Slack, Rationing, Credibility, And Commitment
Accounting and Management Information System 3300
The Ohio State University
David E. Wallin
Version: Autumn 2014
We are going to consider a number of different scenarios over the coming pages. We
will raise issues that address slack, rationing, credibility, and commitment. Through this,
we will also be able to explore issues about the need for firms, time horizons, and ethics.
The Basic Firm
Laura operates a business that does one “project” per period. A project will always
have revenue of 100.1 However, the cost of the project has a 50% chance of being 86 and a
50% chance of being 68. She learns the cost at the beginning of the period and invests the
required amount. She gets the revenue at the end of the period. Half the time she earns 14
(100 – 86), and half the time she earns 32 (100 – 68). Her expected return (i.e., average) is
23 per period.
Laura has decided to hire Rob to run this business. The revenue, costs, and timing
are unchanged. Assume that Laura and Rob both learn the cost at the beginning of the
period. Laura provides the funding and gets back the revenue. She again earns 14 or 32
each period for an average of 23. From this she must pay Rob. We will assume that he is
paid a flat amount per period. We will proceed as if that amount is zero to make the
calculations simpler, since this won’t change the issues we will explore. However, one can
feel free to assume any wage rate for Rob (though it would surely have to be less than 23).
Additionally, it may appear Rob doesn’t do much for his wage. However, that is only
because we won’t find that part interesting. Hiring Rob may have transferred many tasks
from Laura to Rob.
Importantly, we have separated ownership and control (or
management).
Adding Conflict: The Slack Solution
Let’s change the world in which Laura and Rob find themselves. At the beginning of
the period, Rob learns whether the cost is 86 or 68. This is information private to Rob:
Laura never observes or learns the cost. Rob then asks Laura for funding, the project is
done, and Laura nets 100 less the funding cost. If Rob requests funding of 86 when the cost
is 86 and requests funding of 68 when the cost is 68, Laura and Rob find them in the same
place as before. He earns nothing (in addition to any salary), and she earns 23 in
expectation. We will refer to this a “truthful solution.”
Rob need not always be truthful. If the cost is 86, Rob will be forced to ask for 86.
He can only do the project with 86 in funding (he has no resources of his own). If he asked
for 68 when the cost was 86, Laura would provide 68, and Rob could not complete the
project. We could imagine that Rob would get fired or suffer some other sufficiently
We will use these numbers without reference to a currency. One can think of these as being
pounds, or hundreds of euros, or thousands of dollars without a change in the basic conclusions.
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undesirable consequence. Plus, we can see no benefit to Rob of requesting less money than
is necessary to do a project. However, we can see an incentive for Rob to request more
funding.
We have established that Rob would request 86 if the cost is 86. What would he do if
the cost is 68? He could ask for 68 and fund the project. He could ask for 86, and Laura
has no way of knowing he just lied. If she funds the project with 86, Rob then completes the
project for 68. This leaves 18 in unused funds that Rob pockets. The revenue of 100 is
returned to Laura. She makes 14 and Rob makes 18 (beyond his wage). Laura used to
make 32 when the cost was 68. Remember, Laura will never know for sure that a request
for 86 was truthful or not (she will know a request of 68 had to be truthful).
We have arrived at the following solution. If the cost is 86, Rob asks for 86, Rob gets
86, Rob makes 0, and Laura makes 14. If the cost is 68, Rob asks for 86, Rob gets 86, Rob
makes 18, and Laura makes 14. In expectation, Rob makes 9 (50% chance of 0 and a 50%
chance of 18). Laura makes 14 with certainty. This is the “slack solution” (i.e., Rob lies and
builds slack in the budget). You might think of this as nothing more than pure
embezzlement on the part of Rob. We will return to some ethical dimensions of this. You
might also think that Laura must certainly sniff this out in time. If Rob always asks for 86,
won’t she eventually determine he must be lying? We will come back to this issue of
multiple periods. For now it is sufficient to imagine the simplest case: Laura and Rob will
work together for one period only.
