Problem Set ECO 333 (Urban Economics)

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}

g

.5h

.5

subject to: 100 = 10x + P(x)h + g

(3)

Note: Relative to the standard consumer problem studied in other economics courses, there are

two main differences here:

• Residents must choose their location of residence as well as allocate their disposable income

optimally between housing and non-housing goods.

• The price of housing, and thus the budget constraint they face, varies with their location

choice.

Questions

1. Write down the free mobility condition.

2. Derive the indifference curve equation (Express g as a function of h using the free mobility

condition).

3. What is the slope of the indifference curve?

4. Utility maximization requires that the slope of the indifference curve should be equal to the

slope of the budget constraint. Derive the utility maximization condition.

5. In equilibrium, the free mobility condition, the utility maximization condition, and the

budget constraint must hold with equality. Use these three equations to solve for the

equilibrium housing consumption (as a function of x and exogenous parameters of the

model).

6. True or False. Houses are smaller the closer you get to the CBD. Hint: if you live downtown

Toronto, this one should be obvious.

7. Using again the utility maximization condition, derive an expression for the equilibrium

housing prices as a function of x and exogenous parameters of the model.

8. True or False. Housing prices decrease with the distance from the CBD.