Summer 2014 examination
EC202
Microeconomic Principles 2
Suitable for ALL candidates
Instructions to candidates
Time allowed: 3 hours + 15 minutes reading time.
This paper contains seven questions in three sections. Answer QUESTION ONE
of Section A, and THREE other questions, at least ONE question from Section B
and at least ONE question from Section C. Question one carries 40% of the total
marks; the other questions each carry 20% of the total marks.
Calculators are NOT allowed in this examination.
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Section A
1. Answer any FIVE parts from the following eight parts, (a)-(h). Each part carries
eight marks.
(a) A single-output firm can use any of the following three production techniques
1
z1 , z2
Technique #1:
q ≤ min
3
1
Technique #2:
q ≤ min z1 , z2
3
2 2
Technique #3:
q ≤ min
z1 , z2
5 5
where q denotes the quantity of output and z1 , z2 the quantities of two inputs.
The firm can also use combinations of techniques. Draw the isoquant for
q = 1.
Answers: The input requirements set using techniques 1 and 2 is shown by
the shaded area in Figure 1. Clearly technique 3 lies in the interior of this set
and would not be used by the firm (the firm can produce the output with less
input by combining techniques 1 and 2).
Figure 1: Isoquant for q = 1
z2
•
3
z2
•z
2.5
3
z1
•
1
q=1
0
1
2.5
3
z1
8 mks
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(b) A market consists of N identical price-taking firms, where each firm i has the
cost function 16 + qi2 . Market demand is given by N [11 − p] where p is price.
i. What is the supply curve for firm i?
ii. Assuming that N is a large number, carefully describe the equilibrium in
this market
Answers: MC is 2qi and AC is 16/qi + qi so that minimum average cost at
qi = 4 where MC=AC=8.
i. The supply curve is qi = 0 if p < 8, qi ∈ {0, 4} if p = 8, qi = 21 p if p > 8.
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ii. If the price were below 8 there would be zero output and demand would
exceed supply. If the price were 8 + ε the demand per firm would be 3 − ε
and supply by each firm would be 4 + ε: there would be excess supply.
If the price were exactly 8 then there would be an equilibrium quantity of
3 units per firm. No individual firm will produce qi = 3: but equilibrium
is achieved if 0.75N of the firms produce qi = 4 and 0.25N of the firms
produce nothing.
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(c) Mark each of the following true or false; in each case briefly explain your
answer:
i. If an exchange economy is replicated indefinitely the core of the economy
shrinks to a single allocation.
Answers: False: the core shrinks to the set of competitive equilibria.
ii. By Walras’ law, the sum over all goods of price times excess demand
must equal zero, but only in the neighbourhood of equilibrium.
Answers: False: this sum will be equal to zero for all competitive allocations, not just in the neighbourhood of equilibrium.
iii. A general equilibrium will exist only if the weak axiom of revealed preference is satisfied by all excess demand functions.
Answers: False: WARP has to do with uniqueness and stability of equilibrium, not existence.
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iv. In a general equilibrium it is not necessarily the case that excess demand
equals zero in every market.
Answers: True: one could have excess supply in good i if the equilibrium
price of good i is zero.
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(d) Suppose that in the equilibrium of an exchange economy everyone has the
same income. Will the equilibrium be a fair allocation? Explain your answer.
Answers: Consider household h that chooses xh to max U h xh subject to
P h
pi xi ≤ y where y is the same income for every household. Suppose h
chooses x∗h and k chooses x∗k in equilibrium. By definition of optimisation
U h x∗h ≥ U h x∗k and so h would not envy k’s bundle in equilibrium.
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(e) Consider the following normal form game:
1\2
L
R
T
3, 1 0, 2
M
2, 0 1, 1
B
4, 1 3, 2
Does any of the two players has a dominant strategy? Explain your answer.
Answers: Strategy B is a dominant strategy for player 1 while strategy R is
the dominant strategy for player 2, hence this game is dominant solvable and
the Dominant Strategy equilibrium is (B, R).
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(f) Consider a game where player 1 chooses between two strategies labelled U
and D, while player 2 chooses between the two strategies labelled L and R.
