Quasilinear utility
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In
economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...
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 philosopher ...
functions are linear in one argument, generally the numeraire. Quasilinear preferences can be represented by the utility function u(x_1, x_2, \ldots, x_n) = x_1 + \theta (x_2, \ldots, x_n) where \theta 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 * The concavity In ca ...
. A useful property of the quasilinear utility function is that the Marshallian/Walrasian demand for x_2, \ldots, x_n does not depend on wealth and is thus not subject to a
wealth effect The wealth effect is the change in spending that accompanies a change in perceived wealth. Usually the wealth effect is positive: spending changes in the same direction as perceived wealth. Effect on individuals Changes in a consumer's wealth cause ...
; 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 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 ...
\succsim 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 \left ( x+ \alpha e_1 \right ) \sim \left ( y+ \alpha e_1 \right ), \forall \alpha \in \mathbb, e_1= \left ( 1,0,...,0 \right ) * Good 1 is desirable; that is, \left ( x+ \alpha e_1 \right ) \succ \left ( x \right ), \forall \alpha>0 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 ...
; 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 : u \left ( x_1, \dots ,x_L\right ) = x_1 + \theta \left (x_2, ..., x_L \right ) where \theta is an arbitrary function. In the case of two goods this function could be, for example, u \left ( x,y \right ) = x + \sqrt . 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 ''px'' = 1 and ''py'' , if : u ( x,y ) = x + \theta(y) 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 :\theta^\prime (y) = p_y so :y(p,I) = (\theta^\prime)^(p_y), which is independent of the income ''I''. The
indirect utility __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 co ...
function in this case is :v(p,I) = v(p) + I, 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, the ...
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 polytope ...
consumption set with
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** 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 In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form (-\infty,a) is a convex set. For a function of a sing ...
*
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