1.1. The horses
The law of marginal utility is derived over three pages of Man, Economy, and State, and Rothbard derives it on a worked case: a man who owns six horses. This page follows that case throughout, and the Lean returns to the same six horses once the theorems are stated. The point of working that way is that you can hold his paragraph beside the formal statement and judge for yourself whether they say the same thing.
He sets it up twice. First by acquisition (MES p. 24):
The first horse will fulfill the most urgent wants that a horse can
serve; this follows from the universal fact that action uses scarce
means to satisfy the most urgent of the not yet satisfied wants.
Then, for the diagram, by counting the ends (pp. 25–26):
We assume for simplicity that there are 10 ends which the means
could fulfill, and that each unit of means is capable of serving one
of the ends. If the supply of the good is 6 units, then the first
six ends, ranked in order of importance by the valuing individual,
are the ones that are being satisfied. Ends ranked 7–10 remain
unsatisfied.
And then he takes a horse away (p. 25):
Assume that a man has a supply of six (interchangeable) horses. …
Suppose that he is now faced with the necessity of giving up one
horse. … Obviously, he gives up the least urgent of the wants which
the larger stock would have satisfied. Thus, if the individual was
using one horse for pleasure riding, and he considers this the least
important of his wants that were fulfilled by the six horses, the
loss of a horse will cause him to give up pleasure riding.
That is the whole derivation. Everything below is an attempt to say it in a form a machine will check.
Three pieces of vocabulary carry it, and they are worth naming now because the shape of the argument turns on the third. There is a stock — the six horses, and the ends a horse can serve at all. There is a plan — for any number of horses he might have, which wants he would serve with them. And there is the preference the plan expresses, which ranks bundles of wants and is not assumed to do anything else: not transitive, not complete, no numbers attached. The full definitions are under The vocabulary; nothing before it depends on reading them.
The plan is where the trouble starts, and it is worth seeing why now, before the formal statement makes it look inevitable. Rothbard's argument compares six horses with five. But the man has six. What he would do with five is not anything he does — it is a plan covering stocks he does not own. His own words give it away: "which the larger stock would have satisfied".
There is a second horse passage, and it is the one that decides how the plan has to be written down (p. 27):
suppose that the sixth horse that he had previously acquired (named
"Seabiscuit") he had placed in the service of pleasure riding.
Suppose that he now must lose another horse ("Man o' War") which had
arrived earlier, and which was engaged in the more important duty
(to him) of leading a wagon. He will still give up end 6 by simply
transferring Seabiscuit from this function to the wagon-leading end.
Rothbard names two horses in order to insist that which horse goes makes no difference. That is only worth saying if the plan could have depended on which — so here the plan is indexed by which horses, and the claim that only the number matters is kept separate, as a named condition. Both of these conclusions of his are proved below, of the six horses.