EC202 Exam, June 2012
Setup exactly as last year (use the same heading) – Part A and Part B, 3 questions to be
answered from Part A and 1 from Part B. Part A questions are worth 20% and Part B
are worth 40%.
Part A
[Note for Ann: question 1 is from Sayantan, 2-3 from Carlo and 4-6 from Daniel]
1. Suppose an individual consumes two goods
x1
and
x2
. The preferences of this individual
1
2
represented by a utility function u ( x1 , x2 ) = x1 + x2 . The individual has income m > 0 and
p = 1, p2 > 0.
price are such that 1
(i)
For a fixed level of utility c , write down the equation of the indifference curve for
these preferences. [3 marks]
x
(ii)
How does the marginal rate of substitution change as 1 changes? Interpret. (3
marks).
(iii)
Set up the individual’s maximization problem. Does the budget constraint bind at an
optimal solution to the maximization problem and if so, why? [4 marks]
*
*
(iv)
Let x1 , x2 denote a solution to the individual’s maximization problem. Argue that
not all solutions to the individual’s maximization problem have the property that
x1* > 0
. [3 marks]
(v)
State the conditions required for an interior solution precisely. Derive the optimal
solution (whether interior or not) to the individuals maximization problem as a
*
*
m, p2
. What is the effect of an increase in m on x1 , x2 at an interior
function of
solution? [6 marks]
2.
A consumer is facing an uncertain prospect: in state 1, she will receive an income of
100, while in state 2 she will receive an income of 150. The two states are equally likely. Her
utility in each state s = 1, 2 is u(cs) = ln cs , where cs, s =1, 2, represents consumption
(disposable income) in each state, and ‘ln’ denotes the natural logarithm. The consumer is an
expected utility maximiser.
(a)
How much would the consumer be willing to pay with certainty (in both states) in
order to participate in a lottery that increases her gross income by 10 (gross of the cost of
participating in the lottery) in state 2 only?
(b)
How much would the consumer be willing to pay with certainty (in both states) in
order to participate in a lottery that increases her gross income by 50 (gross of the cost of
participating in the lottery) in state 1 only?
((a) and (b) carry equal weight.)
3. Consider a pure-exchange economy with two individuals, A and B, and two goods, 1 and
2. Preferences for two individuals are summarised by the following utility functions:
UA(X1A, X2A) = X1A + X2A
UB(X1B, X2B) = X1B
X2B
Where X1A, X2A, X1B, X2B, are the quantities of each good consumed respectively by A and by
B.
(a)
Using an Edgeworth Box diagram, describe the locus of Pareto efficient allocations
for this economy.
(b)
Market equilibrium prices in this economy can be determined without knowing how
the endowments are initially distributed between consumers. Find equilibrium prices.
(c) Suppose that endowments of good 1 and 2 are distributed as follows: W1A = 2, W2A = 2,
W1B = 1, W2B = 3. Find an equilibrium allocation.
((a), (b) and (c) all carry equal weight.)
4. Consider the following 2-player normal form game where player i's actions are given as Ai
and Bi and player 1’s payoff listed first in each cell of the bimatrix:
Player 1
A1
B1
Player 2
A2
5,5
3, 15
B2
15, 3
10,10
(a). What is the Nash equilibrium in this game? Is there an outcome which Pareto dominates
the Nash equilibrium outcome? [5 marks]
Now imagine that this game can be repeated T times, and denoted GT as the T-period
repetition of this game. After each repetition payoffs are discounted by 0 < δ <1 by both
players.
(b) Consider G100. Can any outcome which Pareto dominates the Nash equilibrium found in
part (a) be supported through a Nash-reversion punishment strategy? Why or why not? [5
marks]
(c) Consider G∞. Consider if any outcome which Pareto dominates the Nash equilibrium
found in part (a) can be supported by a Nash-reversion punishment, and if so under what
condition. [5 marks]
Now consider that after each repetition payoffs are discounted by 0 < δi <1 by player i and
that this discount factor is independently drawn from a uniform distribution bounded above
by 1 and below by 0 for each player.
