Can Quasi-Linear functions be monotonic?

I want to ask that Can Quasi-Linear utility functions satisfy the more is better (monotonicity) assumption?

Yes, a trivial example is the following utility:

$$u(x,y) = x+ \sqrt{y}$$

which is both quasi linear and monotonic.

• I got it, so I want to ask one more Qn. How do I know that does the utility function satisfy monotonicity assumption? Oct 8 '20 at 13:42
• @SteveJosh a monotonic function must be either nondecreasing or nonincreasing (in case of utility it will virtually always be nondecreasing). Hence you have to check if the function is never decreasing (i.e. always increasing or staying constant) over its argument in its domain. A way how to check this is to calculate first derivative with respect to each variable and also cross-derivative and see what is the sign. In this case the function is always increasing in both x and y over its domain (since its only defined for $y>0$)
– 1muflon1
Oct 8 '20 at 13:47
• So, Can I say that $U(x,y) = - \alpha lnx - \beta lny$ is not satisfying this assumption? because I thought that when x and y increase Utility is always decreasing. ($\alpha and \beta$ are given) Oct 8 '20 at 13:53
• @SteveJosh that utility is not quasi-linear but logarithmic. Also that utility will be monotonically always decreasing over domain $x>0,y>0$
– 1muflon1
Oct 8 '20 at 14:01