Is It Always Optimal to “Sell the Firm” to a

Risk-Neutral Agent?

Christopher P. Adams∗

Federal Trade Commission

Email: cadams@ftc.gov

Web: http//www.together.net/∼cpadams

February 17, 2004

Abstract

The paper shows that the answer is no. Holmstrom (1979) and Shavell

(1979) show that the sell the firm contract does not achieve the first best when

the principal and the agent have different preferences over risk. This paper

shows that the sell the firm contract does not achieve the first best when the

principal and the agent have different preferences over time. In a dynamic

decision making problem under uncertainty, if the agent’s discount factor is

less than the principal’s, the agent will choose actions with relatively higher

current payoffs and relatively lower continuation payoffs than the principal

would prefer, even when the agent is sold the firm. When current and future

payoffs are correlated, the principal can do better by offering the agent a

contract that is even higher powered than the “sell the firm” contract. The

paper shows that the principal can align the agent’s incentives over time by

offering the agent stock options. At their exercise date stock options are a

liquid asset that pay the agent in the current period for the future value of

his actions.

∗The paper is based on Chapter 2 of my dissertation (Adams (2001)), and therefore I

would like to thank Larry Samuelson and John Kennan. I would also like to thank KimSau Chung, Gerald Garvey, Luke Froeb, Paul Oyer, Abe Wickelgren and two anonymous

reviewers for the FTC working paper series. Note that this paper does not necessarily

represent the views of the Commission or any individual Commissioners. All errors are

my own.

1
1 Introduction

Obviously, this paper will claim that the answer is no. It is not always optimal

for a risk-neutral principal to “sell the firm” to a risk-neutral agent. A classic

result of principal-agent theory is that it is always optimal for a risk-neutral

principal to offer a contract in which the agent pays the expected value of his

actions and becomes the residual claimant on the outcome of his actions. In

the vernacular of the literature this type of contract is as “high powered” as a

contract can get (Kreps (1990)). Holmstrom (1979) and Shavell (1979) show

that if the agent is more risk averse than the principal then the first best

cannot be achieved and the sell the firm contract is not optimal. This paper

provides an example of a situation in which the agent is risk-neutral but it

is still not optimal to give the agent a sell the firm contract. This paper will

present a simple dynamic environment in which it is not optimal for the riskneutral principal to offer such a contract to the risk-neutral agent. Moreover,

the paper shows that when current payoffs are correlated with future payoffs

it is more efficient to offer a contract in which the power of the incentives is

greater than in the “sell the firm” contract. Finally, the paper suggests that

stock options could be used to align the agent’s preferences over time with

the time preferences of the principal. The value of stock options is that they

are liquid at their exercise date. If the stock price reflects the future value of

the firm, stock options can pay the agent the future value of current choices

at the option’s exercise date. This seems to be the way practitioners view

the value of stock options (NCEO (2003)), the use of which is something of

a puzzle (Hall and Murphy (2003)).

Consider the following illustrative example.1 A senior executive of an oil

company must regularly choose how much money to budget to oil exploration

and drilling and how much money to budget to marketing related activities.

In general assume that the executive prefers that more money be spent on

marketing related activities.2 Both expenditures can increase the firm’s returns, however expenditure on exploration and drilling can also increase the

1Thank you to an anonymous reviewer for suggesting this example.

2Consider former RJR Nabisco executive F. Ross Johnson in Barbarians at the Gate

(2003). Jensen (2001) has a detailed discussion of incentive problems inherent in budgeting.

1
firm’s information about the state of the company’s oil reserves. Obviously

finding oil will increase the company’s current profits and it will lead to

new decisions in the future that increase future expected profits. However,

it is also the case that not finding oil increases the firm’s information and

while it doesn’t increase the firm’s current profits, having better information

about the company’s oil reserves does lead to new decisions in the future that

increase future expected profits. Given the executive’s private information

about the likelihood of finding oil (geological surveys and a like), the executive must choose between spending on oil exploration and drilling or spending

on marketing. There is a moral hazard problem when the senior executive

cares less about the future expected profits of the firm than the representative

shareholder. The problem is that there may be cases where given the same

private information the senior executive will choose to decrease expenditure

on exploration and drilling, while the representative shareholder would have

preferred that the executive increase this expenditure. A contract that “sells

the firm” the executive is not optimal because the senior executive cares less

about the future and thus the executive will still not make decisions that the

shareholder considers optimal. Stock options may help to solve the problem

if the stock market is able to recognize the value of the information discovered

in exploration and drilling operations and incorporate that information into

the stock price. The value that stock options may have over stock is that at

their exercise date they may be more liquid than normal stock allowing the

executive to reap the benefits of her decision to “invest” in the information

discovered during in oil exploration and drilling.

