Question

Consider the following: Suppose that two neighbours with identical nonhomothetic preferences derive utility from the number of flower gardens in their shared community lot, x, and on the number of tomatoes that they eat from their own vegetable gardens, y. The specific form of the utility function is given by: U_i(x,y_i )=x+lny

The price of one flower garden is $1 and the price of a vegetable garden is $2. Each neighbour has a budget of $12.

a) If each neighbour lived in a community that did not have a shared community lot (i.e. had to individually purchase and own both flower and vegetable gardens), what portion of their income would they spend on flower and vegetable gardens, respectively?

b) What utility would they each receive from the allocation in a) above?

c) What is the marginal rate of substitution (MRS) for this utility function? Interpret the meaning of the MRS. Is MRS diminishing?

d) Now, assume the neighbours do, in fact, live in the shared community such that both share and enjoy the number of flower gardens that are available. If neighbour #1 assumes that neighbour #2 will not pay for any flower garden, what will be each neighbour’s utility if neighbour #1 buys the least whole number of flower garden(s) such that he derives utility from the flower garden(s)? Allow for fractions of vegetable gardens.

e) Show that the allocations described in d) are inefficient.

f) Calculate the efficient level of flower garden purchases and that of vegetable gardens.

g) Assume that the neighbour’s split (i.e. 50/50) the cost of efficient number of flower gardens obtained in f) and use the remaining funds to purchase vegetable gardens. Can the neighbours derive any utility from vegetable gardens given nonhomothetic preferences? What utility will each neighbour receive, and is this a Pareto superior solution than your answer in b)? Explain why or why not.

Answer 1

a) U = x + lnY..............(i)

For the equilibrium, MUx/MUy = Px/Py

where MUx is marginal utility of x and MUy is marginal utility of Y. Px is the unit price of the flower garden and Py is the unit price of the vegetable garden.

Px = 1 and Py = 2

The budget equation Px.X + Py.Y = 12 => X + 2Y = 12 ...........(ii)

from equation (i), MUx = 1 (by differentiating with respect to x)

and MUy = 1/Y (by differentiating with respect to Y)

Now, in equilibrium, slope of utility curve = slope of budget line

MUx/MUy = Px/Py

=> 1/ 1/y = 1/2

=> Y = 1/2

from eq (ii) => X = 12- 2 * 1/2 = 12 - 1 = 11

share of income on vegetable garden is Py.Y = 2* 1/2 = $1

and share of income on flower garden is Px.X = 1* 11 = $11

b) Utility derived by each neighbor from above allocation is U = x + lnY = 11+ ln (1/2) = 11 - 0.301 = 10.699

c) MRS is the marginal rate of substitution denoted by MUx/MUy

MRS for this utility function is 1/(1/y) = Y = 0.5

The slope of indifference curve and budget line is equal. Marginal rate of substitution is constant as the Utility function depict two allocation as perfect substitute. It means, the willingness of sacrificing vegetable garden in order to get 1 unit of flower garden is 0.5.

d) If neighbour 1 assumes that neighbour 2 will not pay for any flower garden, and if neighbour 1 purchases the least number of flower garden , then

neighbour 1 has $12 to expend on both allocation. Neighbour will have same utility as above i.e, 10.699

according to the assumpion of neibour 1, neighbour 2 enjoys the same utility as neighbour 1, without expending any dollar on flower garden.