Solutions: Assignment 2 ECON 484/673 Contract Theory

Solutions: Assignment 2
ECON 484/673
Contract Theory
Jean Guillaume Forand ∗
Fall 2011, Waterloo
(Signalling with productive education investments) Consider the education signalling
model from class. That is, a worker has one of two types θ ∈ {θH, θL}, with θH > θL. Given
i ∈ {H, L}, the ex ante probability that the worker is of type θi is βi. Workers have outside
option ū = 0. A firm employs the worker. A worker of type θ produces output θ for the firm.
The firm is willing to employ a worker at wage w if and only if that wage is at least as high
as that worker’s expected productivity within the firm. A worker of type θ can obtain e years
of education at cost c(e, θ) = e
θ . Note that c(e, θ) satisfies the strict single-crossing property in
(−e, θ), that is, if e > e ′ , then c(e, θL) − c(e ′ , θL) > c(e, θH) − c(e ′ , θH). The payoff to a worker
of type θ given wage w and education level e is given by u(w, e|θ) = w − c(e, θ). The timing of
the game is as follows: (i) the worker privately learns its type, (ii) the worker chooses how much
to invest in education, (iii) the firm observes the worker’s education level, but not its type, (iv)
the worker makes a wage offer to the firm and (v) the firm either accepts or refuses to employ
the worker at the proposed wage.
Consider the following modification to this standard setup. Suppose that education can
transform low-type workers into high-type workers. That is, suppose that a θL-type worker
that invests in e years of education becomes a θH-type with probability p(e), with p(0) = 0,
lime→∞ p(e) = 1, pe > 0, pe(0) = ∞, lime→∞ pe(e) = 0 and pee < 0. if a θL-type worker invests
in education and becomes a θH-type worker, it learns its new type before entering the job market.
Again, neither ex ante or ex post types are not observable to the firm.
1. Here, we consider the first-best problem in which worker’s ex ante and ex post types are
observable to the firm. That is, if a θL-type worker invests in education and becomes a
θH-type worker before entering the job market, then the firm observes this.
∗ Room 131, Department of Economics, University of Waterloo, Hagey Hall of Humanities, Waterloo, Ontario,
Canada N2L 3G1. Office phone: 519-888-4567 x. 33635. Email: jgforand@uwaterloo.ca. Website (check for course
materials): http://arts.uwaterloo.ca/~jgforand
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(a) Write down the problem solved by the first-best wage and education levels, wF B and
eF B.
Solution. For θH-type workers, the first-best problem solves
e
max w −
w,e θH
s.t. w ≤ θH.
For θL-type workers, the first-best problem solves
max
wL,wH,e p(e)wH + (1 − p(e))wL − e
θL
s.t. wL ≤ θL and wH ≤ θH.
(b) Solve the first-best problem. In particular, show that the first-best education level for
θH-types is 0 while that for low types is given by eF B > 0, which solves
pe(eF B) =
1
θL(θH − θL) .
Solution. As in class, the solution to the first-best problem for θH-types has them
not obtain any education and receive wage θH. For θL-types, since the individual
rationality constraints of the firm must bind, the first-best problem solves
max
e p(e)θH + (1 − p(e))θL − e
.
θL
Since pee < 0, pe(0) = ∞ and lime ′ →∞ pe(e ′ ) = 0, this problem has a unique solution,
eF B > 0, such that
pe(eF B) =
1
θL(θH − θL) .
(c) Suppose that education investments become more productive. That is, suppose that
a θL worker that invests in e years of education becomes a θH-type with probability
q(e), where q has the same properties as p except that, for all e > 0, q(e) > p(e). What
happens to first-best education levels? What about the payoffs to θL-types under the
first-best? Interpret these results.
Solution. Under productivity q, the first-best education level for θL-types is e ′ F B
such that
qe(e ′ F B) =
1
θL(θH − θL) .
2
> 0

