Attorney
Michael is an attorney
. Let denote his effort and his income.
He has two levels of effort:
and
. There are two levels of income:
and
.
The table below shows the
probability of each income being realized conditional on his effort:
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’s utility function is given by
.
a.
If
Michael
works
by himself
in a one
–
attorney
firm
,
which effort level will he choose? What will
be his expected utility?
b.
Suppose
Michael
forms a partnership with another
attorney
,
Justin
.
Justin
is identical to
Michael
with respect to everything but his risks are independent of
Michael
’s. The
two partn
ers
share equally the
total income of
the firm
. If
Justin
chooses
, what will be
Michael
’s expected
utility if he also chooses
?
(
Hint
: what will be Michael’s income prospect?)
c.
Show that if
Justin
chooses
,
Michael
’s expected utility is higher if he chooses
instead of
.
1
2.
A firm is organized as a 2
–
person partnership
. Each worker has a time endowment T = 24.
Time can
be spent either as leisure (x) or effort (e)
.
The firm’s profit function is
, where
denote
s
Partner
’s effort. Partner
’s
utility function is
given by
, where
denotes
his
income.
a.
Find
the
fa
ir and
efficient outcome.
State the values of the following variables for each partner
in this outcome:
leis
ure,
effort, income and utility.
Draw a
graph
to illustrate
this outcome.
b.
Find the partnership equilibrium.
State the values of the following variables for each partner in
equilibrium:
leis
ure, effort, income and utility.
Illustrate the
equilibrium
in the s
ame
graph
that
you drew in
(a).
c.
Suppose
the
firm has
3
instead of 2
partners
, with the profit function being
3
. A
ll
other information remains the same. The partners
each r
eceive
one third
of the
firm’s
total income. Assuming symmetry
, fin
d
the partnership equilibrium
.
State the values of each
partner’s
leis
ure, effort, income and utility in equilibrium.
Illustrate this equilibrium in the same
graph that you drew in (a).
1
In other words, if
one partner exerts the high
effort,
the other
will have an incentive to shirk.
So it is
not an equilibrium
for
both
to
choose the high effort.
2
3.
Go back
to the
firm
in Problem 2
with 2 workers. Suppose it
is organized as an owner and an
employee. Both workers still contribute to the produc
tion of the firm, but W
orker 1
is the owner
and W
orker 2 th
e employee. The owner offers the following contr
act to the
employee: “You must
exert
units of effort
, and
you will be paid
. If you choose any other level of effort you will be paid
nothing.” The owner incurs a cost of 12 for monitoring the employee’s effort. What will be the
value of
and
in the contract?
You can assume that half of the
monitoring
cost
is subtracte
d
from
each worker
’s income
.
Wha
t will be each person’s leisure, effort,
income
and utility?
(
Hint
:
See
Figure 4.3 on page
210 of the
text which
contains a graph showing how to incorporate the
monitoring cost. Study this graph and figure
out
the new per
–
capita profit function. Note that m is
the
per
–
capita
monitoring cost, so m = 6
here
.)
4.
Prof.
Green
assigns group projects to his students.
Since it is difficult for him to
determine the effort
that each student put in
, he assigns
a total score
to each group,
which is then divided equally among
the
group
members. For example, if a
group of 4 students
get
s
a
total score of 320, each
member
will
get
80. What are the students’ incentives
when they
work on
the
group projects?
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