condition is also sufficient. Additivity with three or more goods When there are three or more commodities, the condition for the additivity of the utility function is surprisingly simpler than for two commodities. This is an outcome of Theorem 3 of Debreu (1960). The condition required for additivity is preferential independence. A subset A of commodities is said to be preferentially independent of a subset B of commodities, if the preference relation in subset A, given constant values for subset B, is independent of these constant values. For example, suppose there are three commodities: x y and z. The subset {x,y} is preferentially-independent of the subset {z}, if for all : . In this case, we can simply say that: for constant z. Preferential independence makes sense in case of independent goods. For example, the preferences between bundles of apples and bananas are probably independent of the number of shoes and socks that an agent has, and vice versa. By Debreu's theorem, if all subsets of commodities are preferentially independent of their complements, then the preference relation can be represented by an additive value function. Here we provide an intuitive explanation of this result by showing how such an additive value function can be constructed. The proof assumes three commodities: x, y, z. We show how to define three points for each of the three value functions : the 0 point, the 1 point and the 2 point. Other points can be calculated in a similar way, and then continuity can be used to conclude that the functions are well-defined in their entire range. 0 point: choose arbitrary and assign them as the zero of the value function, i.e.: 1 point: choose arbitrary such that . Set it as the unit of value, i.e.: Choose and such that the following indifference relations hold: . This indifference serves to scale the units of y and z to match those of x. The value in these three points should be 1, so we assign 2 point: Now we use the preferential-independence assumption. The relation between and is independent of z, and similarly the relation between and is independent of x and the relation between and is independent of y. Hence This is useful because it means that the function v can have the same value – 2 – in these three points. Select such that and assign 3 point: To show that