# Homework Deriving the Bid Rent for Housing Urban and Spatial by eddie21

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```									             Homework #3: Deriving the Bid Rent for Housing
Urban and Spatial Economics
Swiss Federal Institute of Technology
Due October 28, 2008
Prof. Thomas Rutherford
(from Kurt Schmidheiny)

Consider an urban household that derives utility from the consumption of housing, s, and of another good,
z, whose price is 1. In particular, the household’s utility function is given by
U (s, z) = s1/3 z 2/3                                       (1)
The household earns income, Y , and pays transport costs t for each unit of distance r it needs to travel to
the city’s central business district (CBD). For each unit of housing it occupies at distance r from the CBD,
it needs to pay rent R(r). Its budget constraint is thus given by
Y − tr = z + R(r)s.                                           (2)
Let Y = 1600 and t = 50.
a) Maximise the household’s utility subject to its budget constraint and give the optimal demand for
housing s and consumption z.
b) Use the budget constraint (2) to obtain an expression for R(r) in terms of s, z and r. Then,
substitute z using the constraint
s1/3 z 2/3 = u
¯                                          (3)
∗
¯
and maximise R(r) with respect to s. The resulting s will be a function of r and u. This is the
¯
dwelling size function s(r, u); make sure that you understand why. Explain intuitively in one sentence
what the dwelling size function means.
c) Substitute s∗ into the expression for R(r) and show that it has the form
3
4    Y − tr
¯
ψ(r, u) =                     .                              (4)
27     ¯
u

This function of r and u is the bid rent function for housing; make sure that you understand why.
Explain intuitively in one sentence what the bid rent function means.
¯
d) Let u = 400. What is the bid rent at r = 0, 8 and 12? Draw the bid rent curve.
¯
e) Let r = 8. What is the bid rent for u = 200, 400 and 800? Draw the bid rent curves for all three
levels of utility.
f) Look at a household in r = 8 that faces housing rent R(8) = ψ(8, 400), i.e. the second number you
calculated in d). Calculate the optimal consumption of housing s and the other good z. Calculate the
utility level this household reaches.

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