A utility function is separable if it can be written in the formu = U(v1(x1),v2(x2),…, vm(xm)where x1,…, xm form a partition of the available products. (Assume three goods are available and denote the consumption of good i by xi, i = 1, 2, 3. Then x1 = (x1, x2), x2 = (x3) is an example of a partition.) The implication of separability is that the marginal rate of substitution between any two goods in xj is unaffected by the consumption level of any good not in xj. If a consumer has separable preferences then the demand for a good in xj depends on the expenditure allocated to all goods in xj and the prices of goods in xj. This is a form of two-stage budgeting: the consumer first decides how much to spend on each category of goods (meaning, for each element of the partition), and then allocates the expenditure between goods within a category. A utility function is additively separable ifu = v1(x1)+ v2(x2)+…+ vm(xm).This is a special case of separability. Additive separability is frequently used to represent preferences over consumption at different points in time.
u = U(v1(x1),v2(x2),…, vm(xm)
u = v1(x1)+ v2(x2)+…+ vm(xm).