(d) How would your condition in part (c) change? What might you infer from your findings
about the potential for cooperation in an infinitely repeated game when the number of players
rises? [5 marks]
5. Consider an N-firm Cournot model with total consumer demand given by p = 10 - q. The
firms face zero costs of production and set quantities simultaneously.
(a) Solve the model for optimal firm quantities and calculate individual firm profits. [5
marks]
Now imagine that in order to operate in this industry each firm must pay a fixed cost of 5.
(b) What is the maximum number of firms that can compete in this industry and make
positive profits in light of the fixed cost? [5 marks]
Imagine there are currently exactly N = 10 firms in the industry. However, there is the threat
of new entry.
(c). By lobbying the government to insist on higher health and safety standards in
manufacturing the product, the current 10 firms in the industry can raise the fixed cost to 8.
Why would the 10 firms consider lobbying to raise their costs? Should they go through with
their plan to raise fixed costs to 8? [10 marks]
6. Consider the following 2-player normal form game where player i's actions are given as Ai,
Bi ,Ci and Di, and player 1’s payoff listed first in each cell of the bimatrix:
Player 1
A1
B1
C1
D1
Player 2
A2
3,2
0,0
0,0
2,2
B2
1,1
2,3
0,2
0,6
C2
1,0
0,0
0,0
0,2
D2
2,0
2,2
7,1
1,1
(a) Can any strategies for players 1 and 2 be eliminated by dominance arguments? If so,
which ones? [4 marks]
(b) Find all the Nash equilibria in this game (pure and mixed). [8 marks]
(c) Given the payoffs that result from the various Nash equilibria, does it seem reasonable
that players would ever play a mixed strategy Nash equilibrium in this game? Briefly provide
some arguments in defence of mixed strategy Nash equilibria in games in general. [8 marks]
Part B
[Note for Ann: question 7 is from Sayantan, 8 from Carlo and 9-10 from Daniel]
7. Describe a model of allocation of consumption across time that has the property of time
consistency. Discuss, critically, whether such a model can account for stylized facts that
illustrate present-bias. Suggest an alternative formulation of intertemporal choice that could
model such present bias in intertemporal choice and examine whether the solutions are time
consistent. [40 marks]
8. Describe and discuss the notion of “loss aversion” in individual choices. What are its main
behavioural implications and how can they be identified empirically or experimentally? [40
marks]
9. Define a Nash equilibrium and give some examples of its use in Game Theory. Why is
Nash equilibrium used so universally by Game Theorists? [40 marks]
10. The government has decided to attempt to address the “monopoly problem” that seems to
be evident in a certain industry and have asked you to suggest how best to achieve an
outcome that is closer to what might emerge under perfect competition. Set up a simple
mathematical model and using that model outline what the monopoly problem might be and
how it might be corrected. [40 marks]
Answers
Part A
[1 by Sayantan, 2-3 by Carlo, 4-6 by Daniel]
1. (i) x2 = ( c − x1 ) .
2
(ii) MRS x2 x1 =
MU 2 1 − 12
= x2 so it doesn’t depend on x1 at all.
MU1 2
(iii) The individual’s maximization problem is
1
2
2
Maxx1 ≥0, x2 ≥0 x1 + x
s.t.
p1 x1 + p2 x2 ≤ m
The budget constraint binds at an optimal solution to the maximization as preferences are
monotone: if there is an underspend, the individual could increase his consumption of either
one of the two goods marginally, continue to satisfy the budget constraint and gain in utility.
(iv) The point is that with these preferences there might be corner solutions so first order
conditions may not characterize an optimal solution; a diagrammatic exposition that uses
carefully drawn indifference curves and budget line should suffice.
(v)The necessary and sufficient condition for an interior optimum is 1 < mp2 : at an interior
optimum x1* = m −
1 * 1
m
. At an interior
, x2 = . Whenever 1 ≥ mp 2 , x1* = 0 and x2* =
p2
p2
p2
solution, an increase in m has no effect on x2* but
2.
(a) ln 100 + ln 150 = ln (100 − x) + ln (160 − x)
(b) ln 100 + ln 150 = 2 ln (150 – x)
(a) and (b) carry equal weight
3.
(a) Points for which X1B = X2B
(b) p1 = p2
∂x2*
= 1.