The paper presents a principal-agent problem in which the contract is negotiated ex ante to the agent receiving his private information. The agent’s

decision making problem is similar to the monopoly pricing problem described by Rustichini and Wolinsky (1995). The paper takes an insight of

Rustichini and Wolinsky (1995) and shows that the agent will choose the

more costly action more often when his discount factor is higher. The reason

is that given a set of beliefs about the current state of the world the more

costly action leads to a higher future expected payoff. If the principal cares

more about the future than the agent, the principal may want to give the

agent incentives to choose the more costly action. In the example presented,

2
the “good” outcome in the first period leads to higher returns in the second

period. However the action which increases the likelihood of the good outcome has higher current costs. Therefore by paying the agent more today

if the good outcome occurs, the agent will choose the correct action today.

Increasing pay for the good outcome increases the power of the incentive

contract beyond that of the “sell the firm” contract. Similarly, the paper

shows that a contract that models the way stock options work can be used

to provide the first best incentives to the agent. It is shown that stock options can solve the problem by paying the agent for the future value of his

current choices in the current period.

The result occurs because current and future payoffs are positively correlated and so the fact that the agent doesn’t care about future payoffs can be

offset by increasing the importance of current payoffs. This is only a simple

example, but it may be a good model of situations where costly current actions may lead to high current payoffs and high future payoffs. For example,

investing in winning a drug development race to be first to market. If initial

market success leads to long term success the Board of Directors may want

the CEO earn higher returns for the initial successes. Another example may

be winning a “standards” race in a technology market with network externalities. Or investing to become the internet auctioneer of choice, as eBay

did (Cohen (2002)). It is interesting to note the importance of stock options

in sectors of the economy where such races are important (NCEO (2003);

Lazear (2003)).

A standard result of the principal-agent literature is that if the agent is

risk-neutral the first best can be achieved by using a simple “sell the firm”

contract (Kreps (1990)). Holmstrom (1979) and Shavell (1979) show that the

sell the firm contract can be improved upon when the principal and the agent

have different risk preferences. This paper similarly shows that the sell the

firm contract can be improved upon when the principal and the agent have

different time preferences. A number of papers have considered the principal

agent problem in a dynamic setting, most notably Holmstrom and Milgrom

(1987). The important characteristic of the dynamic model presented here

is that the agent’s time preferences affect his choices, and more specifically,

they distort his actions away from those that the principal would prefer.

3
In this paper it is assumed that the agent has private information and that

the contract is negotiated ex ante to this information being revealed.3 This

assumption follows, for example Holmstrom (1979), Adams (2002), Baker

and Jorgensen (2003) and Raith (2004), all of which consider the case in a

static setting. Adams (2002) and Raith (2004) argue that this assumption is

what distinguishes principal-agent models where the agent makes decisions to

models where the agent does not. A decision maker is an agent that observes

private information and makes his choice contingent upon that information.

To the author’s knowledge this paper is first to consider the problem when the

agent faces a dynamic decision making problem under uncertainty. That is,

the paper analyzes a situation where the agent’s private information changes

from period to period and is affected by the action choices of the agent. The

model is based on the model analyzed by Rustichini and Wolinsky (1995),

and is but one example of the models analyzed in the literature on learning

and dynamic decision making under uncertainty (Keller and Rady (1999);

Mirman et al. (1993); Rothschild (1974)). This paper brings some of the

insights of this literature to the principal-agent problem. The results suggest

that the dynamic aspects of the agent’s choices have important implications

for contract theory if the principal and the agent have different preferences

over time.

In recent years stock options have become quite popular, particularly in

the high tech sector (NCEO (2003); Hall and Murphy (2003)). However,

the use of such schemes represents something of a puzzle. Hall and Murphy

(2003) discuss a number of explanations for why stock options are used, these

can be categorized as follows; tax, accounting, borrowing, retention, and incentives. Stock options are taxed differently than normal stock grants, and,

as many now realize, they have traditionally been accounted for differently

in the company reports. The authors argue that even though the market can

efficiently account for these reporting differences, managers seem to ‘falsely

perceive stock options to be inexpensive.’ According to Lazear (2003) stock

options do not seem to be a good way to finance a risky project as it would

be cheaper for the firm to borrow from venture capitalists. However, stock

options may be fine if employees have beliefs that are more optimistic than

3Note that all the analysis is in regards to the ex ante expected value of the contract.

4
the VCs (Hall and Murphy (2003)). Oyer (2003) argues that stock options

may be a useful retention device because their value is correlated with the

outside option of the employees, and so such a scheme would save on renegotiation costs. In regards to the incentive effects of options, the use of which

is generally analyzed using standard static principal-agent model, the literature has found them to be a very costly method of providing incentives (Oyer

and Schaefer (2003)). This paper presents a dynamic model and shows that

stock options may be preferred to restricted stock grants because they can

be immediately sold on the market. If the stock price provides a good signal of the future value of the employee’s current actions, then stock options

may be an efficient method of providing the appropriate incentives. According to NCEO (2003) the most important difference between stock options

and ordinary stock is that the stock is sold immediately the option is exercised. Options allow the employee to “cash out” during a specific period of

time, allowing employees to share in the “future growth of the firm” (NCEO

(2003)).

The rest of the paper proceeds as follows. Section 2 presents a twoperiod model and shows that a “sell the firm” contract is not optimal. The

section shows how stock options can be used to achieve the first best outcome.

