Consider the following principal agent problem with moral hazard (effort is not contractible). The firm's profits are given by: π = e+z, where e is the manager's effort and z is a random variable with E(2) = 0, Var(z) = o². Assume that the contract is given by the linear payment scheme: w = a + bà, where a is a fixed salary and b is the share of profits. The agents's cost of supplying effort is c = 5e², so his net income is y =W- c. Let his utility, u, be given by the linear mean-
Consider the following principal agent problem with moral hazard (effort is not contractible). The firm's profits are given by: π = e+z, where e is the manager's effort and z is a random variable with E(2) = 0, Var(z) = o². Assume that the contract is given by the linear payment scheme: w = a + bà, where a is a fixed salary and b is the share of profits. The agents's cost of supplying effort is c = 5e², so his net income is y =W- c. Let his utility, u, be given by the linear mean-
Chapter16: Labor Markets
Section: Chapter Questions
Problem 16.9P
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![Question:
=
Consider the following principal agent problem with moral hazard (effort is not
contractible). The firm's profits are given by: π = e+ z, where e is the manager's
effort and z is a random variable with E(2) = 0, Var(z) = 0². Assume that the
contract is given by the linear payment scheme: w = a + bí, where a is a fixed
salary and b is the share of profits. The agents's cost of supplying effort is c = .5e²
so his net income is y = wc. Let his utility, u, be given by the linear mean-
variance utility function, u = E(y) — .5R Var(y), where R > 0 is a parameter
(capturing risk aversion). Find the solution to this principal agent problem (the
optimal values of a, b and e).
Use below Methods to solve above question:
3 A Simple Example of a Principal-Agent Problem
• Profits:
where
● Payment: linear in profits:
with
● where, for simplicity, we take:
E(w)
Var(w)
• Utility: depends on income w (good) and effort (bad):
1.
2.
● 3. On the other hand:
if b=1⇒e=
● From the PC, we get:
• Solve FOC:
• Now, consider the principal's problem:
. Moreover,
● By the way, what are the units of measurement of w - .58e²?
• We use the standard approximation of the expected utility function:
E[u(w-.58e²)]
e,
8
• We assume that R is constant (constant absolute risk aversion).
JE (π)
მხ
e
• This is the manager's effort supply function or the ICC.
• Note:
w = a + bπ = a + b(e+z)
=
=
u[E(w) — .5de² — .5RVar(w)]
u[(a + be — .5de²) — .5Rb²o²]
= u(q), where q = (a + be- .5de²) - .5Rb²²
3.2 -
Case 2
Unobserved Effort
• Since e is unobservable, we cannot contract on it or impose it.
● This means that we have to consider also the ICC (the manager's effort choice: his effort supply function),
not only the PC.
So, consider the manager's problem.
• Taking a, b as given, the manager's choice of e is given by the following problem:
max u[a + be — .5Rb²²
e
0
E(z)
Var(z) 0²
.5de²
U
b 6²
max{
¹8
b
if
π = e + z
=
16
0<b=
=
e,
max{-
b
8
U =
b
S
=
de
ab
де
მა
de
θα
• Given the effort supply function, the manager's utility is:
6²
u[a — .5Rb²o² +.5.
Rbo²
= u(w, e)
u(w - .5de²)
= cost of effort
.5Rb²o².56e²] ⇒ FOC
u[a+b=- — .5Rb³²0² — .58(²)²] = u[a — .5Rb²o² + .5%
-
62
max{E(e + 2) − a − b(e + z) : u[a − .5Rb³²o² + .5-—] ≥ uº°, e = } }
62
-
−
=
=
u'(.) (b-de) = 0
b
8
^
=
=
e*, the same as in the full information case
< 0
= 0
max{e(1-b) - a}
b
-6²
max{-(1 − b) — [wº +.5Rb²o² — .5—]}
b
S
if b=0 e=0
a + be
6²0²
0
a,b
a =
S
b>1⇒
b≤0⇒
(29)
• If we plug this solution for a and the solution for e from the ICC (e=) into expected profits, we are left with
b as the only variable we need to choose:
=
|
მხ
OR
Əb
до2
მხ
მა
w⁰ — .5Rb²o² + .5.
u(wº)
w⁰ - .5Rb²6² .5
(1
62
+.5Rb²0² - .5]
1
<1, for R>0, o² > 0, 6 > 0
1+ Rdo²
• Thus, the optimal contract balances incentive and insurance considerations.
• It is a compromise between full insurance (b=0) and full incentives (b= 1).
Finally, note that:
JE(T)
26
JE(T)
JE(™)
set
= 0
0 < b < 1
cannot have b≥ 1, b ≤0
მხ
6²
57]
– b) – Rbo²
< 0
62
< 0
<0 ⇒set b lower
>0 set b higher ⇒
0
FOC
(2)
(3)
(4)
(5)
(6)
max{E(e + z) − a − b(e + z) : u(.) ≥ u e =
− '/ }
(7)
. Hence, except for the value of u, the solution will be the same as above.
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
if R, o², or d are 0⇒b=1
● In the above problem, we assumed that the principal makes the contact decision (and extracts the full surplus
over and above the manager's opportunity cost).
