UNIVERSITY OF WARWICK
Summer Examinations 2021/22
SOLUTION – Mathematical Economics - EC301
1. Consider the signalling game below:
(a) Let player 2’s beliefs be denoted as: α = Pr (1x | L) and β = Pr (1x | R). Derive
player 2’s best response at each information set as a function of α and β. (7 marks)
ANSWER: At L-info set
U2 (u) ≥ U2 (d) ⇐⇒ −3α + 2 (1 − α) ≥ 2α − (1 − α)
3
⇐⇒α ≤
8
This gives the best response correspondence of:
BR2|L



u
α<
= {λu + (1 − λ) d, λ ∈ [0, 1]} α =



d
α>
3
8
3
8
3
8
Similar working at the R-info set gives
BR2|R



d
β<
= {λu + (1 − λ) d, λ ∈ [0, 1]} β =



u
β>
1
4
1
4
1
4
Give 3 marks at each info set for correct working/answer and the last mark for getting
correct notation of BR correspondence at either.
(b) We look for a PBE where both types of player 1 mix. In other words, let 1X choose the
strategy xL + (1 − x) R and 1Y choose the strategy yL + (1 − y) R; we are looking for
PBE where x, y ∈ (0, 1). Find the strategy of player 2 in any such PBE. (8 marks)
ANSWER: For both 1X and 1Y to mix, we require that both are indifferent. Let player 2 do
qu + (1 − q) d at L info set and ru + (1 − r) d at R info set. We require player 2’s
strategies to be such that 5q + 3r = 3 to make 1X indifferent and 3q + r = 1 to
make 1Y indifferent. This pair of simultaneous equations has the unique solution
(q, r) = (0, 1). This means player 2 does d at the L info set and u at the R info set.
Give 5 marks for finding the correct equations and 3 marks for solving them. Can
still give some credit for method if student fails to get right equations but notices
they should be solved simultaneously.
(Continued overleaf)
1
(c) Let player 2’s strategy be d at the L information set and u at the R information
set. Find all PBE of this form as a function of p ∈ (0, 1). You should consider the
following categories: i) Separating PBE, ii) Pooling PBE, iii) Hybrid PBE (one type
of player 1 mixes; the other plays a pure strategy). iv) Completely mixed PBE (both
types of player 1 mix). Make sure for each of the 4 categories, you mention for which
p ∈ (0, 1) those PBE exist. (35 marks)
ANSWER: Given this strategy of player 2, 1X is indifferent between his two strategies as both
give him utility of -2. Similarly 1Y is indifferent between his two strategies as both
give him utility of 1. So both 1X and 1Y can do any mix, including pure strategies
and be acting sequentially rationally. But the strategy of player 1 must be such that
player 2 is best responding by playing du, which requires that α ≥ 83 and β ≥ 14 .
(4 marks for recognising this).
Taking the categories in turn:
i) There are no separating PBE as these would have α = 0 or β = 0. This is true for
all p ∈ (0, 1). (4 marks).
ii) To have pooling PBE where both play L, we need p ≥ 38 , then we will get α = p and
we need the off equilibrium path beliefs to be β ≥ 41 . To have pooling PBE where
both play R, we need p ≥ 14 , then we will get β = p and we need the off equilibrium
path beliefs to be α ≥ 38 . (6 marks).
iii) We cannot have hybrid PBE where player 1X plays a pure strategy and player 2
mixes because then we get α = 0 or β = 0. This is true for all p ∈ (0, 1). But we
can have hybrid equilibria where 1X mixes. There are two such types: where 1Y does
L and when 1Y does R. If 1Y does L and 1X mixes, doing xL + (1 − x) R for some
px
and β = 1. So to have a PBE of this form we need
x ∈ (0, 1), then α = px+(1−p)
px
3
≥ ⇐⇒ 8px ≥ 3px + 3 (1 − p)
px + (1 − p)
8
3 (1 − p)
3
⇐⇒x ≥
. x < 1 ⇐⇒ p >
5p
8
α=
So for p > 38 we have PBE where 1X does xL + (1 − x) R for some
px
1Y does L with beliefs α = px+(1−p)
and β = 1.
