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as follows. He is directed to alter this consumption combinationby arranging his quantities A and B of the twb selected commodities(a) and ( b) in all possible ways, but Without changing the quantitiesC, D, etc. of other commodities. The marginal utility of each willvary not only in relation to its own quantity but also the quantityof the other commodity. Thus,
d UdA 1dJJ
dB>
F(A„B,)
FiB.A)
These may be regarded as derivatives with respect to A and B of
U.^A.B,)
where U, is the total utility to I of the consumption combinationA, and B,.
In fig. 18 let the abscissa OX represent the quantities B, of (b) andthe ordinates (OY) the quantities A, of (a).Any point P by its co-ordinates representsa possible combination of quantities A,and B, consumed by I. By varying pointP all possible combinations of A, and Bare represented. At P erect a perpen-dicular to the plane of the page whoselength shall represent the marginal utilityof A, for the combination, that is, thedegree of utility of a small addition ofA,, (B K remaining the same). If P as-sumes all possible positions, the locus of the extremity of this per-pendicular will be a surface.
Again at P erect a different perpendicular for the marginal utility'of B,; its extremity will generate another surface. The first sur-face takes the place of a utility curve for (a), the second for (b).These two surfaces may be regarded as the derivative surfaces(with respect to the variation of A, and of B,), from a primitivewhose ordinate (perpendicular at P), is the total utility of the com-bination of A, and B, represented by the point P. This surface isusually convex like a dome with a single maximum part, but it neednot always be. There may be two maxima as will presently appeal*.In such a case it cannot be everywhere convex.
If a plane be drawn tangent to this last surface at a point over P,the slope of the plane parallel to the A direction will be the ordinate
18