Question

larry haspreferences represented by utility U=a ln x +b ln y +y^c where a,b and care non negative . show what restrictions should be pit on a b and c   by each of the folow
homethetic
homothetic nd comvex
quasilinear and convex

Answer 1

1) Homothetic - MRS should be homogeneous of degree 0 or depends only on the ratio of the amounts of the two goods, not on the quantities of the goods.

Here if c=0, and a,b > 0 then the preferences will be homothetic.

2) Preferences are convex iff the corresponding utility function is quasi-concave. MRSxy is diminishing as y is given up and x is acquired which means the utility function is strictly quasi-concave. Similar restrictions would accomplish the above -  if c=0, and a,b > 0 then the preferences will be homothetic and convex.

3) Quasilinear preferences are linear in y, so the marginal utility is constant. MRS only depends on f(x)= alnx

a>0 , b=0 and c=1 would give us a utility function of the type,  

U= alnx + y