What kind of preferences are represented by a utility...

What kind of preferences are represented by a utility function of the form
$u\left(x_{1}, x_{2}\right)=x_{1}+\sqrt{x_{2}} ?$ Is the utility function $v\left(x_{1}, x_{2}\right)=x_{1}^{2}+2 x_{1} \sqrt{x_{2}}+x_{2}$
a monotonic transformation of $u\left(x_{1}, x_{2}\right) ?$

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Key Concepts

Quasilinear Preferences

Quasilinear preferences are represented by utility functions where one variable enters linearly and the other nonlinearly, as in u(x1, x2) = x1 + ?x2. This structure implies that one good (here x1) has a constant marginal utility, while the other (x2) exhibits diminishing marginal utility, meaning that changes in income do not affect the consumption of the good that enters linearly. These preferences often simplify analysis by eliminating income effects for one of the goods.

Monotonic Transformations

Monotonic transformations refer to operations on a utility function that preserve the ordering of preferences without altering the underlying choice behavior. When a utility function is transformed by a strictly increasing function, as in v(x1, x2) = (x1 + ?x2)², the new utility function represents the same ordinal preferences as the original one. This property emphasizes that utility is an ordinal measure rather than an absolute one, and only the ranking of different bundles matters in decision making.

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