∂m
=>
=>
x ≈ 3.9
x ≈ 27.53
(c) X1A = 1, X2A = 3, X1B = 2, X2B = 2
(a), (b) and (c) carry equal weight
4. (a). (A1, A2) is the unique Nash equilibrium. It is Pareto dominated by (B1, B2).
(b). A 100 period repetition does not allow the enforcement of any equilibrium other than
(A1, A2) since backward induction unravels any attempt to enforce any other outcome. The
classic description of why would be along the lines: consider trying to enforce (B1, B2) with a
punishment threat of Nash-reversion in the following period and thereafter. This punishment
has no meaning in the final period, and so deviation to Ai by player i is expected. Given this
deviation in the final period (100), there is no reason to play Bi in period 99, so we can expect
deviation in period 99 also. Carrying through this process of backward induction we see that
even in the first period we cannot enforce anything other than (A1, A2). Key phrases like
backward induction and punishments should be present, and terms like unravelling would
also be good.
(c). Under G∞ there is no prospect of unravelling as we have no final period. Hence it might
be possible to enforce (B1, B2). In order for this to work under Nash-reversion it must be the
case that the payoff under an infinite sequence of (B1, B2) is higher than under a deviation.
The best possible deviation involves switching to Ai when the other player plays B-i, though
this would then immediately result in (A1, A2) thereafter. To stave off this deviation we
simply need 10 + 5/(1-δ) > 10/(1-δ) which after some algebra results in a condition of δ > ½,
so the players need to be sufficiently patient.
(d). The calculation is the same as in (c) (the students need not repeat this) but now we need
that the least patient player must meet the condition (or in notation, we need min(δi) > ½).
The suggestion (this should be very informal) is that with randomly allocated discount factors
as the number of players rises it becomes less and less likely that the lowest (independently
drawn) discount factor happens to be sufficiently high to ensure cooperation. In essence since
we need sufficient patience, this seems increasingly unlikely as the number of players rises.
5. (a). Profit is simply price multiplied by quantity as we have no costs, so for firm i:
πi = p qi = (10 - Σq) qi = (10 – qi - Σj≠iqj) qi
Partially differentiating qi and setting this equal to zero and then applying symmetry yields:
10 - 2q* - (N-1)q* = 0
Hence q* = 10/(N+1) and p* = 10N/(N+1) and so the optimal profit for any individual firm
is simply π* = 100 N/(N+1)2. Students may wish to verify the second order conditions.
(b). We need π* = 100 N/(N+1)2 ≥ 5 for profits to be positive. Rewriting in terms of N this
and solving a simple quadratic yields the highest whole number of firms as 17.
(c). Current profits are found by inserting 10 into 100 N/(N+1)2 which yields 8.26 (to 2
decimal places) net of the fixed cost. Including the fixed cost they make 3.26 (to 2 decimal
places). Raising fixed cost to 8 seems idiotic since holding N = 10 fixed, it simply reduces
their profits to 0.26. However, with possible entry, and assuming firms are rational and entry
continues so long as positive profits are available, we know from (b) that we will eventually
get to 17 firms. Inserting 17 into the individual profit function 100 N/(N+1)2 yields a per firm
profit of just under 0.25 once fixed costs of 5 have been taken into account and so this is what
each of the incumbent 10 firms can expect once entry has pushed the number of firms to 17.
Raising fixed costs to 8 will completely stop entry since the 11th firm would make negative
profits, so keep the industry at N = 10. Therefore since 0.26 > 0.25, it is in the best interests
of the 10 firms to raise fixed costs and thereby blockade entry.
6. (a). Yes – C1, D1, C2 and D2 can be eliminated by the iterated deletion of strictly dominated
strategies. Students should go through the logic carefully, for example as follows. D1 is
strictly dominated by A1, and D2 is strictly dominated by B2, so both of these can be
immediately eliminated. We are then left with:
Player 1
A1
B1
C1
Player 2
A2
3,2
0,0
0,0
B2
1,1
2,3
0,2
C2
1,0
0,0
0,0
We can then remove C1 as it is strictly dominated by A1 and C2 is strictly dominated by B2.