In general, both e ′ F B ≥ eF B and e ′ F B ≤ eF B are possible. However, note that
q(e ′ F B)θH + (1 − q(e ′ F B))θL − e′ F B
θL
where the first inequality follows since e ′ F B
≥ q(eF B)θH + (1 − q(eF B))θL − eF B
θL
> p(eF B)θH + (1 − p(eF B))θL − eF B
,
is optimal under q and the second since,
for all e, q(e) > p(e). Hence, the first-best payoff to θL-types under q is higher than
under p. When education is more productive, ex ante θL-types are always better off
under the first-best. However, exactly because education is more productive, it could
be that this leads ex ante θL-types to reduce their first-best education levels.
2. We now consider the second-best problem in which neither ex ante or ex post types are
observable to the firm. We focus on perfect Bayesian equilibria in pure strategies.
(a) Consider separating equilibria, that is, equilibria in which eH �= eL.
i. Show that in any separating equilibrium, θL-types set eL = eF B.
Solution. Consider a separating equilibrium in which eL �= eF B and a deviation
by a θL-type worker to eF B. Let βJ(e) denote the equilibrium belief of the firm
that the worker’s ex ante type was θJ following an education level of e. Then the
payoff to this deviation satisfies
[βH(eF B) + p(eF B)βL(eF B)]θH+(1 − p(eF B))βL(eF B)θL − eF B
θL
≥ p(eF B)θH + (1 − p(eF B))θL − eF B
θL
> p(eL)θH + (1 − p(eL))θL − eL
,
contradicting the claim that eL is an equilibrium choice, since the final line corresponds
to the equilibrium payoff of θL-types.
ii. Characterise the set of education levels for θH-types, eH �= eF B, that can be
achieved in a separating equilibrium.
Solution. Let êH > 0 be the unique solution to
pe(êH) =
1
θH(θH − θL) .
Note that êH > eF B. I claim that eH �= eF B is a separating equilibrium education
level for θH-types if and only if
� �
eH ∈ θL θH − [w(eF B) − eF
� �
B
] , θH θH − [w(êH) − êH
��
] , (1)
3
θL
θH
θL
θL

where w(e) = p(e)θH + (1 − p(e))θL. To show this, first note that condition (1)
is necessary for an education level eH to be sustained in a separating equilibrium.
That is, under a separating equilibrium with education eH for θH-types, we must
have that
and that
θH − eH
θH
w(eF B) − eF B
θL
≥ [βH(êH) + p(êH)βL(êH)]θH + (1 − p(êH))βL(êH)θL − êH
≥ w(êH) − êH
,
θH
≥ θH − eH
.
θL
Second, note that condition (1) is sufficient for an education level eH to be sustained
in a separating equilibrium. That is, if eH satisfies condition (1), the
following strategies and beliefs form a separating equilibrium: θL types obtain education
level eF B, θH-types obtain education level eH, βH(eH) = 1 and βL(e) = 1
for all e �= eH. Given any e �= eH, we have that
w(eF B) − eF B
θL
≥ w(e) − e
,
θL
by the definition of eF B. Also, we have that
w(eF B) − eF B
θL
≥ θH − eH
,
θL
since eH satisfies (1). Hence, eF B is optimal for θL-types. Similarly, given any
e �= eH, we have that
θH − eH
θH
≥ w(êH) − êH
≥ w(e) − e
θH
θH
,
where the first inequality follows from (1) and the second from the definition of
êH. Hence, eH is optimal for θH-types.
iii. Does a separating equilibrium always exist?
Solution. This question reduces to whether the set of effort levels in (1) is
nonempty for any values for the model’s parameters. It turns out that a separating
equilibrium always exists, although the argument establishing this is more
complicated than I anticipated. For any θ ∈ [θL, θH], let
e
U(θ) = max w(e) −
e θ .
4
θH

By the envelope theorem, we have that U ′ (θ) = êθ
θ 2 , where êθ is the solution to
the maximisation problem above (note that êθL = eF B and êθL = êL). Hence, we
have that
U(θH) = U(θL) +
� θH
θL
> U(θL) + eF B
= U(θL) + eF B
êθ
dθ
θ2 � θH
θL
1
dθ
θ2 θH − θL
θHθL
,
where the inequality follows from the fact that êθ is increasing in θ. Using these
results, we have that
�
θH θH − [w(êH) − êH
� �
] −θL θH − [w(eF B) − eF
�
B
]
as desired.
θH
θL
= θH [θH − U(θL)] − θL [θH − U(θL)]
�
> (θH − θL) θH − U(θL) − eF
�
B
= (θH − θL) [θH − w(eF B)]
≥ 0,
iv. Suppose, as in question 1(c), that education investments become more productive,
that is, that a θL worker that invests in e years of education becomes a θH-type
with probability q(e) instead of p(e). What is the effect of this on the education
levels for θH-types that can be sustained in separating equilibria? Interpret this
result.
Solution. We can rewrite the set of θH-type education levels from condition (1)
as [θL[θH − U(θL)], θH[θH − U(θH)]]. Since, as shown in question 1(c), U(θ) is
higher under q than under p for any θ, we have that both the minimal and the
maximal level of education attained in a separating equilibrium are lower under
q than under p. When education is more productive, ex ante θH-types have less
incentives to obtain high levels of education just to separate themselves from θLtypes,
since they can use their lower costs to education to obtain higher wages by
convincing the firm that they are likely to be ex-post θH-types through education
investments. Similarly, when education is more productive, ex ante θL-types have
less incentives to mimic θH-type education investments since their wages increase
even when firms know (in equilibrium) their initial types.
(b) Consider pooling equilibria, that is, equilibria in which eH = eL = e ∗ .
5
θL

