## Quick Reference

A utility function is separable if it can be written in the form*u* = *U*(*v*_{1}(*x*^{1}),*v*_{2}(*x*^{2}),…, *v** _{m}*(

*x*

*)where*

^{m}*x*

^{1},…,

*x*

^{m}form a partition of the available products. (Assume three goods are available and denote the consumption of good

*i*by

*x*

_{i},

*i*= 1, 2, 3. Then

*x*

^{1}= (

*x*

_{1},

*x*

_{2}),

*x*

^{2}= (

*x*

_{3}) is an example of a partition.) The implication of separability is that the marginal rate of substitution between any two goods in

*x*

^{j}is unaffected by the consumption level of any good not in

*x*

^{j}. If a consumer has separable preferences then the demand for a good in

*x*

^{j}depends on the expenditure allocated to all goods in

*x*

^{j}and the prices of goods in

*x*

^{j}. 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 if

*u*=

*v*

_{1}(

*x*

^{1})+

*v*

_{2}(

*x*

^{2})+…+

*v*

*(*

_{m}*x*

*).This is a special case of separability. Additive separability is frequently used to represent preferences over consumption at different points in time.*

^{m}*u* = *U*(*v*_{1}(*x*^{1}),*v*_{2}(*x*^{2}),…, *v** _{m}*(

*x*

*)*

^{m}*u* = *v*_{1}(*x*^{1})+ *v*_{2}(*x*^{2})+…+ *v** _{m}*(

*x*

*).*

^{m}*Subjects:*
Economics.