We note here that all solutions to this point have 100% economic efficiency. The
project must cost less than 100 to complete. Therefore, there is wealth (of 14 or 32)
generated each time a project is funded. All projects are funded, so all possible wealth is
available to the economy. In expectation, that wealth is 23 per period. In the truthful
solution, all of the 23 goes to Laura. In the slack solution, Rob gets 9, and Laura gets 14.
The “pie” is as big as it can be; we see two ways the pie may be cut.
Attempting to Improve on Slack: Rationing
Can Laura improve on this return of 14? Imagine Laura makes the following
announcement: “I will only fund projects that cost 68.” Imagine Rob believes this is how
Laura will act. As before, if he observes a cost of 86, he can only request 86. When he does,
Laura does not fund the project. Both Laura and Rob get zero. When he observes a cost of
68, he could ask for 86. Laura wouldn’t fund that, and both would get zero. However, if he
requested 68, the project would be funded. Laura would make 32, and Rob would get zero.
This suggests Rob has no monetary incentive to ask for 68 when it is 68. However, he has
no monetary incentive to ask for 86 when it is 68, given our current assumptions. If Laura
sticks to her statement, Rob is indifferent between reporting 86 or 68 when the cost is 68.
We will follow a tradition of assuming that an indifferent Rob will select the alternative
that makes Laura better off. Rob requests 68 when the cost is 68. We call this the
“rationing solution.” You might find it strange that we assume Rob will cheat Laura if it
makes him better off, but is willing to make her better off if he can at no cost. We will
return to this soon.
In the rationing solution, the project is only pursued when the cost is 68. Rob makes
zero regardless of cost. Laura makes zero when the cost is 86 and 32 when the cost is 68.
In expectation, she makes 16. Laura is better off than the 14 she makes in the slack
solution. Note that Laura could offer Rob as bonus of, say, 2 whenever a project is funded.
She would make 30 (100 – 68 – 2) net of the bonus when the cost is 68. Her expected value
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is 15, which is still better than the 14 in the slack solution. Thus, we have an alternate
solution to dealing with Rob’s indifference in reporting a cost of 68. She might be able to
provide the inducement to with a smaller bonus. We proceed without such bonuses for now.
Economic efficiency has dropped with rationing. The available increased wealth is
either 14 or 32 each period, with an expected average of 23. The actual increase in wealth
is zero when the cost is 86 and 32 when it is 68. The actual expected wealth increase is 16.
Economic efficiency is 70% of what it could be. They do half the projects, but they do the
higher-profit half. Thus, they earn 70% of the wealth available by doing 50% of the projects
(and, absent a bonus, this all goes to Laura).
You may have noticed that the rationing solution is only slightly better for Laura
than the slack solution (16 versus 14). This may cause you to wonder how important the
chosen parameters are to the solution. We can explore this by changing parameters, and a
good first step is often big changes. What if the costs were 86 and 10? Laura earns 52, 14,
and 45 in the truthful, slack, and rationing solutions. This raises the cost of slack and
makes the rationing solution considerable better than the one with slack. What about costs
of 86 and 80? Laura earns 17, 14, and 10 in the truthful, slack, and rationing solutions.
This shows that rationing is not always better than slack. Indeed, in such a case, slack is
not particularly costly.
Is Rationing Stable?
Is there a threat to this rationing solution? Imagine you are Laura, and Rob has
just requested funding of 86. You have a choice of sticking to your statement that you won’t
fund that or fund it anyway. In the former case you get zero; in the latter you get 14.
Laura has clear monetary incentive to violate her stated rule. In the rationing solution,
Rob reports truthfully, and Laura only funds a request for 68. Can we have a solution
where Rob reports truthfully, and Laura ignores her statement and always funds? This
would extract 100% efficiency and give it all to Laura. It is difficult to imagine that would
be stable over multiple periods. Once Rob sees Laura fund a request for 86, he knows she
will violate her rule. Rob would know always request 86, and we’re back at the slack
solution.2
A problem with the rationing solution becomes apparent. Laura has an incentive to
violate her rule. But, if she does the rationing solution degrades to the less desirable (for
her) slack solution. If this game is played over multiple periods, it is clear why Laura has a
long-term incentive to obey her rule. The one-time benefit of violating her rule would be
expected to have a long-term loss. She can “sneak in” and get 14 extra when 86 is
requested, but now she would have to settle for 2 less (in expectation) each period if this
violation forced the slack solution.3
Imagine the challenge Laura and Rob have in a one-period world. If Laura
announces she won’t fund a request for 86, what would she do if 86 is requested? In a
single-period world, Laura can get 14 by funding or zero if not. Laura has no reputation to
be concerned about; she and Rob will never interact again. So, why not take the “free” gain
of 14? There is no reason for Laura to stick to her rule. Thus, we’d expect her to break it.