Assume that the strategy U is a dominant strategy for player 1 while player
2’s payoffs are such that he is indifferent whatever the strategy choice of his
opponent. What is the best reply of player 1? What is the best reply for player
2? What is the Nash equilibrium strategy profile in this game? Explain your
answers
Answers: The best reply for player 1 is B1 (s2 ) = {U } for all player 2’s strategies s2 ∈ {L, R}. The reason is that it is never a best reply for a player to
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choose a strictly dominated strategy. The best reply for player 2 is B2 (s1 ) =
{L, R} for all player 1’s strategies s1 ∈ {U, D}. The reason is that when indifferent between two strategies both strategies are part of a player’s best reply
correspondence. Clearly this game has two pure strategies NE: (U, L) and
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(U, R).
(g) Mark each of the following statement true or false; in each case briefly explain
your answer:
i. Nash equilibrium strategies may be such each player is completely indifferent among all his available strategies.
Answers: True, an example is a non-degenerate mixed strategy Nash
equilibrium.
ii. In a dynamic game Nash equilibrium strategies cannot be supported by
non-credible threats on the part of the players.
Answers: False, it is possible to construct a Nash equilibrium of a dynamic game where strategy choices that are off-the-equilibrium-path are
non-credible. This is the reason for restricting attention to Subgame Perfect Equilibria.
iii. In a dynamic game Subgame Perfect equilibrium strategies cannot be
supported by non-credible threats on the part of the players.
Answers: True, the reason is that Subgame Perfect Equilibria are, by
definition, Nash equilibria of every proper subgame.
iv. It is possible to construct a mixed strategy equilibrium of a normal form
game where one of the players randomizes with strictly positive probability
on a strategy that is strictly dominated.
Answers: False, it is never optimal to play a strictly dominated strategy
with strictly positive probability. A simple contradiction argument shows
that such a mixed strategy cannot be best reply to any strategy of the
opponent.
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(h) What is the set of feasible and individually rational payoffs associated with the
infinitely repeated game with stage game:
L
1\2
R
U
1, 1 0, 0
D
0, 0 4, 4
Answers: The set of feasible and individually rational payoffs V is the segment connecting point (1, 1) to point (4, 4) in the graph below, if the minmax
payoff for both player is computed restricting attention to pure strategies. In
this case the minmax payoff are π 1 = π 2 = 1.
If instead the minmax payoff is computed considering mixed strategies the
minmax payoffs are: π 01 = π 02 =
4
5
and the set of feasible and individually
rational payoffs V 0 is the segment connecting point (4/5, 4/5) to point (4, 4, ) in
the graph.
6
Π1
r (4, 4)
4
5
r
(0, 0)
4
5
r (1, 1)
-
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Π2
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Section B
Answer at least ONE and no more than TWO questions.
2. Anne lives for three periods and her lifetime utility function is given as
3
X
log (xt − at ) ,
t=1
where xt is Anne’s consumption in period t and the at are parameters such that
a2 > a3 > a1 > 0. The price of consumption goods may be different in each period.
Anne receives exogenous money income in each of the three periods and can
borrow or lend freely at the per-period interest rate r.
(a) Solve Anne’s optimisation problem and interpret the solution.
Answers: The lifetime budget constraint is
p 1 x1 +
p 2 x2
p 3 x3
≤ y,
+
1 + r [1 + r]2
where
y := y1 +
y2
y3
.
+
1 + r [1 + r]2
Note that the at are the minimum consumption requirement (“needs”) in each
of the three periods: needs in middle age are higher than those in old age
which, in turn are higher than those in youth. The Lagrangean for the optimisation problem can be written:
3
X
"
log (xt − at ) + µ y −
t=1
3
X
#
p0t xt
(1)
t=1
where
p01 := p1 ,
p2
,
p02 :=
1+r
p3
p03 :=
.
[1 + r]2
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Differentiating (1) with respect to xt and µ and setting the differentials equal
to zero we get the following FOCs
x∗t
1
− µ∗ p0t = 0, t = 1, 2, 3
− at
3
X
y−
p0t x∗t = 0
(2)
(3)
t=1
where the asterisks denote values at the optimum. Substituting from (2) into
(3) we find
µ∗ =
where
y0 :=
3
,
y − y0
3
X
p0t at .
(4)
t=1
Substituting µ∗ into (2) we find that optimal consumption in period t is
x∗t = at +
y − y0
.
3p0t
(5)
The sum y0 represents the present value in period 1 of the minimum consumption requirements. Call the amount consumed in any period above minimum
requirements “discretionary” consumption; then, in present-value terms, the
expenditure on discretionary consumption in each period is exactly one third
of discretionary income y − y0 .
