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relative amounts of the three articles (that is on the position ofthe point of contact) can readily be discerned by its aid. Far morecomplicated cases are supposable and exist in reality.
§3.
If we suppose for an instant that there are but three commoditiesin the market, the preceding analysis yields a complete account ofthe equilibrium in that market.
To sketch this briefly let us suppose the space to be filled with autility density for I, another superposed but different distributionfor II, and so ou. Let us include production. If one man shouldbe both a consumer and producer of the same article, the net con-sumption or production is now to be taken, and the total utility or dis-utility of this net amount is the density. The planes before referredto as partial income planes may now be called “total income andexpenditure planes,” and they must each pass through the origin *
37.
(01, fig. 27 for two dimensions). Since the “maximum directions”(normal to their planes) are parallel, these planes must all coincide.The point in this plane selected by I will be that of tangency to anindifference surface for I. Likewise for II, HI, etc. Such points
# For since income balances expenditure, if A,, B,, C,, represent the (net)amounts consumed or produced by I, those consumed being treated as positive,and those produced as negative, the whole money value must be zero: i. e.
A . p a + B . p s + C . p c - 0,
which is the equation of a plane passing through the origin.