# Why's A's expected profit \$.04?

Paul Davies. JC Smith's The Law of Contract (2018 2 ed). p. 6.

On a slightly diﬀerent tack, but in a similar vein, Judge Richard Posner has written:17

Suppose I sign a contract to deliver 100,000 custom-ground widgets at \$.10 apiece to A, for use in his boiler factory. After I have delivered 10,000, B comes to me, explains that he desperately needs 25,000 custom-ground widgets at once since otherwise he will be forced to close his pianola factory at great cost, and oﬀers me \$.15 apiece for 25,000 widgets. I sell him the widgets and as a result do not complete timely delivery to A, who sustains \$1000 in damages from my breach. Having obtained an additional profit of \$1250 on the sale to B, I am better off even after reimbursing A for his loss. Society is also better off. Since B was willing to pay me \$.15 per widget, it must mean that each widget was worth at least \$.15 to him. But it was worth only \$.14 to A—\$.10, what he paid, plus \$.04 (\$1000 divided by 25,000), his expected profit. Thus the breach resulted in a transfer of the 25,000 widgets from a lower valued to a higher valued use.

17 R Posner, Economic Analysis of the Law (8th edn, Aspen, 2011) 151.

1. Pls see the emboldenings. Whence did the \$1000 hail?

2. Whence did the 25,000 hail? This is B's quantity demanded, not A's?

• This seems to be a question of either economics or arithmetic rather than law, and perhaps should be migrated. – Nate Eldredge Apr 19 '19 at 22:46
• I'm voting to close this question as off-topic because it is a question about economics, not the law or legal process. – Nij Apr 20 '19 at 1:06
• @NateEldredge Although purely making sense of the numbers requires arithmetic and algebra, this is on topic because it premises Posner's questionable --albeit legal-- proposition that a breach of contract makes society "better off". – Iñaki Viggers Apr 20 '19 at 7:08

Why's A's expected profit \$.04?

Without criticizing the other answer, I think an alternative approach illustrates—and helps assess—Posner's legal argument that "Society is also better off" with a breach of contract. To outline A's profit, one needs to consider the proceeds A would obtain for each widget. Thus, this alternative approach consists of comparing A's profit for various sales prices (meaning sales by A) with and without breach of contract, and then find the break-even sales price from A's perspective (which turns out to be \$0.14 = \$0.04 + \$0.10).

At \$0.10 apiece, A's total cost for 100,000 widgets is \$10,000.

But due to the supplier's breach of contract, A experiences a total cost of \$0.1*75,000 – \$1,000 = \$6,500 for the 75,000 delivered units (recall that the supplier pays A \$1,000 for the breach). To A, the breach results in a unit cost of \$0.086667.

At A's sales price of \$0.10 per unit, the scenario with no breach of contract leads to profit = (\$0.10 – \$0.10)*100,000 = \$0.00. With breach of contract, A's profit would be (\$0.10 – \$0.086667)*75,000 = \$1,000 (>\$0.00). Therefore, A is better off with the breach.

At A's sales price of \$0.12 per unit, the scenario with no breach of contract leads to profit = (\$0.12 – \$0.10)*100,000 = \$2,000. With breach of contract, A's profit would be (\$0.12 – \$0.086667)*75,000 = \$2,500 (>\$2,000). Therefore, A is better off with the breach.

Now suppose A's sales price is \$0.20 per unit. The scenario with no breach of contract leads to profit = (\$0.20 – \$0.10)*100,000 = \$10,000. With breach of contract, A's profit would be (\$0.20 – \$0.086667)*75,000 = \$8,500 (< \$10,000). Thus, in this case, the breach hurts A's profit.

In this example, it turns out that the break-even sales price is \$0.14 per unit. Indeed, the scenario with no breach leads to profit = (\$0.14 – \$0.10)*100,000 = \$4,000, and with breach of contract, A's profit (\$0.14 – \$0.086667)*75,000 = \$4,000. That is, at a sales price of \$0.14 per unit, A is indifferent between the breach and the no-breach scenarios.

Essentially, it is a matter of solving for x the equation (x – \$0.10)*100,000 = (x – \$0.086667)*75,000.