Think of this issue in light of an often issued statement by governments (or their leaders): “We do
not negotiate with terrorists.” The same, but more dire, issues pop up.
3 You might think Laura would be willing to take the 14 this period by funding 86 and settle for 2
less in future periods because of the chosen parameters. In a world where the costs are 86 and 10,
she would net 14 extra this period, but lose 31 (45 – 14) each remaining period.
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But, if we can anticipate she will fund 86 even though she said she would not, then Rob can.
If Rob observes 68, why would he not request 86? He expects she’ll take the 14 rather than
zero and fund it. We would not expect the rationing solution to hold in a single-period
world. Rob is no better or worse off reporting 86 when the cost is 68, if Laura obeys her
rule. But, he can be better off and can’t be worse off if he reports 86 when the cost is 68, if
there is any chance Laura will violate her rule. Even in the case where Rob receives a
bonus for reporting 68, the bonus won’t be big enough if Rob thinks there is a sufficiently
high probability that Laura will fund 86. And, why would that probability be less than
100%.4
Credibility and Commitment
Can Laura rescue the rationing solution in a single-period world? Her basic problem
is one of credibility. Laura needs to be able to make a credible commitment that she won’t
fund a project at a cost above 68. She might try writing in into a contract with Rob, but this
creates problems. Parties to the original contract can agree to modify the contract. Laura
could write in the employment contract that she will not fund projects at more than 68.
However, if she funded a request for 86, Rob would have no legal basis to bring suit (he had
no loss from her violation of the clause). Laura can write an employment contract where
she agrees to pay Rob a “penalty” of 15 (or anything more than 14), if she funds a project for
more than 68. Laura would never violate that clause in the contract to earn 14 from the
project, if she then must pay Rob 15. But, she would be willing to renegotiate the penalty
in the contract to 1 (or anything less than 14). Rob would renegotiate, because he knows he
won’t get 15 in any case. Now, we just speculating on what positive penalty below 14 would
come out of negotiations. Since Rob could anticipate her willingness to renegotiate a
penalty of 15, he would request 86 when the cost is 68. With renegotiation, Rob would end
up with the slack of 18, plus the renegotiated penalty: a situation worse than not even
trying rationing. Laura would anticipate the she could and would renegotiate the penalty,
so she would never try that type of contract.
Are there other ways to gain credibility? It is easy enough for a military leader to
attempt to motivate his troops by claiming retreat is not an option. William the Conqueror
burned his ships during the Norman Invasion of 1066. Cortés followed the same strategy in
his conquest of Mexico. It is one thing to claim retreat will not occur; it is another to gain
credibility by burning (or disabling) your only means of retreat. Interestingly, the Trojans
got it backwards when the Greeks landed to rescue Helen: they tried (but failed) to burn
the Greek ships. Your enemy having a way to retreat may be something you desire.
An (Almost) Continuous Case
Mary owns and Lou operates a business similar to Laura’s. One project may be
pursued per period, which returns revenue of 300 with certainty. The cost may be any one
of the hundred values of 250, 249, 248, 247, … 153, 152, 151. To obtain the truthful
250−151
solution, we calculate the expected cost of a project as 100.5 (100.5 =
). Thus, Mary
2
In the single-period world, Laura has no reason to pass up on the 14 she gets by violating her
declaration. That is the last choice anyone makes. She decides if she wants zero or 14. If she played
a single period with Rob, then an additional period(s) with a new employee(s), she could have
incentive to pass on the 14 to signal to the next employee she sticks to her policies.
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will have an expected profit of 99.5 (300 – 100.5). With slack, Lou would always report the
highest cost of 250, and Mary would make 50 each period. The rationing solution is more
complicated to calculate.