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(b) Under what conditions will Anne save in period 1? If the price of period-2
goods were expected to rise, what effect would there be on period-1 savings?
Answers: Anne will want to save in period 1 if
p1 x∗1 < y1 .
Denote net savings in period 1 by
s1 := y1 − p1 x∗1
(6)
(if s1 is negative then −s1 is the amount of borrowing in period 1). From (5)
and (6) we have
s 1 = y 1 − p 1 a1 −
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y − y0
.
3
(7)
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Note that, if we differentiate (4) we have
∂y0
at
=
, t = 1, 2, 3.
∂pt
[1 + r]t−1
So, differentiating (7) with respect to p2 we have,
∂s1
a2
=
.
∂p2
3 [1 + r]
(8)
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(c) If the price of period-3 goods were expected to rise, what effect would there
be on period-1 savings? Is this greater than the effect in part (b)?
Answers: Differentiating (7) with respect to p3 , we have
∂s1
a3
.
=
∂p3
3 [1 + r]2
(9)
Given that a2 > a3 and the interest rate is positive it must be the case that
∂s1
∂p2
>
∂s1
.
∂p3
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(d) What would be the effect on period-1 savings of a rise in the price of period-1
goods? Explain why this effect is different from those analysed in parts (b)
and (c).
Answers: Differentiating (7) with respect to p1
∂s1
a1
= −a1 +
∂p1
3
2
= − a1 .
3
(10)
(11)
If there is a rise in pt for any t this raises the cost of the minimum lifetime expenditure y0 and so there is an effect of lowering lifetime discretionary expenditure; by itself this effect raises period-1 savings by an amount 13 at [1 + r]1−t
– see equations (8), (9) and the second term in (10). For price rises in middle
age or old age there is no other effect. But if p1 rises one has to spend more
to satisfy needs during youth – this extra amount is exactly a1 ; so there is an
additional effect of −a1 on period-1 savings, the first term in (10); this is larger
than the second term in (10) and thus overall period-1 savings must decrease. 5 mks
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3. Fred’s current wealth y is given by y0 +y1 where y0 is the value of his financial assets
and y1 is the current value of his house. Unfortunately Fred’s house was built in
an area liable to flooding; if there is a flood his house will be worth nothing, but his
financial assets will remain unaffected; if there is no flood the value of all his wealth
remains unchanged; the probability of flood is π. A company offers insurance
against flood: if Fred buys an amount x of insurance the insurance company will
charge a premium κx where π < κ < 1 and x ≤ y1 .
(a) If Fred buys an amount x of insurance, find his ex-post wealth y in two cases:
(i) where there is no flood (ii) where there is a flood.
Answers: Consider Fred’s wealth using the two-state model (NO FLOOD,FLOOD).
If Fred remained uninsured it would be (y0 + y1 , y0 ); if he insures fully it is
(y0 + [1 − κ] y1 , y0 + [1 − κ] y1 ); if he insures an amount x his ex-post wealth in
the two states will be
(y0 + y1 − κx, y0 + [1 − κ] x)
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(b) Fred’s utility is Eu (y) where u is an increasing, concave function and E is the
expectations operator. Is Fred risk averse? Will he take out full insurance on
his house?
Answers: Expected utility is given by
Eu = [1 − π] u (y0 + y1 − κx) + πu (y0 + [1 − κ] x)
Therefore
∂Eu
= − [1 − π] κuy (y0 + y1 − κx) + [1 − κ] πuy (y0 + [1 − κ] x)
∂x
Consider what happens in the neighbourhood of x = y1 (full insurance). We
get
∂Eu = − [1 − π] κuy (y0 + [1 − κ] y1 ) + [1 − κ] πuy (y0 + [1 − κ] y1 )
∂x x=y1
= [π − κ] uy (y0 + [1 − κ] y1 )
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We know that uy (·) > 0 and π < κ. Therefore this expression is strictly negative which means that in the neighbourhood of full insurance the individual
could increase expected utility by cutting down on the insurance cover.
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(c) If u(y) = log (y) find x∗ , the optimal value of x.