Section 3 presents a infinite-period model. Section 4 concludes.

2 Two-Period Model

This section presents a simple two-period model and gives an example of a

situation in which it is not optimal to “sell the firm” to the agent. Proposition

1 shows that if the agent is less patient than the principal, it is not optimal

for the principal to “sell the firm”. Proposition 2 shows that the first best

contract will be higher powered than “sell the firm” contract. The results

follow in a straightforward manner from the realization that under some set

of beliefs the value of choosing the costly action in the first period is that it

leads to a higher expected payoff in the second period. The agent’s valuation

of the second period is lower than the principal’s so there are a set of beliefs

where the agent does not choose the costly action, even when the principal

prefers that he did. The principal can solve the problem by paying the

5
agent more if the “good” outcome occurs (less if the “bad” outcome). Note

that the agent is risk-neutral so fixed amounts can be varied to achieve the

optimal allocation between the principal and the agent over the two periods.

Proposition 3 shows that a practical way to implement the first best contract

is to offer the agent an option contract. This contract pays the agent a large

positive amount if the good outcome occurs when the market “believes” that

the agent chooses the costly task.

2.1 The Model

Consider a situation in which the principal and the agent are risk-neutral.

The agent faces a two-period problem, t ∈ {1, 2}. The timing is as follows.

• Period 0.

– The principal and the agent have a common ex ante distribution

f(w1) on the agent’s period 1 belief, w1, of the state of the world

s1 ∈ {0, 1}, where w1 = P r(s1 = 1).

– The principal makes a take-it-or-leave-it offer, {π0t

, π1t}t∈{1,2}. The

agent’s outside option is 0. The agent’s discount factor is δ ∈ [0, 1)

and the principal’s discount factor is 1.

• Period 1.

– The agent chooses an action a1 ∈ {0, 1}, such that the agent bears

the private cost c > 0 of choosing at = 1 and no cost of choosing

a1 = 0.

– The outcome y1 ∈ {0, 1} is observed where

y1 =







1 w/ prob 1 if a1 = 1 and s1 = 1

1 w/ prob 0 if a1 = 1 and s1 = 0

1 w/ prob p if a1 = 0

(1)

where p ∈ (0, 1).

– The agent is paid π11 if y1 = 1 and π01 if y1 = 0.

6
• Period 2.

– The agent chooses a2 given his updated belief (w2) about the state

of the world, s2.

w2 =







α if a1 = 1 and y1 = 0

1 − α if a1 = 1 and y1 = 1

(1 − α)w1 + α(1 − w1) if a1 = 0

(2)

where α ∈ (0, .5).

– The agent is paid π12 if y2 = 1 and π02 if y2 = 0.

To give this simple two-period model some of the flavor of the infinite

period model analyzed in the next section, it is assumed that the principal

and the agent have a common ex ante belief regarding the distribution of

the agent’s first period belief (signal) of the state of the world (f(w1)). The

principal is assumed to choose an incentive contract that maximizers her ex

ante expected profits. This assumption is implicit in the analysis that follows.

Note that the agent’s individual rationality constraint is also assumed to be

ex ante. The agent’s incentive compatibility constraint is defined as the

optimal choices a1 and a2 given beliefs w1 and w2 and payoffs c, π01, π11, π02,

and π12.

The important and interesting feature of this model is that the agent’s

belief (w2) is a function of the agent’s action in the previous period (a1). By

choosing a1 = 1 the agent learns the state s1 after observing the outcome y1.

If a1 = 0, no information is revealed about the state. If the agent chooses the

low cost action (a1 = 0), the agent’s belief at the beginning of period 2 will be

w2 = (1 − α)w1 + α(1 − w1). While no additional information is learned, the

agent’s beliefs must adjust to account for the probability that the state of the

world has changed, this probability is denoted, α.

4

If the agent chooses the

high cost action then the agent “learns” the state by observing the outcome

y1. If the good outcome occurs (y1 = 1) then it is known that the state in

period 1 was s1 = 1. Therefore, the agent’s belief in period 2 is w2 = 1 − α,

where the probability that the state of the world is still good (s2 = 1) is

4Below it is assumed that α is “small”.

7
decreased by α to account for the probability that the state of the world

changes between the two periods. If the bad outcome (y1 = 0) is observed

the agent’s belief in period 2 is w2 = α. The value of learning the state in

period 1 is that it may increase the expected value of the choice in period 2.

It does so because the principal is better off if the high cost action is chosen

only in the good state of the world. It is this “investment” in information

that provides the link between today’s choices and future payoffs. Thus, it

is this investment that is subject to the vagaries of the agent’s preferences.