Imagine Mary establishes a “cutoff value” or 𝑐, such that the claim before the period
is: “I will not fund any project that cost more than 𝑐.” To introduce the math, consider the
case where 𝑐 = 240. When the project has a cost higher than 240, Lou must request at least
that value and Mary (since we’ll assume for now she sticks to her statement) won’t fund
those projects and makes nothing. If Lou observes a value of 240, he requests it and it is
funded. However, if Lou observes a value less than 240, he can request 240 and pocket the
slack. Thus, Mary will always make 60 (300 – 240) whenever the cutoff is 240 and the
240−150
project cost 240 or less. The project will cost 240 or less 90% of the time ( 100 ). Mary will
make 54 in expectation (60 × 90%).
𝑐−150
For a general rationing solution, Mary will make 300 − 𝑐 with a frequency (
).
Thus, Mary will have expected earnings (𝜋) of: 𝜋 =
𝑐−150
(300 − 𝑐) (
)
100
100
for values of 𝑐 that
conform to 250 ≥ 𝑐 ≥ 151. Mary would never have a cutoff above the maximum cost; that is
just a license for Lou to take more than the worst slack case. Mary would make zero profit
if 𝑐 < 151, since no projects would ever qualify. We can simplify the expected earnings to
300𝑐−45,000−𝑐 2 +150𝑐
𝜋=
, and then to 𝜋 = 4.5𝑐 − 450 − 0.01𝑐 2. The first derivative of 𝑐 with
100
respect to 𝜋 is 𝜋 ′ = 4.5 − 0.02𝑐. Setting 𝜋 ′ to zero, we calculate the profit-maximizing cutoff
value is 225. (Calculus fans will note that since 𝜋 ′′ = −0.02, we have confirmed this is a
225−150
maximum, not minimum). The expected profit is calculated from 𝜋 = (300 − 225) (
)
100
and is 56.25. Mary makes 75 (300 – 225), 75% of the time.
A few technical points are necessary about the above calculations. We described
Mary’s world as one with 100 different whole number values for costs. The cost could be,
say, 207 or 208, but could not be 207.5, 20757, or 66𝜋. The distribution is discrete. The
calculations actually assume a continuous distribution (where the cost is greater than 150
and less than or equal to 250 and can include any of those numbers noted above). That is
why 150, not 151, is used in the calculations determining the cutoff. This convenience of
using a continuous approximation of the discrete function will often provide a close, or by
chance (as here) an exact, estimate of the correct value. This method would provide a
terrible estimate of the cutoff for Laura to use.5
In Mary’s world, expected profits are 99.5, 50, and 56.25 in the truthful, slack, and
rationing solutions. If Mary’s revenue was 250, the optimal cutoff for rationing would be
200. She would make 49.5, 0, and 25 in the truthful, slack, and rationing solutions. If
Mary’s revenue was 350, the optimal cutoff for rationing would be 250 (essentially there is
no cutoff; all projects are funded). She would make 149.5, 100, and 100 in the truthful,
slack, and rationing solutions. Thus, we can see that, depending on the parameters
selected, we can draw different conclusions. With revenue of 300, Mary will lose about half
her earnings to slack and regain about one-eighth of those lost earnings with rationing.
With revenue of 250, Mary will lose all her earnings to slack and regain about half of those
lost earnings with rationing. With revenue of 350, Mary will lose about a third of her
earnings to slack and can’t use rationing to regain any of that.
Think about what these observations about different revenues essentially detail.
When we have a world with “slim” profit margins (revenue is 250, potential profit ranges
5
The approximation will tend to be better when there are more observations (here of costs).
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from 0 to 99), Mary can be devastated by slack (wiping out all her profit), and therefore, can
be helped a lot by successful rationing. In a world with “medium” profit margins (revenue
is 300, potential profit ranges from 50 to 149), Mary is hit hard slack (wiping half her
profit), and can be helped a bit by successful rationing. Move to a world with “large” profit
margins (revenue is 350, potential profit ranges from 100 to 199), Mary suffers a lesser, but
significant, blow from slack (wiping a third of her profit), but cannot be helped by rationing.
