The model begins with a standard CES utility function:
where N is the number of available goods, xi is the quantity of good i, and σ is the elasticity of substitution. Placing the restriction that σ > 1 ensures that preferences will be convex and thus monotonic for over any optimising range. Additionally, all CES functions are homogeneous of degree 1 and therefore represent homothetic preferences.
Additionally the consumer has a budget set defined by:
For any rational consumer the objective is to maximise their utility functions subject to their budget constraint (M) which is set exogenously. Such a process allows us to calculate a consumer's Marshallian Demand. Mathematically this means the consumer is working to achieve:
Since utility functions are ordinal rather than cardinal any monotonic transform of a utility function represents the same preferences. Therefore, the above constrained optimisation problem is analogous to:
since is strictly increasing.
By using a Lagrange multiplier we can convert the above primal problem into the dual below (see Duality)
Taking first order conditions of two goods xi and xj we have
dividing through:
thus,
summing left and right hand sides over 'j' and using the fact that we have
where P is a price index represented as
Therefore, the Marshallian demand function is:
Under monopolistic competition, where goods are almost perfect substitutes prices are likely to be relatively close. Hence, assuming we have:
From this we can see that the indirect utility function will have the form
hence,
as σ > 1 we find that utility is strictly increasing in N implying that consumers are strictly better off as variety, i.e. how many products are on offer, increases.
The derivation can also be done with a continuum of varieties, with no major difference in the approach.[2]