In the first best case, the firm would be solving the following problem. In

Period 2, the problem is

maxa2∈{0,1} (1 − a2)p + a2(w2 − c) (3)

If the agent chooses a2 = 0, the expected return is p and there is no cost to

the agent of choosing the action. If the agent chooses a2 = 1, the expected

return is w2, the agent’s belief about the state in period 2, minus the private

cost of choosing the action (c). Let WP

2

characterize the agent’s first best

strategy, where P denotes “principal”, such that if w2 ≥ WP

2

, then a2 = 1

and if w2 < WP

2

, a2 = 0. From Equation (3) we have that WP

2 = p + c. The

following three assumptions guarantee the interesting case, where the action

choice in period 1 (a1) affects the agent’s information and his optimal action

choice (a2) in period 2.

Assumption 1 The Interesting Case:

1. Let 1 − c > p

2. Let w1 < p + c

3. Let α be small.

In this case, we see from Equation (2) that the second period’s action is

a function of the first period’s action and the observed outcome y1. Given

Assumption (1), in the first best case the high cost action is chosen if the

following inequality holds.

w1 − c + w1(1 − α − c) + (1 − w1)p ≥ 2p (4)

8
If the low cost action (a1 = 0) is chosen in the first period nothing is learned

about the state of the world and under Assumption 1, the optimal choice in

period 2 is the low cost action (a2 = 0) again. The expected payoff from the

low action in period 1 is 2p. However, if the high cost action is chosen there

is some possibility that the good state occurs (s1 = 1) and there will be a

high payoff (y1 = 1) and the beliefs will be such (given Assumption 1) that

it will be optimal to choose the high cost action (a2 = 1) in period 2. The

expected payoff in this case is 1−α−c where 1−α is the belief that the state

is still good (s2 = 1). The other possibility is that the state is bad and the

bad outcome occurs (y1 = 0). Given Assumption 1, the best choice is the low

cost action in period 2 (a2 = 0). The expected payoff in this case is p. Given

ex ante beliefs (f(w1)) and Assumption 1, the expected first best profits are

maximized when the low cost action is chosen for the agent’s belief below a

cutoff WP

1

and the high cost action is chosen for the agent’s belief above the

cutoff, where WP

1

is such that Equation (4) holds with equality.

2.2 Results

Let W1 characterize the agent’s optimal strategy given the contract, where

a

∗

1 = 1 if w1 ≥ W1 and a

∗

1 = 0 if w1 > W1. Proposition 1 is the main result

of the paper. It states that if the agent’s discount factor is less than 1, the

cutoff belief W1 is greater than the first best cutoff belief. This means that

there is a set of beliefs w1 ∈ (WP

1

, W1), such that the agent chooses the low

cost action but ex ante the principal would prefer he choose the high cost

action.

Proposition 1 Given Assumption 1, if πt1−πt0 = 1, δ < 1, then W1 > WP

1

.

Proof. From Equation (4),

WP

1 =

p + c

2 − α − c − p

(5)

For the agent, the equivalent inequality is

w1 − c + δ(w1(1 − α − c) + (1 − w1)p) ≥ p + δp (6)

9
and so

W1 =

p + c

1 + δ(1 − α − c − p)

(7)

If δ = 1 then W1 = WP

1

. We have the result if ∂W1

∂δ < 0. The derivative is

negative if 1 − c > p and α is small, which they are by Assumption 1. QED.

The proposition shows that in this example the first best cannot be

achieved with the “sell the firm” contract if δ is less than 1. The intuition

is the same as for the similar result in the infinite-period problem shown

below. The sell the firm contract is represented by the difference between

the payoffs (π1t − π0t = 1). The agent prefers to choose the low cost action

“more often” than the principal would like because for some set of beliefs

the action has a greater current period payoff and a smaller second period

payoff. Given the “sell the firm” contract, the agent doesn’t fully account for

the extra “investment” value of choosing the high cost action. In expectation

this extra amount is 1 − α − c − p. As long as this is positive (which it is

by assumption) and δ < 1 the agent will choose a1 = 0 more often than the

principal would like. Proposition 2 states that the first best can be achieved

with a contract that is higher powered than the “sell the firm” contract.

Proposition 2 Given Assumption 1, if p is small, the first best can be

achieved if π11 − π01 > 1.

Proof. Let π11 − π01 = D and π12 − π02 = 1, and so the agent’s problem

is

w1D − c + δ(w1(1 − α − c) + (1 − w1)p) ≥ pD + δp (8)

and so

WD

1 =

pD + c

D + δ(1 − α − c − p)

(9)

We have the result if ∂WD

1

∂D < 0. The derivative has the same sign as

p(D + δ(1 − α − c − p)) − pD − c = pδ(1 − α − c − p) − c (10)

which is negative if p is small enough. QED.

10
The intuition is that the principal has the possibility of earning an extra

1 − α − c if the agent chooses the high cost action in the first period and

the outcome is y1 = 1. However the agent does not value this possibility as

highly as the principal. The principal can replicate the first best incentives

by increasing the relative value of choosing the high cost action in the first

period, in this model this is done by putting a relatively higher payoff on the

outcome y1 = 1 (lower payoff on the outcome y1 = 0). This works because

the payoffs are positively correlated (α < .5). This suggests that one solution

to the incentive problem is to use the time series correlation in the payment

instruments (the signals). Stock options may be a particular way to do this.5

2.3 An Option Contract

An option contract pays the agent the market’s expected value of the firm in

some future period less the strike price of the option, with a minimum value

of zero. By paying the agent the expected value of the future value of the

firm, the principal can solve the problem that the agent’s time preferences

distort their choices to actions that weight present returns greater than the

principal would like. It is assumed that the stock bought in the option is

immediately resold to the market when the option is executed.6

Consider a two-period problem similar to the problem described above.

