||on the use of the differential calculus etc.
conditions of equilibrium which have been already established, all the economic particles are disposed at points such that no indefinitely small change of the system is possible; the final step which any one X can be induced to make in conjunction with one Y having the same slope for all, corresponding to a rate of exchange for small quantities Δη:Δx. Now by the assumption introduced in this paragraph, it is open to any individual on either side of the market, say Xr (or, mutatis mutandis, Yr) to supply himself by exchanging small quantities with a great number of dealers. In other words, his whole course from zero to the point (xr, ηr) may be made up of steps taken in conjunction with different Ys, each small step in the direction of the common slope Δη:Δx. If now Xr by multiplying steps of this kind can reach a point, say (x'r, η'r) which represents greater advantage to him than the point at which he was just now supposed to be at (xr, ηr), he will tend to proceed to that point. He cannot do so, it may seem, because there will not be increments of y forthcoming on the terms offered to proceed to the new point; since they are as advantageously employed by their owners in dealing with other Xs. But it will be worth the while of Xr to offer rather better terms (with respect to short terminal steps), to a number of Ys than those which are represented by the given slope; since if it is advantageous for Xr to move from (xr, ηr) to (x'r, η'r) it will in general1 be advantageous for him to move to some point in the neighbourhood of (x'r, η'r). Thus the system cannot be in equilibrium unless any individual dealer Xr (and similarly each Y) whose dealing consists of ηr received in return for xr given, obtains as much y as he is willing to take at the rate of exchange defined by the ratio ηr:xr. In other words, the point (xr, ηr) is on the demand-curve of the individual Xr, and accordingly the point of equilibrium for the system is on the collective demand curve which represents the total demand (on the part of the Xs) for y at each compared rate of exchange between x and y. Likewise the point of equilibrium is on the collective demand-curve which represents the total demand on the part of the Ys for x; or in other words, the curve representing the supply of y. Therefore
- ↑ In the absence of singularities; the usual continuity in the functions with which we are concerned may postulate.