Consider the following: Suppose that two neighbours withidentical nonhomothetic preferences derive utility from the numberof flower gardens in their shared community lot, x, and on thenumber of tomatoes that they eat from their own vegetable gardens,y. The specific form of the utility function is given by:Ui(x,yi )=x+lny
The price of one flower garden is $1 and the price of avegetable garden is $2. Each neighbour has a budget of $12.
a) If each neighbour lived in a community that did not have ashared community lot (i.e. had to individually purchase and ownboth flower and vegetable gardens), what portion of their incomewould they spend on flower and vegetable gardens, respectively?
b) What utility would they each receive from the allocation ina) above?
c) What is the marginal rate of substitution (MRS) for thisutility function? Interpret the meaning of the MRS. Is MRSdiminishing?
d) Now, assume the neighbours do, in fact, live in the sharedcommunity such that both share and enjoy the number of flowergardens that are available. If neighbour #1 assumes that neighbour#2 will not pay for any flower garden, what will be eachneighbour’s utility if neighbour #1 buys the least whole number offlower garden(s) such that he derives utility from the flowergarden(s)? Allow for fractions of vegetable gardens.
e) Show that the allocations described in d) areinefficient.
f) Calculate the efficient level of flower garden purchases andthat of vegetable gardens.
g) Assume that the neighbour’s split (i.e. 50/50) the cost ofefficient number of flower gardens obtained in f) and use theremaining funds to purchase vegetable gardens. Can the neighboursderive any utility from vegetable gardens given nonhomotheticpreferences? What utility will each neighbour receive, and is thisa Pareto superior solution than your answer in b)? Explain why orwhy not.