Answers: For an interior maximum we have
∂Eu
=0
∂x
which means that the optimal x∗ is the solution to the equation
− [1 − π] κuy (y0 + y1 − κx∗ ) + [1 − κ] πuy (y0 + [1 − κ] x∗ ) = 0
If u (y) = log (y) this becomes
−
[1 − κ] π
[1 − π] κ
+
=0
∗
y0 + y1 − κx
y0 + [1 − κ] x∗
− [1 − π] κ [y0 + [1 − κ] x∗ ] + [1 − κ] π [y0 + y1 − κx∗ ] = 0
x∗ =
[] y0 + [1 − κ] πy1
.
κ [1 − κ]
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(d) How will x∗ change if (i) the value of Fred’s financial assets increases and (ii)
the value of Fred’s house increases? Briefly comment on your answer.
Answers: Differentiating the above equation with respect to y0 and y1 respectively:
∂x∗
π−κ
=
< 0,
∂y0
κ [1 − κ]
∂x∗
[1 − κ] π
=
> 0.
∂y1
κ [1 − κ]
In the first case we have a pure endowment effect. Given DARA richer people
take more risks.
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4. In a two-commodity exchange economy there are two groups of people: type a
have the utility function 2 log(xa1 )+log(xa2 ) and an endowment of ` units of commodity 1 and k units of commodity 2; type b have the utility function log(xb1 ) + 2 log(xb2 )
and an endowment of 3 − ` units of commodity 1 and 7 − k units of commodity 2,
where 0 ≤ ` ≤ 3 and 0 ≤ k ≤ 7.
Answers: The incomes for the two types of people are:
y a = ρ` + k
(12)
y b = [3 − `] ρ + [7 − k]
(13)
where ρ is the price of commodity 1 in terms of commodity 2.
(a) Show that the equilibrium price of good 1 in terms of good 2 is
7+k
.
6−`
Answers: We have a Cobb-Douglas utility function and so the expenditure
shares are constant at 32 , 13 for aand 13 , 23 for
b. Demand for the two
comρ[3−`]+7−k 2ρ[3−`]+14−2k
2ρ`+2k ρ`+k
for a and
for b.
, 3
,
modities is therefore
3ρ
3ρ
3
This means that the excess demand for commodity 2 at price ρ is
ρ` + k 2ρ [3 − `] + 14 − 2k
+
−7
3
3
which simplifies to
[6 − `] ρ − k − 7
3
The equilibrium price is found by setting this equal to 0. Doing this we get
ρ=
7+k
.
6−`
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(b) What are the individuals’ incomes (y a , y b ) in equilibrium as a function of k and
`? Assume that it is possible to carry out lump-sum transfers of commodity 2
(k can be varied), but impossible to transfer commodity 1 (` is fixed). Draw a
diagram of the set of attainable income distributions
i. in the case where ` = 0;
ii. in the case where ` = 3.
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yb
(0,10.5)
ℓ=0
(7, 7)
ℓ=3
ya
Figure 2:
0
(21,0)
Answers: Substituting back into (12) and (13) we get:
ya
=
7+k
`
6−`
y b = [3 − `] 7+k
6−`
7` + 6k
6−`
63 − 14` − 3k
+ [7 − k] =
6−`
+k
(14)
=
(15)
Equations (14) and (15) imply that, if ` is fixed, there is a straight-line frontier
on the set of income pairs (y a , y b ) mapped out by letting k vary from 0 to 7
i. If ` = 0 the boundary of the income possibility set is
ya = k
21 − k
yb =
2
a straight line from (0, 10.5) to (7, 7) in Figure 2..
ii. If ` = 3 the boundary of the income possibility set is given by
y a = 7 + 2k
yb = 7 − k
the straight line from (7, 7) to (21, 0) in Figure 2.
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The government seeks to maximise the welfare function y a + y b .
(c) Is the government inequality-averse?
(d) What would be the optimal distribution of income in the case ` = 0? Comment
on the result.
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(e) What would be the optimal distribution of income in the case ` = 3? Comment
on the result.
Answers: Given the social-welfare function y a + y b :
(c) The government is indifferent to inequality; the contours of the welfare function
are given by the broken lines in Figure 2.
(d) In case ` = 0, the optimum is at (7,7). Even though the government is indifferent to income inequality the optimum is perfect equality!
(e) In case 3, the optimum is at (7,7). This is an unattractive outcome since it
involves an extreme outcome where the b people have zero income.
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Section C
Answer at least ONE and no more than TWO questions.
5. Two players, labelled i ∈ {1, 2} play the following game. Both players choose
between two alternative actions simultaneously and independently.
