Dixit–Stiglitz model is a model of
monopolistic competition
Monopolistic competition is a type of imperfect competition such that there are many producers competing against each other, but selling products that are differentiated from one another (e.g. by branding or quality) and hence are not perfec ...
developed by
Avinash Dixit
Avinash Kamalakar Dixit (born 6 August 1944) is an Indian-American economist. He is the John J. F. Sherrerd '52 University Professor of Economics Emeritus at Princeton University, and has been Distinguished Adjunct Professor of Economics at Lin ...
and
Joseph Stiglitz (1977).
It has been used in many fields of economics including
macroeconomics,
economic geography
Economic geography is the subfield of human geography which studies economic activity and factors affecting them. It can also be considered a subfield or method in economics.
There are four branches of economic geography.
There is,
primary secto ...
and
international trade theory. The model formalises consumers' preferences for product variety by using a
CES function. Previous attempts to provide a model that accounted for variety preference (such as
Harold Hotelling's
location model) were indirect and failed to provide an easily interpretable and usable form for further study. In the Dixit-Stiglitz model, variety preference is inherent within the assumption of
monotonic preferences because a consumer with such preferences prefers to have an average of any two bundles of goods as opposed to extremes.
Mathematical Derivation
The model begins with a standard
CES utility function:
where N is the number of available goods, x
i is the quantity of good i, and σ is the
elasticity of substitution Elasticity of substitution is the ratio of percentage change in capital-labour ratio with the percentage change in Marginal Rate of Technical Substitution. In a competitive market, it measures the percentage change in the two inputs used in respons ...
. Placing the restriction that σ > 1 ensures that preferences will be
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
and thus
monotonic
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
for over any optimising range. Additionally, all
CES functions are
homogeneous of degree 1 and therefore represent
homothetic preferences
In consumer theory, a consumer's preferences are called homothetic if they can be represented by a utility function which is homogeneous of degree 1. For example, in an economy with two goods x,y, homothetic preferences can be represented by a ut ...
.
Additionally the consumer has a
budget set
In economics, a budget set, or the opportunity set facing a consumer, is the set of all possible consumption bundles that the consumer can afford taking as given the prices of commodities available to the consumer and the consumer's income. Let the ...
defined by:
For any rational consumer the objective is to maximise their utility functions subject to their budget constraint (M) which is set
exogenously
In a variety of contexts, exogeny or exogeneity () is the fact of an action or object originating externally. It contrasts with endogeneity or endogeny, the fact of being influenced within a system.
Economics
In an economic model, an exogen ...
. Such a process allows us to calculate a consumers
Marshallian Demands. 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 decreasing.
By using a
Lagrange multiplier
In mathematical optimization, the method of Lagrange multipliers is a strategy for finding the local maxima and minima of a function subject to equality constraints (i.e., subject to the condition that one or more equations have to be satisfied e ...
we can convert the above primal problem into the dual below (see
Duality)
Taking
first order conditions of two goods x
i and x
j 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
In microeconomics, a consumer's Marshallian demand function (named after Alfred Marshall) is the quantity they demand of a particular good as a function of its price, their income, and the prices of other goods, a more technical exposition of the ...
is:
Under
monopolistic competition
Monopolistic competition is a type of imperfect competition such that there are many producers competing against each other, but selling products that are differentiated from one another (e.g. by branding or quality) and hence are not perfec ...
, 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
__NOTOC__
In economics, a consumer's indirect utility function
v(p, w) gives the consumer's maximal attainable utility when faced with a vector p of goods prices and an amount of income w. It reflects both the consumer's preferences and market con ...
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.
References
Further reading
*
Monopoly (economics)
Economics models
{{econ-stub