3(1−p)
5p
≤ x < 1,
The other hybrid PBE is 1Y does R and 1X mixes, doing xL + (1 − x) R for some
p(1−x)
x ∈ (0, 1), then α = 1 and β = p(1−x)+(1−p)
. So to have a PBE of this form we need
p (1 − x)
1
≥ ⇐⇒ 3p (1 − x) ≥ (1 − p)
p (1 − x) + (1 − p)
4
4p − 1
1
⇐⇒x ≤
. x > 0 ⇐⇒ p >
3p
4
β=
(Continued overleaf)
2
So for p >
1
4
we have PBE where 1X does xL + (1 − x) R for some 0 < x ≤
does L with beliefs α = 1 and β =
4p−1
,
3p
1Y
p(1−x)
.
p(1−x)+(1−p)
Give 13 marks for part iii) and be reasonably generous for students who struggle.
iv) In a completely mixed PBE 1X does something of the form xL + (1 − x) R for some
x ∈ (0, 1) and 1Y does something of the form yL + (1 − y) R for some y ∈ (0, 1). As
found in part b):
α=
px
px + (1 − p) y
β=
p (1 − x)
p (1 − x) + (1 − p) (1 − y)
We now show that a completely mixed equilibrium exists if and only if p > 14 ; that
is, we can find (x, y) combinations (infinitely many) such that α ≥ 38 and β ≥ 41 if
and only if p > 14 . Note first that if x ≤ y then p ≥ α and hence it must be p ≥ 38 .
Second, if x > y then 1 − x < 1 − y and thus p > β; hence it must be p > 41 . Thus
the lower bound on p is 14 . Finally note if p > 41 then we can get β ≥ 38 by setting,
y. To see that the
as just shown, x > y and we can get α ≥ 83 by setting x ≥ 35 1−p
p
1−p
1
second constraint can be satisfied, note that p > 4 implies p < 3.
Give 8 marks for part iv).
2. There are two equally likely states of nature, B for bad and G for good. A profit maximizing, risk neutral, firm has assets already in place at date 0 whose value is 3 in state B
and x > 3 in state G. At date 0, the firm may invest in a new project at a cost of 10. The
new project is successful with probability ηB = 1/2 in state B and ηG = 6/11 in state G.
If successful the new project yields an additional value of 22; if unsuccessful it yields 0.
Suppose the state is known to the firm before it decides whether to invest. Note that
if the firm had 10 in cash, it would always invest and start the new project. Suppose
instead that the project has to be fully funded by a risk-neutral bank who cannot observe
the state. After observing the state, the firm offers the bank a contract that specifies as
repayment a share s of the firm’s total ex post value V , if the bank lends 10 and finances
the new project. (V is the sum of the value of the assets already in place and the value
of the new project if it is undertaken.) The bank accepts the contract and lends 10 with
probability α(s) when the offer is s.
(a) As a benchmark, suppose the bank can observe the state, for what values of s would
it accept to lend in the two states? (5 marks)
ANSWER: In the bad state total expected value is 3 + 11 = 14; hence the bank accepts if
s ≥ sB = 10/14. In the good state the total expected value is x + 12; hence the bank
accepts if s ≥ sG = 10/(x + 12).
(Continued overleaf)
3
From now on, suppose the investor does not observe the state.
(b) Define the three components of a perfect Bayesian equilibrium (PBE) of this game.
