The market for residential land in an urban area can be described by the following set of equations:

(1) lD(x) ¼ BR(x)
u Individual household’s demand curve for land at distance x (2) LD(x) ¼ N(x)lD(x) Market demand for land at distance x (3) R0 (x)lD(x) + t ¼ 0 Condition for locational equilibrium of the household (4) R(x) ¼ R Land rent at edge of city (5) LS(x) ¼ 2Px Supply of land at distance x, circumference of a circle (6) LS(x) ¼ LD(x) Market equilibrium at each distance x It is further assumed that the total number of households in the urban area is fixed at N.
The variables in the model are:
lD(x) ¼ quantity of land demanded by a household R(x) ¼ land rent at distance x u ¼ price elasticity of demand for land B ¼ a constant N(x) ¼ number of households located at distance x R0 (x) ¼ dR/dx ¼ slope of the rent function t ¼ commuting cost incurred by residing one mile farther from the CBD The model incorporates the following assumptions:
Households consume land, and consume less land per household where rent is higher [equation (1)].
All households must commute to the CBD, a dimensionless point at which all employment is located.
Alllandis consumed by households. Noland is needed fortransportation facilities [equation (5)].
The market for land is perfectly competitive. In equilibrium the rent at each distance [R(x)] equates supply and demand in that annulus.
Have you got all of that? Now here are the questions.

a. What is the meaning of equation (3)? Where does this equation come from?

b. Assuming that u ¼ 1 (unitary price elasticity of demand for land), derive the land-rent gradient R(x). This is actually a very simple question if you understand how to do it. If you find yourself performing an elaborate mathematical derivation, you are on the wrong track.

c. Derive the population density gradient N(x)/L(x).

d. What happens to the distribution of population over space if t (marginal transportation cost) rises?

e. How does this little mathematical model relate to the graphical model in Figure 7.7?