Lucetta consumes only two goods, AAA and BBB$.$ Her only source of income comes from gifts of these two goods from her many admirers. She does not always receive these goods in the proportions she desires to consume, but she can sell or buy AAA at a price of PA$=1$P$\_$A $=1$PA$=1$ and BBB at a price of PB$=2$P$\_$B $=2$PB$=2.$ Her utility function is U$($a$,$b$)=$abU$($a$,$ b$)=$ abU$($a$,$b$)=$ab$,$ where aaa is the quantity of AAA she consumes and bbb is the quantity of BBB she consumes.