In
economics and
consumer theory
The theory of consumer choice is the branch of microeconomics that relates preferences to consumption expenditures and to consumer demand curves. It analyzes how consumers maximize the desirability of their consumption as measured by their pref ...
, quasilinear
utility
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosophe ...
functions are linear in one argument, generally the
numeraire. Quasilinear preferences can be represented by the utility function
where
is strictly
concave
Concave or concavity may refer to:
Science and technology
* Concave lens
* Concave mirror
Mathematics
* Concave function, the negative of a convex function
* Concave polygon, a polygon which is not convex
* Concave set
In geometry, a subset o ...
.
A useful property of the quasilinear utility function is that the Marshallian/Walrasian demand for
does not depend on wealth and is thus not subject to a
wealth effect;
The absence of a wealth effect simplifies analysis
and makes quasilinear utility functions a common choice for modelling. Furthermore, when utility is quasilinear, compensating variation (CV), equivalent variation (EV), and
consumer surplus
In mainstream economics, economic surplus, also known as total welfare or total social welfare or Marshallian surplus (after Alfred Marshall), is either of two related quantities:
* Consumer surplus, or consumers' surplus, is the monetary gain ...
are algebraically equivalent.
In
mechanism design
Mechanism design is a field in economics and game theory that takes an objectives-first approach to designing economic mechanisms or incentives, toward desired objectives, in strategic settings, where players act rationally. Because it starts a ...
, quasilinear utility ensures that agents can compensate each other with side payments.
Definition in terms of preferences
A
preference relation The term preference relation is used to refer to orderings that describe human preferences for one thing over an other.
* In mathematics, preferences may be modeled as a weak ordering or a semiorder, two different types of binary relation. One speci ...
is quasilinear with respect to commodity 1 (called, in this case, the ''numeraire'' commodity) if:
* All the indifference sets are parallel displacements of each other along the axis of commodity 1. That is, if a bundle "x" is indifferent to a bundle "y" (x~y), then
* Good 1 is desirable; that is,
In other words: a preference relation is quasilinear if there is one commodity, called the numeraire, which shifts the indifference curves outward as consumption of it increases, without changing their slope.
In two dimensional case, the indifference curves are
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of IB ...
; which is useful because the entire utility function can be determined from a single indifference curve.
Definition in terms of utility functions
A
utility function
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
is quasilinear in commodity 1 if it is in the form
:
where
is an arbitrary function.
In the case of two goods this function could be, for example,
The quasilinear form is special in that the
demand function
In economics, a demand curve is a graph depicting the relationship between the price of a certain commodity (the ''y''-axis) and the quantity of that commodity that is demanded at that price (the ''x''-axis). Demand curves can be used either for t ...
s for all but one of the consumption goods depend only on the prices and ''not'' on the income. E.g, with two commodities with prices ''p
x'' = 1 and ''p
y'' , if
:
then, maximizing utility subject to the constraint that the demands for the two goods sum to a given income level, the demand for ''y'' is derived from the equation
:
so
:
which is independent of the income ''I''.
The
indirect utility function in this case is
:
which is a special case of the
Gorman polar form Gorman polar form is a functional form for indirect utility functions in economics.
Motivation
Standard consumer theory is developed for a single consumer. The consumer has a utility function, from which his demand curves can be calculated. The ...
.
Equivalence of definitions
The
cardinal
Cardinal or The Cardinal may refer to:
Animals
* Cardinal (bird) or Cardinalidae, a family of North and South American birds
**'' Cardinalis'', genus of cardinal in the family Cardinalidae
**'' Cardinalis cardinalis'', or northern cardinal, t ...
and
ordinal definitions are equivalent in the case of a
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 polyto ...
consumption set with
continuous preferences that are
locally non-satiated
In microeconomics, the property of local nonsatiation of consumer preferences states that for any bundle of goods there is always another bundle of goods arbitrarily close that is strictly preferred to it.''Microeconomic Theory'', by A. Mas-Colel ...
in the first argument.
See also
*
Quasiconvex function
*
Linear utility In economics and consumer theory, a linear utility function is a function of the form:
::u(x_1,x_2,\dots,x_m) = w_1 x_1 + w_2 x_2 + \dots w_m x_m
or, in vector form:
::u(\overrightarrow) = \overrightarrow \cdot \overrightarrow
where:
* m is the n ...
function - a special type of a quasilinear utility function.
References
{{Reflist
Financial economics
Utility function types