(b). After the process outlined in (a) we are left with a simple Battle of the Sexes game:
Player 1
A1
B1
Player 2
A2
3,2
0,0
B2
1,1
2,3
There are two pure strategy Nash equilibria: (A1, A2) and (B1, B2). Students are expected to
go through the game explaining why these are the only two pure strategy Nash equilibria. The
unique mixed strategy Nash equilibrium can be found easily by noting that mixing requires
indifference and so the payoff from selecting A1 must equal the payoff from selecting B1 for
player 1. Setting the mixture probabilities for player 1 as p and 1-p for A1 and B1, and for
player 2 as q and 1-q for A2 and B2 respectively, and looking at player 1’s payoffs we have:
U1(A1) = U1(B1) which implies 3q + 1(1-q) = 2(1-q) which in turn implies that q must be
equal to ¼. Similarly we require U2(A2) = U2(B2) which implies 2p = p +3(1-p) and so
p=3/4.
(c). This question essentially has three parts and suggested breakdowns within the 8 marks
available are give as follows. (i) First, examine the payoffs. For player 1: U1(A1, A2) = 3,
U1(B1, B2) = 2 and U1(mix) = 1.5. For player 2: U2(A1, A2) = 2, U2(B1, B2) = 3 and U2(mix) =
1.5. So player 1 prefers the pure strategy Nash equilibrium (A1, A2) and player prefers (B1,
B2). Both players like the mixed strategy Nash equilibrium least of the three Nash equilibria.
[2 marks]
(ii). Then note that since the mix generates a worse payoff to both players than their least
preferred pure strategy Nash equilibrium it seems like they would rather settle for 2 than
entertain the prospect of the mix. However, it is not clear how they would coordinate to get
this, and should they fail to coordinate they might do worse (for instance they might get 1 or
0). On the other hand coordinating on the mix is not trivial either. However, making the bold
assumption that they can simply pick the Nash equilibrium they want it does seem unlikely
they would go for the mix. It does have the nice feature that they get the same payoff – but
alternating between the two pure would also ensure that and the payoff would be better (2.5 >
1.5). [2 marks]
(iii). Finally the students need to consider (briefly) arguments for mixed strategies in general.
Nash equilibrium exists (and then only in a finite game) so more generally, many games do
not have a pure strategy Nash equilibrium (and students may wish to mention matching
pennies, rock-paper-scissors, etc.) and so mixing may be the only way to ensure a Nash
equilibrium in some contexts. Mixed strategies may also have a nice interpretation in terms of
beliefs over types (a la Harsanyi) and students have covered this in lectures. [4 marks]
Part B
[7 by Syantan, 8 by Carlo, 9-10 by Daniel]
7. (There are conceivably several ways of answering this in a satisfactory manner. The
following outlines points that a good answer might address.)
(Based on lecture notes)
Statement of problem with three time periods i.e. t=0 a planning period (no consumption),
t=1 and t=2 as periods where consumption occurs with instantaneous utility and exponential
discounting. Wealth is realised at t=1. Solve for the maximization problem at t=0; state the
FOC; check that the FOC remain unchanged if the agent re-optimzes at t=1. Discuss evidence
on present-bias presented in the lecture eg. payday loans, gym membership etc. Restate the
three period problem with quasi-hyperbolic discounting; state the FOC; show that the FOC
are different if the agent re-optimizes at t=1. A good answer at this point should have a
discussion of the role of commitment devices as addressing the time inconsistency problem.
8. (There are conceivably several ways of answering this in a satisfactory manner. The
following outlines points that a good answer might address.)
Statement of representation of preferences by a value function m(x-r) defined in terms of a
gap between the outcome x and a reference point r. Main characteristics of value function: (i)
Loss Aversion: attitudes towards gains and losses are different because people hate loosing
more than they like winning; formally slope gets flatter at x=r; (ii) Diminishing Sensitivity:
the further you are from the reference point the less intense the experience of departing from
the reference point is; formally, value function is convex below zero and concave above zero.
Extension to uncertainty involves using value function to define an expected reference
dependent utility function. Implications: people value gains more than they value losses of
equal size, or equivalently they value something more if they are endowed with it to begin
with (endowment effect). Experimental evidence: divergence between willingness to pay and
willingness to accept in experiments with random assignment of endowments. Other evidence
from asking prices in the housing market. With reference to the uncertainty case, evidence for
the rejection of small, better-than fair gambles.
