5 Convex preferences

(a)

Show that the quasi-linear utility function given by

u(x, y) = logx+y

is a concave function. [Hint: Use the convex combination definition from the slides. You can use calculus to show thatlogis a concave function.]

(b)

i. Showthatu:R+R

u(x) =x2

is a convex function but it is quasi-concave.

ii. Is this true if we include negative numbers in the domain ofui.e.u:RR

and

u(x) =x2.

You can use a diagram to argue your case.