In economics and consumer theory, quasilinear utility functions are linear in one argument, generally the numeraire. Quasilinear preferences can be represented by the utility function
u(x1,x2,\ldots,xn)=x1+\theta(x2,\ldots,xn)
\theta
x2,\ldots,xn
\succsim
\left(x+\alphae1\right)\sim\left(y+\alphae1\right),\forall\alpha\inR,e1=\left(1,0,...,0\right)
\left(x+\alphae1\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.
A utility function is quasilinear in commodity 1 if it is in the form
u\left(x1,...,xL\right)=x1+\theta\left(x2,...,xL\right)
\theta
u\left(x,y\right)=x+\sqrt{y}.
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)=py
y(p,I)=(\theta\prime)-1(py),
The indirect utility function in this case is
v(p,I)=v(p)+I,
The cardinal and ordinal definitions are equivalent in the case of a convex consumption set with continuous preferences that are locally non-satiated in the first argument.