Dixit–Stiglitz model

The model begins with a standard CES utility function:

${\displaystyle u=\left[\sum _{i=1}^{N}x_{i}^{\frac {\sigma -1}{\sigma }}\right]^{\frac {\sigma }{\sigma -1}}}$

where N is the number of available goods, x_i 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:

${\displaystyle B=\{{\boldsymbol {x}}:\sum _{i=1}^{N}p_{i}x_{i}\leq M\}}$

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:

${\displaystyle \max\{u=[\sum _{i=1}^{N}x_{i}^{\frac {\sigma -1}{\sigma }}]^{\frac {\sigma }{\sigma -1}}\}\ st.\ {\boldsymbol {x}}\in B}$

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:

${\displaystyle \max\{u=\sum _{i=1}^{N}x_{i}^{\frac {\sigma -1}{\sigma }}\}\ st.\ {\boldsymbol {x}}\in B}$

since ${\displaystyle f(u)=u^{\frac {\sigma -1}{\sigma }}}$ is strictly increasing.

By using a Lagrange multiplier we can convert the above primal problem into the dual below (see Duality)

${\displaystyle \nabla =\sum _{i=1}^{N}x_{i}^{\frac {\sigma -1}{\sigma }}-\lambda [\sum _{i=1}^{N}p_{i}x_{i}-M]}$

Taking first order conditions of two goods x_i and x_j we have

${\displaystyle \nabla x_{i}={\frac {\sigma -1}{\sigma }}x_{i}^{-{\frac {1}{\sigma }}}-\lambda p_{i}=0}$

${\displaystyle \nabla x_{j}={\frac {\sigma -1}{\sigma }}x_{j}^{-{\frac {1}{\sigma }}}-\lambda p_{j}=0}$

dividing through:

${\displaystyle ({\frac {x_{i}}{x_{j}}})^{-{\frac {1}{\sigma }}}={\frac {p_{i}}{p_{j}}}}$

thus,

${\displaystyle p_{j}x_{j}=p_{i}^{\sigma }x_{i}p_{j}^{1-\sigma }}$

summing left and right hand sides over 'j' and using the fact that ${\displaystyle \sum _{j=1}^{N}p_{j}x_{j}=M}$ we have

${\displaystyle M=p_{i}^{\sigma }x_{i}P^{1-\sigma }}$

where P is a price index represented as ${\displaystyle P=(\sum _{j=1}^{N}p_{j}^{1-\sigma })^{\frac {1}{1-\sigma }}}$

Therefore, the Marshallian demand function is:

${\displaystyle x_{i}={\frac {M}{P}}({\frac {p_{i}}{P}})^{-\sigma }}$

Under monopolistic competition, where goods are almost perfect substitutes prices are likely to be relatively close. Hence, assuming ${\displaystyle p_{i}=p}$ we have:

${\displaystyle x_{i}^{m}(\mathbf {p} ,M)={\frac {M}{Np}}}$

From this we can see that the indirect utility function will have the form

${\displaystyle v(\mathbf {p} ,x_{i}^{m})=\left(\sum _{i=1}^{N}\left({\frac {M}{Np}}\right)^{\frac {\sigma -1}{\sigma }}\right)^{\frac {\sigma }{\sigma -1}}}$

hence,

${\displaystyle v(\mathbf {p} ,x_{i}^{m})={\frac {M}{p}}N^{\frac {1}{\sigma -1}}}$

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.

• Brakman, Steven; Heijdra, Ben J., eds. (2001). The Monopolistic Competition Revolution in Retrospect. Cambridge University Press. ISBN 0-521-81991-1.