Rationing would never help Mary whenever the revenue is 350 or more (assuming the same
cost possibilities). With revenues of 400, 450, and 500, Mary would lose about 25%, 20%,
and 17%, respectively, of her profit, with no help available from rationing.
A Reconsideration of Time Horizons
Notice an important issue in arriving at our solution rested on the time horizon.
Assume Rob wants to take as much slack as he can. One strategy would be for him to
report 86 every period. How long could that last? Laura realizes that if Rob reports 86
each of the first four periods they are together, there in only a 6.25% chance that 86 was
observed each period. A streak of 5, 6, 7, or 8 observations of 86 would occur naturally with
probabilities of 3.13%, 1.56%, 0.78%, and 0.39%, respectively. If Rob always reports 68 as
86, it won’t take long to catch him. Of course, if the only punishment is termination and
Rob does not suffer from the job loss (e.g., he can easily get as good a job), he might be just
fine with the tradeoffs. However, what if Laura can affect Rob’s life after termination? For
example, she gives him poor recommendations when a future employer asks about him.
You might think it is not a great strategy to always report 68 as 86. Let’s say you
recommend to Rob that he flip a coin when the cost is 68. If it comes up heads, report 68,
otherwise report 86. Rob is harder to catch. The simple “lie all the time” strategy seems
easy to catch. This new strategy would be more difficult to determine without more
observations. Even though Rob is still getting slack, it is happening at a reduced rate.
Laura makes 18.5 in expectation, clearly less than a truthful and more than a slack
solution. Still, it is an improvement over the rationing solution.
And, of course you see Rob’s problem. If he is going to report 68 as 86 with
frequency 𝑥, he must carefully chose that value. A value of 𝑥 = 100 is highly profitable to
Rob, highly costly to Laura, and can be detected over a short horizon. Values of 𝑥 near zero
build little slack for Rob, cost Laura little, but are difficult to detect, possibly even over
many periods.
Note the even greater challenge facing Lou. Theoretically, he should report 250 as
the cost in a single-period world. The odds of Lou being truthful if he reports 250 each of
1
1
the first two periods is 0.01% (there are 100 possible costs: 0.01% = 100
× 100
). Lou has to be
much crafter than always reporting the highest cost each period in a multi-period world if
he doesn’t want to get caught.
The new challenges to Rob/Lou when the game expands to one with multiple periods
work well for Laura/Mary. Laura/Mary can run simple statistical tests on requests of
Rob/Lou. She can determine a 𝑝-value for the stream of requests relative to the known
distribution of costs. If the value of 𝑝 crosses some threshold, she can fire him. Of course, if
he does not suffer (or suffer sufficiently) from being fired, he might not have even an
incentive to try and hide his slack (i.e., request the highest cost every period). But, should
he suffer sufficiently upon being fired (e.g., has costs to move to a new job, is unemployed
for some time, can’t get a job that pays this well again, etc.), he would have an incentive to
keep the slack requests down. Since more observations allow Laura/Mary to do statistical
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tests with less α (type I) and less β (type II) error, Rob/Lou would be forced to have very low
levels of slack. Simply, long-term relationship can have value.
Ethical Considerations
So far we have had Rob/Lou embezzle money. One hopes a student reading this
feels that is clearly unethical, even before considering that Rob/Lou may end up with jail
time and a requirement to repay the money. As one looks at the history of accounting, it
started about the time that businesses moved from owner controlled to manager controlled.
Accounting’s earliest role was centered on stewardship: how well the person in control
protected the firm’s assets.
However, there are other ways Rob/Lou can consume firm resources short of
embezzlement: perquisite consumption. Firms regular offer employees perquisites. This
can include reduced prices on products/services they make/sell, health benefits, and
company cars. However, the issue here is not perquisites that the firm has designed for the
benefit of employees. From this point, the only perquisites that concern us are
unauthorized perquisites. This can range from things many would consider trivial, like
personal use of an office copier, to far more serious issues, like having the firm pay for an
employee’s personal airfare. We include the previous thought that Rob/Lou might embezzle
funds as part of this perquisite consumption.