In period 2, the contract pays π12 − π02 = 1. In the first period the contract

has two parts. First π

p

11 −π

p

01 = 1, this is the profit sharing part. The option

part of the contract is, π

o

.

7

π

o =

(

r(1 − α − c) − rp) if w1 > WO

1

and y1 = 1.

rp − rp otherwise (11)

5Note that stock options may also work when there is no positive time series correlation

if the stock market is able to observe other (more direct) signals of the agent’s actions and

information.

6This seems to be a reasonable characterization of what actually happens (Hall and

Murphy (2003); NCEO (2003)). Note that what distinguishes stock options from normal

stock is this characteristic that the stock is immediately sold when the option is exercised.

7The stock option gives the profit sharing contract an extra boost, and is thus a way

of implementing the first best contract described above.

11
The option pays the agent the principal’s “belief” about the value of the firm

less a discount factor r and the strike price, which is assumed to be rp. The

cutoff belief Wo

1

is defined below (Equation (13)). The option is exercised

after the action is chosen and the outcome is observed. The principal’s belief

about the value of the firm is consistent with the agent’s equilibrium action

if, given this contract, the agent chooses a1 = 1 when w1 > WO

1

. Note that

the agent may choose a1 = 0 and be paid r(1 − α − c − p) if y1 = 1. That

is, the agent can cheat and choose the low cost action. If the good outcome

(y1 = 1) occurs the principal will still believe that the high cost action was

chosen and that the good state (s1 = 1) has been revealed. Therefore we have

an equilibrium if the following inequality holds (the incentive compatibility

constraint).

w1−c+δ(w1((1−α−c)+(1−w1)p)+w1r(1−α−c−p)) ≥ p+δ(p+pr(1−α−c−p))

(12)

The cutoff value, WO

1

, is the w1 such that this equation holds with equality.

The following proposition shows that there exists an r such that WO

1 = WP1

.

That is, the first best can be achieved with an option contract of this form.

Proposition 3 For small p, there ∃r such that WO

1 = WP1

.

Proof. From Equation (12) we have

WO =

p + pr(1 − α − c − p) + c

1 + δ(1 − α − c − p) + r(1 − α − c − p)

(13)

From Proposition 1, if r = 0, WO

1 > WP1

. We have the result if ∂WO

1

∂r is

negative. For small p this follows from Equation (13). QED.

The option contract pays the agent today as a function of the firm’s future

expected value. However, the option only pays out a positive amount if the

good outcome occurs. It is not optimal to sell the firm to the agent, but the

first best can be achieved by paying the agent more for the choice of a1 = 1

in the first period. As shown above, the option contract is a practical way to

do that. The option pays the agent today the discounted present extra value

of choosing the costly action, r(1 − α − c − p).

12
3 Infinite-Period Model

This section presents a more general model using results from Rustichini and

Wolinsky (1995). Much of the intuition from the two-period model carries

over to the infinite-period model. Although this model is more complicated

than the model presented above, the results are more general than those

for the particular case analyzed in Section 2. Proposition 4 states that as

the agent’s discount factor increases, the agent is more likely to choose the

costly action (at = 1). The proposition also states that there is a positive

relationship between the power of the incentives and the number of times the

costly action is chosen in the optimal strategy. Corollary 1 shows that the

first best can be achieved with a contract that is higher powered than a “sell

the firm” contract.

3.1 The Model

Consider a situation in which the principal and the agent are risk-neutral.

The agent faces a discrete but infinite period problem, t ∈ {1, 2, ...}. The

timing is as follows.

• Period 0.

– The principal and the agent have a common ex ante belief, w0, of

the state of the world s1 ∈ {0, 1}, where w0 = P r(s1 = 1).

– The principal makes a take-it-or-leave-it offer, {π0t, π1t}t∈{1,2,...}.

The agent’s outside option is 0. For simplicity it is assumed that

the both the agent and the principal negotiate the contract over

the expected value of the stationary distribution generated by the

actions of the agent.8

• Period 1.

– The state of the world follows a simple Markov process.

Pr(st+1 = 1) = α(1 − st) + (1 − α)st (14)

8Given that the principal can offer contracts that pay the agent more in earlier periods,

this assumption does not seem to matter.

13
where α ∈ (0, .5).

– At the beginning of each period t, the agent has a belief wt ∈ [0, 1]

about the state of the world st ∈ {0, 1}, such that wt = Pr(st =

1).9

– The agent chooses an action at ∈ {0, 1}, such that the agent bears

the private cost c > 0 of choosing at = 1 and no cost of choosing

at = 0.