i. Given a pooling equilibrium with education level e ∗ , what is the wage received by
the worker?
Solution. We have that w P (e ∗ ) = [βH + p(e ∗ )βL]θH + βL(1 − p(e ∗ ))θL.
ii. Characterise the set of education levels e ∗ that can be achieved in a pooling
equilibrium.
Solution. I claim that e∗ is an education level in a pooling equilibrium if and
only if
e ∗ � �
≤ min θH w P (e ∗ �
) − w(êH) − êH
�� �
, θL w
θH
P (e ∗ �
) − w(eF B) − eF
���
B
(2)
θL
To show this, first note that condition (2) is necessary for an education level e ∗ to
be sustained in a separating equilibrium. That is, under a separating equilibrium
with education e ∗ , we must have that
w P (e ∗ ) − e∗
θH
≥ [βH(êH) + p(êH)βL(êH)]θH + (1 − p(êH))βL(êH)θL − êH
≥ w(êH) − êH
,
θH
and that, following a similar argument
w P (e ∗ ) − e∗
θL
≥ w(eF B) − eF B
θL
.
Second, note that condition (2) is sufficient for an education level e ∗ to be sustained
in a pooling equilibrium. This follows by constructing equilibria in a manner very
close to that in question 2(a)ii.
iii. Does a pooling equilibrium at e ∗ = 0 always exist? What about at e ∗ = eF B?
Solution. A pooling equilibrium at e ∗ = 0 does not always exist. To see this,
suppose that p is such that eF B, êH ≈ 0 and p(eF B), p(êH) ≈ 1. Then condition
(2) requires that
e ∗ = 0 ≤ min{θH[βHθH + βLθL − θH], θL[βHθH + βLθL − θH]} < 0,
a contradiction. If e ∗ = eF B, then since w P (eF B) > w(eF B), condition (2) reduces
to
eF B ≤ θH[w P (eF B) − w(êH)] + êH,
which is necessary and sufficient for such a pooling equilibrium to exist.
iv. Suppose, as in question 1(c), that education investments become more productive,
that is, that a θL worker that invests in e years of education becomes a θH-type
with probability q(e) instead of p(e). What is the effect of this on the education
levels e ∗ that can be sustained in pooling equilibria? Interpret this result.
6
θH

Solution. As in question 2(a)iv., we can rewrite the righthand side of condition
(2) as min{θH[w P (e ∗ ) − U(θH)], θL[w P (e ∗ ) − U(θL)]}. Since U(θH), U(θL) and
w P (e ∗ ) are higher under q than under p, the effect of this change on the education
levels sustained in pooling equilibria is ambiguous. When education is more
productive, wages rise in a pooling equilibrium, but so do the value of workers’
‘outside options’, which is to invest in enough education that firms believe them
to be of ex post θH-type with high probability.
v. With productive investments in education, do any pooling equilibria satisfy the
intuitive criterion?
Solution. As in the simple model in class, no pooling equilibria survive the intuitive
criterion when education investments are productive. Note that the most
favourable beliefs about the worker that the firm can hold following any education
level is that the worker was an ex ante θH-type. Hence, we can follow the same
argument as in class for the simple model: given favourable beliefs by the firm,
the education level that keeps the ex ante θL-type worker indifferent between its
pooling equilibrium payoff and a deviation to this education level is strictly preferred
by the ex ante θH-type worker. Hence, the intuitive criterion requires that
equilibrium beliefs be such that (nearby) such deviations are attributed to ex ante
θH-types, which is inconsistent with pooling equilibria incentive constraints.
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