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 observesor 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 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 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 10in 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. 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 solution, we calculate the expected cost of a project as 200.5 (). Thus, Mary will have an expected profit of 99.5 (300 – 200.5). 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 , such that the claim before the period is: “I will not fund any project that cost more than .” To introduce the math, consider the case where . 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 project cost 240 or less. The project will cost 240 or less 90% of the time (). Mary will make 54 in expectation (60 × 90%).
For a general rationing solution, Mary will make with a frequency (). Thus, Mary will have expected earnings () of: for values of that conform to . 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 , since no projects would ever qualify. We can simplify the expected earnings to , and then to . The first derivative of with respect to is . Setting to zero, we calculate the profit-maximizing cutoff value is 225. (Calculus fans will note that since , we have confirmed this is a maximum, not minimum). The expected profit is calculated from 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, , or . 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 25in 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). Shewould make 149.5, 100, and 100in 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.