– The outcome yt ∈ {0, 1} is observed where

yt =







1 w/ prob 1 if at = 1 and st = 1

1 w/ prob 0 if at = 1 and st = 0

1 w/ prob p if at = 0

(15)

where p ∈ (0, 1).

– The agent is paid π1tif yt = 1 and π0tif yt = 0.

• Period 2.

– The agent chooses a2 given his updated belief (w2) about the state

of the world, s2.

wt+1 =







α if at = 1 and yt = 0

1 − α if at = 1 and yt = 1

(1 − α)wt + α(1 − wt) if at = 0

(16)

where wt ∈ B.

10

• Period t ∈ {3, 4, ...}

– Follow in the same manner as 1 and 2.

9Unfortunately some clarity is lost because there are two different states, one is the

state of the world st (in Rustichini and Wolinsky (1995) this is demand (dt)), and the

other is the state of the Markov decision process which is denoted σ.

10Note that B is an infinite and countable set given w0. To see this let a1 = 1 and

at = 0 for all t ∈ {2, 3, ...}.

14
– At each period t the agent has a belief wt and the agent chooses

a

∗

t

such that

a

∗

t

(wt) = arg maxat∈{0,1}(1 − at)((1 − p)π0 + pπ1)

+at((1 − wt)π0 + wtπ1 − c) + δ

P

{wt+1∈B} q(wt+1|wt

, at)vt+1(wt+1)

(17)

where δ ∈ (0, 1) is the agent’s discount factor, q(wt+1|wt, at) is the

transition probability, and v(wt+1) is the expected value of future

choices given wt+1.

Period 0 represents the agent’s individual rationality constraint. It is

assumed that the agent chooses to accept or reject the contract based on

the expected value of the stationary distribution (or the “long run expected

value”). This assumption is made for simplicity and it also represents the

idea that when employees negotiate an employment contract they consider a

longer horizon than when they are making day-to-day decisions. The agent’s

incentive compatibility constraint is discussed below. Note further that in

this case the ex ante beliefs and the principal and the agent are not relevant.

The agent’s belief (wt+1) is a function of the agent’s action in the previous

period (at). By choosing at = 1 the agent learns the state st after observing

the outcome yt. If yt = 1 is observed then the agent knows that the state

of the world is st = 1 and thus his belief about the state of the world in

then next period is wt+1 = 1 − α, where α is the probability that the state

changes from period to period. Similarly if yt = 0 is observed the agent’s

belief in period t + 1 is wt+1 = α. If at = 0, no information is revealed

about the state. Note however that the agent’s belief changes to account

for the probability that the state of the world changes. Therefore his belief

is wt+1 = (1 − α)wt + α(1 − wt). The value of knowing the state is that it

increases the expected value of the choice in period t+1. While the expected

value of choosing the low cost action (at = 0) doesn’t change with the state

of the world (it always equals p), the expected value choosing the high cost

action is dependent on the state of the world. The high cost action pays

more than p when the state of the world is st = 1 and less than p in the

other state of the world.

15
It is straight forward to see that if δ = 0 and π1t > π0t

, that we have

a

∗

t = 1 if wt ≥ W M and a

∗

t = 0 if wt < W M, where W M is the agent’s myopic

cutoff belief

W M =

p(π1t − π0t) + c

π1t − π0t

(18)

We can also see that a

∗

t = 1 is chosen “more often” when π1t −π0t

is larger, p

is smaller and c is smaller. By increasing the difference π1t−π0t the expected

payoff from choosing at = 1 increases. If p increases the expected payoff

from choosing at = 0 increases and if c increases the cost of choosing at = 1

increases relative to the expected value of choosing at = 0. That is, a myopic

agent would react to incentives exactly as we would expect.

Lemma 1 If π1 ≥ π0, then the agent’s optimal strategy exists and is characterized by a cutoff belief, W ∈ [0, 1], such that if wt ≤ W, a

∗

t = 0, and if

wt > W, a

∗

t = 1.

Lemma 1 shows that for all δ there exist a W such that a

∗

t = 1 if wt ≥ W

and a

∗

t = 0 otherwise. While Lemma 1 is similar to Rustichini and Wolinsky

(1995) Claim 1, the proof is written out in full in Adams (2001) because B is

infinite and countable and therefore it is not possible to appeal to results in

Derman (1970). Rather, the proof is based on results presented in Puterman

(1994). Rustichini and Wolinsky (1995) note that given Lemma 1, the agent’s

strategy can be summarized by the variable, N. The smaller N is, the greater

the private cost born by the agent.

Definition 1 (Rustichini and Wolinsky (1995)) N is the smallest non-negative

integer if wt = 1 − α and at = ... = at+N−1 = 0, then wN+t < W. N = ∞ if

there is no such integer.

By this definition, N is the number of times the agent chooses at = 0

after yt = 0 when at = 1. That is, N is the number of times the low cost

action is chosen after the agent receives a signal that the state of the world

is bad. If N is large, a

∗

t = 1 is chosen less often (W is high), while if N is

small, a

∗

t = 1 is chosen more often (W is low). Therefore at anytime t (after

the initial couple of periods), the agent’s incentive compatibility constraint

is characterized by N.

