Suppose we have some quantity of good A on hand, and we will trade it for a suitable quantity of good B. We have a prior notion of the values of A and B by some knowledge of the world or levels of our inventories or our costs or other experience.
Notation:
a = The quantity of A that I offer for trade.
La = prior notion of value of A, expressed as units of essential value per unit of A.
Lb = prior notion of value of B, expressed as units of essential value per unit of B.
P(L) = Lb/La, the “natural” or “fair” price without profit.
b = The quantity of B that I want to receive in trade for a.
r = the rate of profit (or margin) that I intend to realize on the trade, expressed as a fractional increment of a.
p(r,{A,B}) = the price at which I receive a unit of B, that is, the number of units of A per unit of B, given a profit r. p(r,{A,B}) = p(r) if no ambiguity results.
Identities:
La * a = Lb * b for a trade without profit.
La * a * (1+r) = Lb * b for a trade with nonzero profit.
p(r,{A,B}) = a/b = (Lb/La) / (1+r) = P(L) / (1+r)
a = b * p(r)
b = a / p(r)
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