04 Jun Question Problem Set ECO 333 (Urban Economics)
Question
Problem Set
ECO 333 (Urban Economics)
When you write up your answers, your goals should be to (1) be correct, and (2) convince the
TA that your answer is correct. To achieve these goals, your work should be legible and all steps
should be clearly presented. Answers which do not achieve these goals will not be awarded full
credit.
Consider a rectangular city with housing1
:
• Employment and consumption of non-housing goods take place at a single location x = 0,
the Central Business District (CBD).
• Every resident commutes to the CBD everyday and gets an exogenous wage of w = 100.
• In addition to non-housing goods, individuals living in the city consume housing.
• Preferences are represented by a utility function U(g, h), where g ($/month spent on nonhousing
goods) is the consumption of non-housing goods and h(square feet floor area) is
the consumption of housing. This function is assumed to be Cobb-Douglas:
U(g, h) = g
.5h
.5 (1)
• Assume that the cost of commuting is strictly monetary and increases linearly with distance
to the CBD, so that a resident living at distance x (in miles) from the CBD incurs a
commuting cost of ?x. Assume that ? = 10.
• Land covered by the city is endogenously determined in the model and is represented by
the segment on the positive real line between [0, x¯].
• Residents are assumed to be identical, with an exogenous daily utility level U¯ determined
outside the model. Assume that U¯ = 5
Let P(x) be the rental price of housing at a distance x from the CBD, the representative consumer’s
budget constraint is:
w = tx + P(x)h + g
??
100 = 10x + P(x)h + g
(2)
1
(See O’Sullivan p. 139-143 for an introduction to this model)
1
The consumer’s problem is:
Max
{g,h,x}
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