Player 1 chooses between actions {U, D} while player 2 chooses between actions
{L, R}. When the action profile chosen is (U, L) the vector of the players’ payoffs
is (6, 6), where the first number is player 1’s payoff while the second number is
player 2’s payoff. When the action profile chosen is (U, R) the vector of the players’
payoffs is (2, 8). When the action profile chosen is (D, L) the vector of the players’
payoffs is (8, 4) and finally when the action profile chosen is (D, R) the vector of
the players’ payoffs is (4, 4).
(a) Formulate the strategic situation described above as a normal form game.
What are the strategies for the two players?
Answers: The normal form game is:
1\2
L
R
U
6, 6 2, 8
D
8, 4 4, 4
(b) Identify the set of pure strategy Nash Equilibria of this game.
Answers: There exists two pure strategy Nash equilibria of this game: (D, L)
and (D, R) with payoffs (8, 4) and (4, 4) respectively.
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(c) Identify the set of mixed strategy Nash equilibria of this game.
Answers: In addition to the two pure strategy (degenerate mixed strategy)
Nash equilibria derived in (b) above there exists a continuum of additional
mixed strategy Nash equilibria of this game:
(p = 0, q ∈ [0, 1])
with expected payoffs (4 + 4q, 4).
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Assume now that player 1 moves first and chooses between actions {U, D}. Player
2 observes player 1’s action choice and only then chooses between actions {L, R}.
The payoffs are the same described above.
(d) Formulate this new strategic situation as an extensive form game. What are
the strategies for the two players?
Answers: The extensive form of the game is:
1
a
#c
#
c
c
U ##
D
c
#
c
#
c
c
2 ##
c 2
q
#
cq
S
T
L1 T R1
L2 S R2
T
S
q
Tq
Sq
q
(6, 6)
(2, 8)
(8, 4)
(4, 4)
The strategy of player 1 is s1 ∈ {U, D} while the strategy of player 2 is a pair
s2 = (s2 (U ), s2 (D)) where s2 (U ) ∈ {L1 , R1 } and s2 (D) ∈ {L2 , R2 }.
(e) What is the normal form associated with the extensive form of this dynamic
game?
Answers: The normal form is:
1\2
L1 , L2
L1 , R2
R1 , L2
R1 , R2
U
6, 6
6, 6
2, 8
2, 8
D
8, 4
4, 4
8, 4
4, 4
(f) Identify the set of pure strategy Nash Equilibria of this dynamic game.
Answers: There are three pure strategy Nash Equilibria of this extensive form
game: [D, (L1 , L2 )], [D, (R1 , L2 )] and [D, (R1 , R2 )] with payoffs (8, 4), (8, 4) and
(4, 4) respectively.
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(g) Identify the set of Subgame Perfect equilibria of this dynamic game.
Answers: Using backward induction we obtain that only two of the Nash
equilibria above are Subgame Perfect equilibria of this game: [D, (R1 , L2 )]
and [D, (R1 , R2 )].
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6. Consider the following Cournot duopoly game.
Two firms labelled i ∈ {1, 2} simultaneously and independently choose their output
level qi so as to maximize their profits.
Both firms have the same constant returns to scale technology and their constant
marginal cost
c = 1.
(16)
p = 2 − q1 − q2 .
(17)
The inverse market demand is
(a) Represent the Cournot competition described above as a normal form game.
What are the strategies of the two firms?
Answers: Cournot competition can be represented as the following game.
The set of players is N = {1, 2}, the strategy choice for player i ∈ {1, 2} is
{qi ≥ 0}, payoff for player i ∈ {1, 2} is Πi = qi (p − c).
(b) Compute the best replies strategies of firm 1 and firm 2.
Answers: Best reply for firm i ∈ {1, 2} is:
qi =
(1 − q−i )
2
(c) Identify the set of Nash equilibria of this game and the associated equilibrium
strategies and profits for both firms.
Answers: The unique Nash equilibrium of this game is characterized by
strategies:
q1c = q2c =
1
3
Πc1 = Πc2 =
1
9
and profits:
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Consider now the following change in the timing of competition. Firm 1 is a Stachelberg leader and as such chooses its output q1 first. Firm 2 observes the output
choice of firm 1 and only then decides how much to produce. Firms’ technology
and the market demand are the same as in (16) and (17) above.
(d) Represent the Stachelberg competition described above as an extensive form
game. What are the strategies of the two firms?
