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:

??

??

??

??

?????

’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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