9. [There is a suggested breakdown of marks within the essay given in this answer in
bold]
Nash equilibrium can be defined in various ways but in lectures the definition given in two
steps. First we defined best response: player i’s strategy si ∈ Si is a best response to the other
players’ strategies s-i ∈ S-i if Ui(si,s-i) ≥ Ui(si’,s-i) for all si’ ∈ Si. Then we can define a Nash
equilibrium as simply a strategy pair (s1*,s2*) at which s1* is a best response to s2* and s2* is
a best response to s1*.
Notice however that this is done entirely in terms of pure strategies. A better definition in
terms of mixed strategies (also covered in lectures), would be: the mixed strategy profile σ* =
(σ1*,σ2*,...σn*) in MS1×MS2×...×MSn is a Nash equilibrium if every player i∈N is choosing a
best response σi* ∈ MSi to σ-i*∈MS-i. That is Ui(σi*, σ-i*) ≥ Ui(σi, σ-i*) for all σi ∈ MSi. [10
marks]
In terms of its use the most obvious approach is for the student to specify various games and
solve for the Nash equilibrium. Even better is if they then think of real world applications for
these example games. Good examples might be Prisoner’s Dilemma, Battle of the Sexes,
Chicken, Matching Pennies, etc. The first is interesting as it shows that Nash equilibrium can
be Pareto dominated, and does not solve the cooperation problem. The second two are
examples of coordination games and admit a mixed strategy Nash equilibrium which the
students should be expected to be able to solve and discuss. The last has only a mixed
strategy Nash equilibrium. [10 marks]
Good students would discuss Nash’s theorem and the fact that it only specifies that a mixed
strategy Nash equilibrium must exist, not a pure, plus it only applies to finite games and they
should come up with an example of an infinite game (e.g. price setting with no integer
number issues and they might discuss Bertrand competition in this context). [10 marks]
For the second part, again the focus should be Nash’s theorem – the fact that for any finite
game we have some sort of prediction which is useful, and the fact that Nash equilibrium
seems very close to our understanding of rationality (they should discuss best replies at this
point). It also implies no regret once a strategy is chosen and makes sense when coordination
before a game is not possible. It also works well in extensive form games and they should
discuss backwards induction. They may wish to point out weaknesses (uniqueness being a
major one). [10 marks]
10. [There is a suggested breakdown of marks within the essay given in this answer in
bold]
The model is left to the student and so no single model will be specified and solved here, but
rather this answer will attempt to outline the best method (the one used in lectures) to
examine such an issue.
First, the student should start by (a) specifying a cost function and (b) specifying a demand
function. Then they should set up an optimization. [5 marks]
They then need to solve out that problem to find the optimal monopoly quantity. [5 marks]
It is extremely important that they either (a) do something similar for the perfectly
competitive output and show that this involves a higher quantity and then discuss the
monopoly problem in terms of inducing the monopolist to raise his output, or (b) find the
total surplus in a general model and show that the monopolist’s solution does not maximize
total surplus, again with the implication that total output needs to be increased. [10 marks]
[The approach taken in lectures was mathematical and the question makes it clear that
a mathematical model needs to be developed in this question. Diagrams may be used to
supplement this but cannot be used as a replacement.]
At this stage it should be clear that the monopoly problem is that quantity is too low. A
tax/subsidy scheme would involve effectively adding a cost of tq to the cost function. Since
the quantity needs to be raised, the student should then show that this tax needs to be negative
to induce a higher output, in other words the monopolist needs to be subsidized. [10 marks]
The student should comment on this – noting that subsidizing a monopolist is unlikely to
seem like a popular policy however this can be solved by imposing a lump-sum tax on the
monopolist (this would be a constant designed to just cover the costs of the subsidy) and
since it is lump-sum it would not affect the optimization. Hence a lump-sum tax combined
with a subsidy per unit produced designed to just make the monopolist achieve the same
output as a perfectly competitive firm is possible. [10 marks]
[For this second part of the question, again I would expect the model to be used to
specify algebraic solutions, but solid intuition and understanding is also very
important.]