16
As stated above it is assumed that the individual rationality constraint

is defined over the expected value of the stationary distribution given N.

Following Rustichini and Wolinsky (1995) it is straightforward to show that

the total surplus of the firm is

Surplus =

Nαp + 1 − ψN

(N + 1)α + 1 − ψN

(19)

where ψN =

1+(1−2α)N+1

2

. Assuming that the agent’s outside option in terms

of the expected value of the stationary distribution is 0, the first best contract

would be one in which the principal maximizes Equation (19).

The following lemma states that as δ gets near to 1, the optimal solution

to the agent’s problem is equivalent to the N that optimizes the expected

value of the stationary distribution (Equation (19)).

Lemma 2 Let qit be the marginal probability distribution at time, t, for

Markov chain, i ∈ I, qiis the stationary marginal probability distribution

for Markov chain, i, and Rσ is the per period payoff in state σ. Then

arg max

i∈I

lim

δ→1

X∞

t=1

δ

t

ÃX

σ∈Σ

qti(σ)Rσ

!

= arg max

i∈I

X

σ∈Σ

qi(σ)Rσ (20)

Proof.

arg maxi∈I limδ→1

P∞

t=1 δ

t P

σ∈Σ qti(σ)Rσ

= arg maxi∈I limδ→1

P∞

t=1

1

T

δ

t P

σ∈Σ qti(σ)Rσ, when multiplied by a constant

= arg maxi∈I limδ→1 limT→∞ PT

t=1

1

T

δ

t P

σ∈Σ qti(σ)Rσ

= arg maxi∈I limT→∞ PT

t=1

1

T

limδ→1 δ

t P

σ∈Σ qti(σ)Rσ

reversing the order of the limits

= arg maxi∈I limT→∞ PT

t=1

1

T

P

σ∈Σ qti(σ)Rσ, when δ → 1

= arg maxi∈I

P

σ∈Σ(limT→∞ PT

t=1

1

T

qti(σ))Rσ

(21)

The Markov process has a stationary distribution (Rustichini and Wolinsky

(1995)), and so

= arg max

i∈I

X

σ∈Σ

( limT→∞

X

T

t=1

1

T

X

σ0∈Σ

q1i(σ

0

)Q

t−1

(σ, σ0))Rσ (22)

17
where Q(σ, σ0

) is the probability of being state σ in period t + 1 conditional

upon being in σ

0

in period t

= arg maxi∈I

P

σ∈Σ(

P

σ0∈Σ q1i(σ

0

)(limT→∞ PT

t=1

1

T Qt−1

(σ, σ0)))Rσ

= arg maxi∈I

P

σ∈Σ(

P

σ0∈Σ q1i(σ

0

)qi(σ))Rσ

(23)

(Hoel et al. (1972), p. 72)

= arg max

i∈I

X

σ∈Σ

qi(σ)Rσ (24)

QED.

From Lemma 2 we can see that if δ is close to 1 and π1t − π0t = 1 the

agent will choose the N that maximizes Equation (19). Therefore in this

case the “sell the firm” contract achieves the first best. The next section

discusses what happens when δ is not close to 1.

3.2 Results

This section shows that there exists a contract that is more efficient than the

“sell the firm” contract. Proposition 4 presents an insight from Rustichini

and Wolinsky (1995) that N decreases in δ. That is, as the discount rate falls

the agent is “less likely” to choose the high cost action. Corollary 1 shows

that the “sell the firm” contract is not optimal.

Following Rustichini and Wolinsky (1995), define Z as the expected value

to the agent of his optimal strategy given that yt = 0 and at = 1. Define Y as

the expected value given yt = 1 and at = 1. Further, let π1t = π0t + D. Note

that the agent is paid at least π0tin every period so its value is irrelevant for

the maximization problem. Note further that for the case when p =

1

D

and

c = 0, the problem is as presented in Rustichini and Wolinsky (1995).

Z =

δ − δ

N+1(pD)

1 − δ

+ δ

N+1(ψN Z + (1 − ψN )(D + Y ) − c) (25)

and

Y = δ((1 − α)(D + Y ) + αZ − c) (26)

18
Solving out for Y .

Y =

δ((1 − α)D + αZ − c)

1 − δ(1 − α)

(27)

Substituting in, we have an equation that can be solve to give Z.