Hence, x = \$0.14. A benefits (loses) from the breach as long as the sales price is lower (greater) than \$0.14.

Strictly speaking, it is not that a breach itself makes society better off. The so-called "enhancement" of valued use (what Posner describes as "[upgrading] widgets from a lower valued to a higher valued use") due to a breach takes place only inasmuch as B subsidizes that "enhancement" by paying a premium for the breach he incentivized.

But Posner's questionable conclusion (that society is "better off" with the breach) misses one ramification. In exchange for the aforementioned "enhancement", the breach is overall detrimental from the standpoints of subsequent formation of contracts and the predictability of their performance:

1. The supplier's inability to fulfill its contract with A will tend to make A more uncertain about future agreements; and
2. in turn, A's partial inability to deliver (namely, per the missing 25,000 units) tends to have on his customers the same effect that the supplier's breach tends to have on A with respect to the supplier.

This sort of domino effect contravenes or weakens the contract law premise that a party knowingly subjects himself to a set of circumstances.

Also, Posner's term of "expected" profit is somewhat misleading because this has nothing to do with probabilities.

Just to be clear, I don't mean any of the above in a derogatory way toward former Judge Posner. In fact, he is one of the very few members of the judiciary I consider actually honorable. Sadly, it was his integrity and his indefatigable diligence what alienated his crooked peers in judicial office.

• Whoever downvoted this, please explain why. If you cannot articulate that, then it reflects you might not be capable of assessing the answer(s) you impulsively try to suppress. – Iñaki Viggers Apr 20 '19 at 7:05
• I didn't downvote, but your explanation makes the mistake of solving a motivationless equation. The 0.086667 was your calculation, not an original number. Your solution for break-even ought to start with the facts of the situation, which are that A suffered and was reimbursed for a \$1000 loss, that is, indifference occurs when (x - \$0.10) * 100,000=(x - \$0.10) * 75,000 + 1,000 This equation is equivalent to the one you used, but it has the advantage of being directly traceable to the problem statement. – Ben Voigt Apr 20 '19 at 7:52
• @BenVoigt Thanks for the feedback. I concede that your equation appears more directly traceable to Posner's example. But the calculation of \$0.086667 is neither a mistake nor even an artifice [to make numbers "fit in", so to speak]. That calculation reflects --and nowhere departs from-- the situation that A factually experiences a lower cost per unit as a result of the breach & reimbursement. Regardless, a discussion of profits without considering the notion of proceeds or revenues (which here we denote by x) evidently rendered Posner's propositions harder to follow and to assess. – Iñaki Viggers Apr 20 '19 at 11:05
• +1. Thanks a lot as always for your answers! I hope my edit assists. I can do algebra. Thus wouldn't it be easier if we worked only with (x - \$0.10) * 100,000 =(x - \$0.10) * 75,000 + 1,000 ? Perhaps you can write more on its motivation as in how you derived it? We don't truly need the trial and error with x = \$.10, .12. 20, do we? – Ghreu Dec 5 '19 at 10:32
• Downvoted for not being a well constructed answer. @nate's answer arrives at the correct answer in a far more readable manner – Shazamo Morebucks Dec 6 '19 at 5:11
1. \$1000 is the total amount of damages sustained by A due to his delivery being 25,000 widgets short. We presume these damages represent lost profits. So that is 4 cents of profit lost per widget not delivered; in other words, had he received those 25,000 widgets, he would have been able to earn a profit of 4 cents on each one.

2. 25,000 is the number of widgets that were delivered to B, even though A wanted them. Therefore, A received 25,000 widgets fewer than he wanted.

By failing to obtain 25,000 widgets on time, A suffered a loss of profits. A sues the seller and is awarded \$1000 in damages for these lost profits.

25,000 widgets / 1000 in lost profit = 0.04 profit per widget.

Price per widget + profit per widget = what A thought each widget was truly worth.

Therefore A thought each widget was worth \$0.14 in economic value.

Interestingly, the 1000 in lost profits is set by the court, not necessarily A. Therefore 0.14 is what the court thinks they are worth, not necessarily what A thinks they are worth.