● Alternatively, we can examine a case when the contract is determined in bargaining between the firm and the
manager.
(30)
• Since any bargaining solution must be Pareto efficient, it is clear that other than the level of the opportunity
cost, the rest of the solution will be the same.
● Namely, since any bargaining solution must be efficient, it can be obtained by loving the Pareto efficiency
problem for some arbitrary level of u:
(31)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F09238b20-1e79-43dc-b0a6-3d40e032e374%2F14be9d60-7a6a-44d6-aa8d-8d6c36d78132%2Fxvud16k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question:
=
Consider the following principal agent problem with moral hazard (effort is not
contractible). The firm's profits are given by: π = e+ z, where e is the manager's
effort and z is a random variable with E(2) = 0, Var(z) = 0². Assume that the
contract is given by the linear payment scheme: w = a + bí, where a is a fixed
salary and b is the share of profits. The agents's cost of supplying effort is c = .5e²
so his net income is y = wc. Let his utility, u, be given by the linear mean-
variance utility function, u = E(y) — .5R Var(y), where R > 0 is a parameter
(capturing risk aversion). Find the solution to this principal agent problem (the
optimal values of a, b and e).
Use below Methods to solve above question:
3 A Simple Example of a Principal-Agent Problem
• Profits:
where
● Payment: linear in profits:
with
● where, for simplicity, we take:
E(w)
Var(w)
• Utility: depends on income w (good) and effort (bad):
1.
2.
● 3. On the other hand:
if b=1⇒e=
● From the PC, we get:
• Solve FOC:
• Now, consider the principal's problem:
. Moreover,
● By the way, what are the units of measurement of w - .58e²?
• We use the standard approximation of the expected utility function:
E[u(w-.58e²)]
e,
8
• We assume that R is constant (constant absolute risk aversion).
JE (π)
მხ
e
• This is the manager's effort supply function or the ICC.
• Note:
w = a + bπ = a + b(e+z)
=
=
u[E(w) — .5de² — .5RVar(w)]
u[(a + be — .5de²) — .5Rb²o²]
= u(q), where q = (a + be- .5de²) - .5Rb²²
3.2 -
Case 2
Unobserved Effort
• Since e is unobservable, we cannot contract on it or impose it.
● This means that we have to consider also the ICC (the manager's effort choice: his effort supply function),
not only the PC.
So, consider the manager's problem.
• Taking a, b as given, the manager's choice of e is given by the following problem:
max u[a + be — .5Rb²²
e
0
E(z)
Var(z) 0²
.5de²
U
b 6²
max{
¹8
b
if
π = e + z
=
16
0<b=
=
e,
max{-
b
8
U =
b
S
=
de
ab
де
მა
de
θα
• Given the effort supply function, the manager's utility is:
6²
u[a — .5Rb²o² +.5.
Rbo²
= u(w, e)
u(w - .5de²)
= cost of effort
.5Rb²o².56e²] ⇒ FOC
u[a+b=- — .5Rb³²0² — .58(²)²] = u[a — .5Rb²o² + .5%
-
62
max{E(e + 2) − a − b(e + z) : u[a − .5Rb³²o² + .5-—] ≥ uº°, e = } }
62
-
−
=
=
u'(.) (b-de) = 0
b
8
^
=
=
e*, the same as in the full information case
< 0
= 0
max{e(1-b) - a}
b
-6²
max{-(1 − b) — [wº +.5Rb²o² — .5—]}
b
S
if b=0 e=0
a + be
6²0²
0
a,b
a =
S
b>1⇒
b≤0⇒
(29)
• If we plug this solution for a and the solution for e from the ICC (e=) into expected profits, we are left with
b as the only variable we need to choose:
=
|
მხ
OR
Əb
до2
მხ
მა
w⁰ — .5Rb²o² + .5.
u(wº)
w⁰ - .5Rb²6² .5
(1
62
+.5Rb²0² - .5]
1
<1, for R>0, o² > 0, 6 > 0
1+ Rdo²
• Thus, the optimal contract balances incentive and insurance considerations.
• It is a compromise between full insurance (b=0) and full incentives (b= 1).
Finally, note that:
JE(T)
26
JE(T)
JE(™)
set
= 0
0 < b < 1
cannot have b≥ 1, b ≤0
მხ
6²
57]
– b) – Rbo²
< 0
62
< 0
<0 ⇒set b lower
>0 set b higher ⇒
0
FOC
(2)
(3)
(4)
(5)
(6)
max{E(e + z) − a − b(e + z) : u(.) ≥ u e =
− '/ }
(7)
. Hence, except for the value of u, the solution will be the same as above.
(20)
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
if R, o², or d are 0⇒b=1
● In the above problem, we assumed that the principal makes the contact decision (and extracts the full surplus
over and above the manager's opportunity cost).
● Alternatively, we can examine a case when the contract is determined in bargaining between the firm and the
manager.
(30)
• Since any bargaining solution must be Pareto efficient, it is clear that other than the level of the opportunity
cost, the rest of the solution will be the same.
● Namely, since any bargaining solution must be efficient, it can be obtained by loving the Pareto efficiency
problem for some arbitrary level of u:
(31)
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