(5 marks)
ANSWER: A PBE is a triple {s∗ (·), α∗ (s), µ∗ (G|s)}, i.e., a share offer s∗ by the firm as a function
of the observed state; a decision probability α∗ (s) by the bank as a function of the
share offer s; a belief µ∗ (G|s) by the bank about the state as a function of the share
offer s,
(c) Find the pooling equilibrium values of s which the firm would be willing to offer when
the state is B and that the bank, who does not know the state, would be willing to
accept. Explain your answer in two or three sentences. (5 marks)
ANSWER: It must be s∗ (G)=s∗ (B) ∈ [sM = 20/(26 + x), sB = 10/14]
If the bank believes the firm is making the same (pooling) contract offer s irrespective
of the state, it will accept it if and only if [0.5(14) + 0.5(x + 12)]s ≥ 10, or s ≥ sM
Even if the bank believes that the state is B with probability 1, it will always accept
an offer s ≥ sB , hence the firm will always profit from reducing any offer s above sB
to sB .
(d) Derive the values of x for which a pooling equilibrium exists where the bank finances
the new project in both states. Explain your answer in two or three sentences. (10
marks)
ANSWER: x ≤ 9. The firm must prefer to invest when the state is G
Suppose s∗ (G) = s∗ (B) = sM = 20/(26 + x) (lowest possible s).
If obtaining financing issuing equity sM in state G, firm’s expected profits are
20
6+x
= (x + 12)
(x + 12)(1 − sM ) = (x + 12) 1 −
26 + x
26 + x
If not issuing equity in state G, expected profits are x. It must be
6+x
x ≤ (x + 12)
26 + x
or 20x ≤ 12(6 + x)
(e) What are the shares offered by the firm in states B and G in a separating equilibrium?
Derive and explain in a couple of sentences. (5 marks)
ANSWER: Let sB = 10/14 be the lowest offer that bank accepts when the state is B and
sG = 10/(x + 12) be the lowest offer that bank accepts when the state is G. In a
separating equilibrium it is s∗ (B) = sB and s∗ (G) < sG . If the state is B the bank
accepts the share offer and the firm invest; if the state is G the bank rejects the share
offer and there is no investment.
(Continued overleaf)
4
In any separating equilibrium, there are shares s1 6= s2 with s1 being the share offered
when the state is G and s2 the share offered by the firm when the state is B. The
offer by the firm reveals the state.
1. The bank lends in any state if s ≥ sB . Hence firm never offers a share greater
than sB .
2. The offer s2 reveals that the state is B to the bank. Hence the bank does not lend
if s2 < sB , and the firm prefers to deviate and offer sB because:
14(1 − sB ) = 14
4
> 3.
14
3. If the offer s1 made by the firm when the state is G satisfies s1 < s2 = sB and is
accepted by the bank, then when the state is B the firm prefers to deviate and offer
s1 instead of s2 . Hence it must be s1 < sG , so that the bank does not lend if they
observe s1 . In a separating equilibrium the firm only invest if the state is B. Any
offer s1 made by the firm in state G must be rejected to sustain the equilibrium. It
is optimal for the bank to reject as long as s1 < sG .
(f) Derive the values of x for which a separating equilibrium exists. Explain your answer
in a couple of sentences. (10 marks)
ANSWER: In the separating equilibrium it must be the case that the firm does not want to
deviate in either state. In equilibrium only the offer in state B is accepted by the
bank. The bank’s beliefs that support the equilibrium in the stronger way attach
probability one that the state is B for any offer different from the equilibrium offer
s∗ (G) (which could be zero) made by the firm when the state is G. To sustain the
equilibrium: First, we know that when the state is B the firm prefers to offer sB ,
which is accepted, to any offer that is rejected. Second, when the state is G the firm
does not want to deviate and make an offer s∗ (B) = sB which will be accepted:
(x + 12) (1 − sB ) ≤ x,
or equivalently
10
≤ x,
(x + 12) 1 −
14
4 (x + 12) ≤ 14x,
or equivalently
or equivalently
x ≥ 4.8
(g) Suppose x = 10. Compare the market price of the firm (total value minus the
investment cost if the investment takes place) before it goes to the capital market
(i.e., before it knows the state and offers the equity contract) and the market price
of the firm after it that there will be an equity issue, that is after the firm offers a
positive share s∗ (B). Discuss in a couple of sentences. (10 marks)
(Continued overleaf)
5
ANSWER: When x = 10 the only equilibrium is separating and when the state is B the offer
sB = 10/14 is made and accepted, while when the state is G the offer is rejected and
there is no investment. The market price of the firm drops when it is announced that
there will be an equity issue Before the announcement the market price is
1
1
· 10 + · (14 − 10) = 7
2
2
After the announcement the market price is 14 − 10 = 4.