There are many ways an employee can abscond with firm resources through
perquisites. An employee could delay the purchase of an airline ticket for company
business to force the firm to pay a higher price, but giving the employee a fare category that
would allow an upgrade or more frequent-flier mileage. A similar tactic could be used to
secure a better hotel room. An employee consuming a meal on the company’s dime might
choose a very expensive one. What if the project always costs Rob 68? However, half the
time the project is “difficult,” and he must hire an outside firm do some of the work at a cost
of 18 (a net cost of 86). The other half of the time, it is “easy,” and he need not hire the
outside firm (and the cost could be 68). However, when Rob learns the project is easy, he
could still hire the outside firm, surf the internet all day, and Laura would be none the
wiser that she was paying extra for Rob’s leisure.
Consider this alteration. At the beginning of the period, Rob knows if prospects are
“good” or “bad.” When prospects are bad, there will be a project with a cost of 86 available.
Rob will have to work “hard” to find and execute that project. When prospects are good,
Rob will have to work “easy” to find and execute a project costing 86 and will have to work
hard to find and execute a project costing 68. Laura knows the previous information, but
will not know if prospects are good or bad, the actual cost, or if Rob worked hard or easy. If
prospects are bad, Rob must request 86 (since Laura knows there is always a project costing
86 available). If funded, he will have to work hard. If prospects are good and Rob requests
68, Laura will fund him and he will have to work hard. If prospects are good and Rob
requests 86, Rob will work easy if funded. The truthful and slack solutions are the same as
they were before. However, in a rationing solution, we must address one new factor: what
does Rob do when he requests 86 and isn’t funded? To make this parallel the previous
development, we must have Laura assign Rob an alternate task to perform when she does
not fund the project, and that work must be equally hard. Laura cannot profit from this
activity more than she would if she funded a request for 86.
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Profit Sharing as a Bribe
Could we bribe Rob not to steal? Consider how much it would cost a storeowner to
bribe the thief holding a gun on him. The simple answer is whatever is in the cash register.
And, in a similar vein, it would cost Laura all her gains from the lower cost, 18, to bribe Rob
not to report 86 when the cost is 68, if Rob is just pocketing the money. But, what if Rob
isn’t pocketing cash, but consuming perquisites. A thief steals a $4 million painting. The
owner’s insurance company offers a $1 million “no questions asked” reward for the return of
the painting. The thief returns it, since he cannot sell it through legitimate channels and
would get less than $1 million converting it to cash through the black market. The
insurance company is $3 million better off than if the painting is never seen again. Now,
the insurance company did not bribe the thief not to steal the painting. Still, we have
arrived at pretty much the same point: paying ransom and bribing no to steal can lead to
the same outcome.
Imagine when Rob reports 86 when the cost could be 68 and consumes 18 of slack in
the form of perquisites. So, for example, he spends an extra 18 on better lodging and travel.
Rob would never pay 18 for those extra benefits. They are worth to him, say, a third of
that. Rob would gladly report 68 when the cost is 68 if his pay went up by 6 in those cases.
So, Laura offers the following contract. Rob gets his regular salary each period and gets a
bonus equal to one-third of profits that exceed 14. Rob truthfully reports, getting a bonus of
6 half the time. Laura makes 14 half the time (100 – 86) and 26 the other half (100 – 68 –
6), for an expected value of 20. If Rob had been truthful without the bonus/bribe, Laura
would have made 23 in expectation.
The effectiveness of the bonus/bribe depends on how much Rob values the
perquisites consumed. If he valued the 18 in perquisites consumed at 16, the bonus must
be 16, and Laura makes 14 half the time (100 – 86) and 16 the other half (100 – 68 – 16), for
an expected value of 15. This is worse than rationing. Also, what if Laura had multiple
“Robs”? If Laura had five employees, each running a division, and each facing the same
situation as Rob (and each values the 18 in perquisites at 6), Laura could offer each a
contract of a fixed salary plus one-third of the divisional profits and solve slack problems.
But, if Laura could determine firm profit but not divisional profit, she could not motivate
the division managers to remove slack. Well, she could if she gave them each one-third of
firm profits beyond 70 (5 × 14), but that requires she pays out 167% of firm profit beyond
the slack solution.
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