Answers: The extensive form game is characterized by the same set of players as above N = {1, 2} the action space for player i is {qi ≥ 0} the set of
non-terminal histories are H = {∅, q1 }, the player function P (h), h ∈ H such
that P (∅) = 1, P (q1 ) = 2. The strategy for player 1 is q1 while a strategy for
player 2 is s2 (q1 ). Payoffs for both players are then: Π1 = q1 (1 − q1 − s2 (q1 ) and
Π2 (q1 ) = q2 (1 − q1 − q2 ).
(e) Compute the best replies strategies of firm 1 and firm 2 in this dynamic game.
Answers: The best reply for player 2 is:
s2 (q1 ) =
(1 − q1 )
2
while the best reply for player 1 is:
q1 =
(1 − s2 (q1 ))
2
(f) Identify the set of Subgame Perfect equilibria of this game and the associated
equilibrium profits and strategies for both firms.
Answers: Backward induction implies that the unique Subgame perfect equilibrium of this game is:
(1 − q1 )
1 s
s
q1 = , s2 (q1 ) =
2
2
the outcome of the subgame perfect equilibrium strategy are the pair of quantities:
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1
1
q1s = , q2s = ss2 (1/2) =
2
4
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the profits in equilibrium are:
1
1
Πs1 = , Πs2 =
8
16
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(g) Compare the Nash equilibrium quantities and profits you derived in (c) above
with the quantities and profits associated with the Subgame Perfect equilibria
you identified in (f).
Answers: Clearly
q1s > q1c , q2s < q2c , Πs1 > Πc1 , Πs2 < Πc2
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7. Consider the following normal form game:
1\2
L
C
R
T
1, 1
0, 0 1, 0
M
0, 5
4, 4 2, 1
B
0, 0
5, 0 3, 3
(a) Identify the set of pure strategy Nash equilibria of this game.
Answers: The game has two pure strategy Nash equilibria: (T, L) and (B, R).
(b) Identify the mixed strategy Nash equilibria of this game.
Answers: In addition to the two pure strategy (degenerate mixed strategy)
Nash equilibria derived in (a) above the game has the mixed strategy Nash
equilibrium
2
3
σ1 (U ) = , σ1 (M ) = 0; σ2 (L) = , σ2 (C) = 0
4
3
with associated expected payoffs Π1 (σ) = 1 and Π2 (σ) = 3/4.
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Assume now that this game is played in two consecutive periods. Each player
discount the future at the same rate δ. The average discounted payoff of the players
is:
1 gi (a1i , a1−i ) + δgi (a2i , a2−i )
(18)
1+δ
where gi (ati , at−i ) is the stage game payoff of player i if the strategy profile chosen
Πi =
by both players in period t ∈ {1, 2} is (ati , at−i ): at1 ∈ {T, M, B} and at2 ∈ {L, C, R}.
(c) Construct strategies for the two-periods repeated game that support the payoff (3, 3) in each period of the game as a Subgame Perfect equilibrium. For
what values of the discount factor δ are these strategies subgame perfect.
Answers: Te history independent strategies that support the payoff vector
(3, 3) are:
for player 1: play B every period independently of the history,
for player 2: play R every period independently of the history.
c LSE 2014/EC202
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These strategies are a Subgame Perfect equilibrium of the two-periods repeated game for all values of the discount factor δ since each player is choosing the stage game Nash equilibrium strategy in every subgame.
6 mks
(d) Construct strategies for the two-periods repeated game that support the payoff (4, 4) in period t = 1 and the payoff (3, 3) in period t = 2 for both players as
a Subgame Perfect equilibrium. For what values of the discount factor δ are
these strategies subgame perfect.
Answers: Te history dependent strategies that support the payoff vectors
(4, 4) in t = 1 and (3, 3) in t = 2 are:
for player 1: play M in in t = 1.
In in t = 2 play B if the previous period outcome is (M, C), otherwise play T .
for player 2: play C in in t = 1.
In in t = 2 play R if the previous period outcome is (M, C), otherwise play L.
These strategies are a Subgame Perfect equilibrium of the two-periods repeated game since each player is choosing the stage game Nash equilibrium
strategy in every subgame at t = 2. Moreover, both players do not want to
deviate from these strategies if
4 + δ3 ≥ 5 + δ1
or
δ≥
1
2
9 mks
c LSE 2014/EC202
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