There is a negative stock-price reaction to the announcement, because the announcement reveals that the state is B. It is in the firm’s best interest to issue equity in
state B even though it knows the stock-price will drop.
3. A company’s manager may exert high (eH ) or low (eL ) effort. Exerting low effort has a
disutility to the manager of (cL = 0), while exerting high effort has a disutility of (cH = 10).
If the manager exerts high effort, then shareholders’ revenue is either 2R > 100 or 100 with
equal probability. If the manager exerts low effort, then revenue is 100 with probability
1. When exerting effort ej , j ∈ {H, L} and being paid a wage w, the manager’s payoff is
u(w) − cj , with u(0) = 0, u0 (w) > 0 and u00 (w) ≤ 0 for all w. The manager has an outside
option; they can take a job at another company that gives a payoff of 20. If indifferent
between the outside option and the contract offered by the shareholders, we assume the
manager accepts the contract. Shareholders are risk neutral and their goal is to maximise
expected profit (expected revenue minus expected wage payments). They observe the
revenue realization, but not the effort level exerted by the manager. They must offer a
wage which may depend on the revenue realization.
(a) Suppose the manager is risk neutral, i.e., u(w) = w. For what values of R does
exerting low effort maximise expected revenue minus the disutility of effort? What
flat-wage contract (i.e., wage not depending on the revenue realisation) could the
shareholders offer if they want the manager to exert low effort? Explain in two to
three sentences the intuition behind your answer. (5 marks)
ANSWER: Welfare is R + 50 − 10 with high and 100 with low effort. Hence low effort is optimal
if R ≤ 60. To induce low effort shareholders may offer a contract with flat wage
w = 20, the outside option payoff. If accepted, this contract induces the manager to
exert low effort and makes the manager indifferent between staying and taking the
outside option. The contract allows shareholders to extract as much rent as possible
from the manager while inducing low effort.
(b) Suppose the manager is risk neutral. Are there contract offers that the shareholders
can make that would induce the manager to choose the effort level that maximises
the difference between expected revenue and the disutility of effort? If yes, which
(Continued overleaf)
6
of these offers is best for the shareholders? Explain in two to three sentences the
intuition behind your answer. (5 marks)
ANSWER: Shareholders could “sell” the firm to the manager by offering a wage as a function
of whether revenue is 2R or 100 as follows: w2R = 2R − α; w100 = 100 − α, with α
a constant. The best such contract for the shareholders gives the manager a total
payoff equal to 20. If it is optimal (maximizes total welfare) to induce low effort (i.e.,
R ≤ 60), this requires α = 80. If it is optimal to induce high effort (i.e. R ≥ 60),
this requires α = R + 50 − 20.
(c) For what values of R do shareholders prefer a contract that induces the manager to
exert high effort over the one that induces low effort? (5 marks)
ANSWER: When it is efficient (maximizes total welfare) to exert low effort, i.e., R ≤ 60.
(d) Now suppose the manager is risk averse u00 (w) < 0 for all w. Let φ(u) be the inverse of
the function u(u); that is, φ(u) = w if and only if u(w) = u. What is the best contract
shareholders may offer if they wants the manager to exert low effort? Explain your
answer in two to three sentences. (5 marks)
ANSWER: The shareholders may offer a flat wage w0 , independent from the revenue realisation.