Z =

δ−δN+1(pD)

1−δ

+δ

N+1 3

ψN Z + (1 − ψN )

3

D +

δ((1−α)D+αZ−c)

1−δ(1−α)

 ́

− c

 ́ (28)

Define the implicit function, f.

f = −Z +

δ−δN+1(pD)

1−δ

+δ

N+1 3

ψN Z + (1 − ψN )

3

D +

δ((1−α)D+αZ−c)

1−δ(1−α)

 ́

− c

 ́ (29)

Using the implicit function theorem, the optimal solution for N is the N such

that ∂f

∂N = 0, and define f

0 as the implicit function of the solution.

f

0 = −

log(δ)δN+1(pD)

1−δ

+ log(δ)δ

N+1 3

ψN Z + (1 − ψN )

3

D +

δ((1−α)D+αZ−c)

1−δ(1−α)

 ́

− c

 ́

+.5δ

N+1 log(1 − 2α)(1 − 2α)N+1 3

Z − D −

δ((1−α)D+αZ−c)

1−δ(1−α)

 ́

(30)

The following proposition which is similar to Rustichini and Wolinsky (1995)

Claim 3, characterizes the agent’s strategy given parameters of the contract

and the model.11

Proposition 4 (i) For sufficiently small α and sufficiently large δ, N decreases in D. (ii) For sufficiently small α and sufficiently small p, N decreases in δ.

Proof. (i) We have the result if ∂f0

∂N and ∂f0∂D are the same sign. Let α = 0,

then f

0

is

f

0 = −

log(δ)δ

N+1(pD)

1 − δ

+ log(δ)δ

N+1(Z − c) (31)

Therefore

∂f0

∂N = −

log2(δ)δ

N+1(pD)

1 − δ

+ log2(δ)δ

N+1(Z − c) (32)

11The proposition is not the same is Rustichini and Wolinsky (1995) Claim 3 because

of differences in the model specification, but the proof follows the same logic used by the

authors.

19
and

∂f0

∂D = −

log(δ)δ

N+1(p)

1 − δ

(33)

We can see that for large enough δ both equations are negative and we have

the result. (ii) In this case we have the result if ∂f0

∂N and ∂f0∂δ are the same

sign.

∂f0

∂δ =

3

−

pD

1−δ + Z − c

 ́

(N + 2)δ

N +

log(δ)δN+1pD

(1−δ)

2 (34)

We see that if p is sufficiently small, then both Equation (32) and Equation

(34) are positive. QED.

The proposition characterizes the agent’s optimal strategy. It shows that

as the power of the incentive increases, N decreases, meaning that the agent

is going to choose the more costly action more often. Part (ii) shows that N

decreases as the discount factor rises. This result has the same intuition as

Proposition 1 in the two-period problem. When the agent cares more about

the future, he will choose the costly action more often because for certain

beliefs, it increases expected payoff in future periods.

The following corollary shows that the principal will get higher profits

with π1t − π0t > 1 than she would get if π1t − π0t = 1.

Corollary 1 If α and p are sufficiently small, and δ is sufficiently large, the

first best contract is such that π1t − π0t > 1.

Proof Let the contract be such that π1t − π0t = 1, and the agent optimal strategy of the agent be characterized by N1. Given the agent’s outside

option is 0, the principal receives the expected value of the stationary distribution given N1. Let N2 be the N that optimizes the expected value of

the stationary distribution. Given π1t − π0t = 1, by Lemma 2 N2 is also the

optimal choice for the agent as δ gets close to 1. By Part (ii) of Proposition

4, N2 < N1. By Part (i) of Proposition 4 for N2 to be implemented the

difference between π1t − π0t must be increased. QED.

The proof of the corollary is based on the insight of Rustichini and Wolinsky (1995) that the N describing the optimal strategy of the agent is decreasing in the discount factor of the agent. The higher the discount factor, the

20
greater the value to the agent of choosing the costly action at = 1. Although

the agent bears higher current cost, the action increases the agent’s information and thus the expected value of future choices. Given that the agent’s

discount factor is less than 1, under the “sell the firm” contract the agent

does not reap the full benefits of choosing the costly action and “investing”

in his information. The principal can achieve the first best by increasing the

power of the incentive. In particular, by paying the agent more for the good

outcome today the principal can align the agent’s incentives to account for

the higher future expected payoff. This result occurs because current and

future payoffs are positively correlated, so the principal can replicate the first

best be increasing the importance of current payoffs in the agent’s decision

making.

4 Conclusion

A standard result of the principal-agent literature is that it is optimal to sell

the firm to a risk-neutral agent. Holmstrom (1979) and Shavell (1979) show

that this isn’t true when the agent is risk averse. This paper shows that it

isn’t true either when the agent and the principal have different preferences

over time and the agent’s choices are a function of the agent’s discount factor.

The agent doesn’t care about the future as much as the principal would

like, however because current and future payoffs are positively correlated the

principal can replicate the first best by increasing the relationship between

the agent’s pay and current payoffs.

This paper presents an explanation for why a firm may use broad based

stock options rather than ordinary stock to motivate its workers. The value

of stock options is that they can exercised at a given date, paying the agent

the expected future value of his current choices. In this way stock options

can align the agent’s and the principal’s preferences over time, and enable

the first best to be achieved. This explanation seems to line up with the

way practitioners view the value of using broad based stock options (NCEO

(2003)). In the model presented, the principal benefits in the future from

costly actions. In particular the future payoffs are highly correlated with

current payoffs. This may be a good representation of situations like R and

21
D races or “standards” races, and may account for the use of options in

sectors where such races are important.

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23