Consider a linear city with housing:

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 y = 100.

In addition to non-housing goods, individuals living in the city consume housing.

Preferences are represented by a utility function U(c, q), where c is the consumption of non-housing goods and q is the consumption of housing. This function is assumed to be Cobb-Douglas: U(c, q) = c^(5)*q^(5)

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 km) from the CBD incurs a commuting cost of tx. Assume that t = 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. This is an open city. 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:

y = tx + P(x)q + c 100 = 10x + P(x)q + c

The consumer's problem is: Max {c,q,x} c^(5)*q^(5) subject to: 100 = 10x + P(x)q + c.

- Solve the optimal budget allocation between housing and non-housing goods at each location (for a given x).

- Obtain housing prices by ensuring that, with each consumer allocating optimally his disposable income, utility is equalised across locations in the city and is equal to their outside option U = 5 (this is the free mobility condition).