Consider Jen, a consumer with preferences U(H,F) = F1/3H2/3, where H is the quantity of housing and ...

Consider Jen, a consumer with preferences U(H,F) = F1/3H2/3, where H is the quantity of housing and F is the quantity of food (per month). Suppose Jen has a stipend of $600/month which she uses to purchase food at a price of $1/unit and housing at a price of $10/unit.
a.   Compute Jen's utility-maximizing bundle of goods.
b.   Suppose that Jen's employer subsidizes housing by paying 50% of her total housing costs, thereby effectively lowering the price Jen pays for housing to $5/unit. Compute Jen's new optimal consumption bundle.
c.   How much does Jen's employer pay in total for this subsidy? How much utility does Jen enjoy with this subsidy (compute her utility at the optimal bundle).
d.   Suppose that her employer simply gave Jen the dollar cost you found in (c) as a lump sum (instead of subsidizing housing). Will Jen gain a higher utility from the housing subsidy or the lump-sum equivalent transfer?