Quasilinear Utility
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In
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics analy ...
and consumer theory, 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 philosoph ...
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. 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 caus ...
; 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, quasilinear utility ensures that agents can compensate each other with side payments.


Definition in terms of preferences

A preference relation \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; 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 functions 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 function in this case is :v(p,I) = v(p) + I, which is a special case of the Gorman polar form.


Equivalence of definitions

The cardinal 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 polytop ...
consumption set with continuous preferences that are locally non-satiated in the first argument.


See also

* Quasiconvex function * Linear utility function - a special type of a quasilinear utility function.


References

{{Reflist Financial economics Utility function types