
to buy one roll of toilet paper in period 2 (in the order described at the beginning of the
problem: i20 first, then i
2
1, etc). Given your answer in (a), what are i
3
0, i
3
1, and i
3
2?
(c) Suppose that, as in lectures: (i) in period 1, consumers anticipated that toilet paper
may not available in period 4 with probability p, but (ii) at the beginning of period 2,
they have learned that there will be no shortages in period 4. Would it be an equilibrium
if:
• Each consumer in i10 attempts to buy two rolls in period 1
• Each consumer in itk, with t = 2, 3 and k = 0, 1, 2, attempts to buy one roll in period
t.
Assume that the cost of storage c is small relative to p and w. Provide an explanation,
but you do not need to derive conditions on c, p and w.
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