Quasilinear utility function

Quasilinear (and therefore parallel) indifference curves in the two-dimensional case.

A quasi-linear utility function is a special mathematical function that is used in economics, and in particular in microeconomics , to model the preferences of economic subjects (households, individuals).

Description and meaning

The quasi-linear utility function describes the special case that the utility function is a quasi-linear function . This means that the function is linear in one argument ( called Numéraire ). The function can depend on other arguments in different ways, but this only affects the ordinate intersection. Such a function would have the form: . The influence of the other goods on good 1 is therefore additively separable. ${\displaystyle u(x_{1},x_{2},\ldots ,x_{n})=x_{1}+V(x_{2},\ldots ,x_{n})}$

A possible justification for the assumption that a good is used linearly as numéraire could be that this good takes on the function of a medium of exchange (these goods are also often referred to as money). ${\displaystyle x_{1}}$

In the special case of quasi-linear utility, there is no income effect , i.e. H. Changes in income do not affect demand. Quasilinear utility functions are used, among other things, to model subsistence goods .

If the function is differentiable and the function and the preferences are monotonic, then for the two-dimensional case it can be shown that the marginal rate of substitution does not depend on the amount of consumption of the numéraire good. ${\displaystyle V}$

example

An example of such a function is shown in. The corresponding indifference curves in the goods diagram are . In the two-dimensional case, this also means that the corresponding indifference curves are parallel. In this example, this is the case for various constant utility levels (see figure). The marginal rate of substitution can be expressed by the ratio of the individual marginal utility , in this case it follows: ${\displaystyle U(x,y)={\sqrt {x}}+y}$ ${\displaystyle y={\overline {U}}-{\sqrt {x}}}$

${\displaystyle GRS={\frac {GN_{x}}{GN_{y}}}={\frac {\frac {d{\sqrt {x}}}{dx}}{1}}={\frac {1}{2{\sqrt {x}}}}.}$

As described, the marginal rate is independent of the Numéraire property.

Definition via preferences

The preference relation to is called quasi-linear with regard to good 1 (called Numéraire), if the following applies: ${\displaystyle \precsim }$ ${\displaystyle X=(-\infty ,\infty )\times \mathbb {R} _{x}^{L-1}}$

1. All indifference curves are parallel to each other with respect to the axis of good 1. That is, for applies to and everything . ${\displaystyle x\sim y}$ ${\displaystyle (x+ae_{1})\sim (y+ae_{1})}$ ${\displaystyle e_{1}=(1,0,\dots ,0)}$ ${\displaystyle a\in \mathbb {R} }$
2. Good 1 is desirable (not an evil), that is, it applies to everyone . ${\displaystyle x+ae_{1}\succ x}$ ${\displaystyle x,a>0}$

Individual evidence

1. ↑ Hens, Thorsten, and Paolo Pamini. Basics of analytical microeconomics. Springer-Verlag, 2008. p. 179.
2. ↑ Varian, Hal R. Fundamentals of Microeconomics. Walter de Gruyter GmbH & Co KG, 2011. p. 282.
3. ↑ Wiese, Harald. Microeconomics: an introduction to 376 problems. Springer-Verlag, 2010. p. 54.
4. ↑ Mas-Colell, Andreu, Michael Dennis Whinston, and Jerry R. Green. Microeconomic theory. Vol. 1. New York: Oxford university press, 1995. p. 45.