Under this contract the manager exerts low effort, as they thus save the cost 10 of
exerting high effort. The owner extracts all rent by setting w0 = φ(20).
(e) If shareholders want to offer a contract that induces the manager to exert high effort,
what is the incentive constraint that the contract must satisfy? (5 marks)
ANSWER: Let w2R and w100 be the wage when the revenue realisations are 2R and 100, respectively. The manager’s incentive constraint is:
0.5u(w2R ) + 0.5u(w100 ) − 10 ≥ u(w100 )
which can be written as:
u(w2R ) − u(w100 ) ≥ 20
(1)
(f) If shareholders want to offer a contract that induces the manager to exert high effort,
what is the participation constraint that the contract must satisfy? (5 marks)
ANSWER: Let w2R and w100 be the wage when the revenue realisations are 2R and 100, respectively. The manager’s participation, or individual rationality, constraint is:
0.5u(w2R ) + 0.5u(w100 ) − 10 ≥ 20
which can be written as:
u(w2R ) + u(w100 ) ≥ 60
(2)
(g) Suppose shareholders want to offer a contract that induces the manager to exert high
effort. What is the maximization problem shareholders must solve to select the best
(Continued overleaf)
7
contract offer? Derive the solution to the problem; that is, derive the shareholders’
optimal contract offer. Explain in two or three sentences the economic significance
of your conclusion. (15 marks)
ANSWER: The shareholders’ problem is:
max R − 0.5w2R + 50 − 0.5w100
w2R , w100
subject to (1), (2)
It is immediate to see that constraint (2) must bind, otherwise shareholders could
lower w100 without violating either constraint and increase their payoff. Hence it
must be:
u(w100 ) = 60 − u(w2R )
or
w100 = φ(60 − u(w2R ))
and thus we can write the shareholders problem as:
2R − w2R + 100 − φ(60 − u(w2R ))
max
w
subject to:
2R
u(w2R ) − 60 + u(w2R ) ≥ 20
with the constraint that can be written as:
u(w2R ) ≥ 40
or
w2R ≥ φ(40)
Letting λ be the multiplier on the constraint, the Lagrangean is:
2R + 100 − w2R − φ(60 − u(w2R )) + λ (w2R − φ(40))
Differentiating wrt to w2R gives the first order conditions (foc):
−1 + φ0 (60 − u(w2R ))u0 (w2R ) + λ = 0
λ (w2R − φ(40)) = 0
w2R − φ(40) ≥ 0
Differentiating the first foc again shows that the second order condition holds, as:
2
φ0 (60 − u(w2R ))u00 (w2R ) − φ00 (60 − u(w2R )) (u0 (w2R )) < 0
The first foc can be written as
−1 +
φ0 (60 − u(w2R ))
+λ=0
φ0 (u(w2R ))
(Continued overleaf)
8
and since u(w2R ) > u(w2R ) − 20 ≥ 60 − u(w2R ) and φ is a convex function (i.e.,
φ00 > 0), it must be λ > 0 and the incentive constraint must bind. It follows that the
solution is:
φ0 (20)
w2R = φ(40),
λ=1− 0
φ (40)
Thus, the best contract to induce the manager to work hard offers revenue-contingent
wages of w2R = φ(40) and w100 = 60 − φ(40). There is an efficiency loss when the
manager is risk averse, as they are not fully insured; this is the typical inefficient
insurance outcome due to moral hazard.
(h) What is the shareholders’ profit when inducing the manager to exert low and high
effort? For what values of R do shareholders prefer the contract that induces the
manager to exert high effort? (5 marks)
ANSWER: The best contract offer that induces low effort yields shareholders an expected profit
of 100 − φ(20). The best contract offer that induces high effort by the manager yields
shareholders an expected profit of R + 50 − 0.5φ(40) − 0.5(60 − φ(40)) = R + 20.
Shareholders prefers the contract that induces high effort if R + 20 ≥ 100 − φ(20), or
R ≥ 80 − φ(20).
(End)
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