A city is comprised of firms in a competitive industry (with no
externalities) that produced a good or services that sells for
price of p, which is taken as given by firms and does not vary with
location. Output of the firms is represented by a Cobb-Douglas
production function:

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Q(x) = A(x)E^alpha*L^(1-alpha)

where x is the distance to the city center, E is labor (# of
workers), L is land and A represents “total factor productivity“-it
represents the state of technology and therefore determines the
productivity of labor and land together. We assume that A'(x)<0,
so that due to knowledge spillovers and other agglomeration
effects, technology is more productive the closer the firm is to
the city center. This generates a mechanism whereby firms will have
incentives to locate closer to the city center, as households do in
the textbook version of the monocentric city model.

We also assume that labor earns wage w and that land earns rent
v(x). Note that we assume that v depends on x, allowing land rent
to vary by distance from city center. Since this is a competitive
industry, in long-run equilibrium, firms must earn zero economic
profit. So total revenue and total cost must equal:

a) Note that since this is competitive industry with constant
returns to scale, the problem facing a representative firm is
identical to the problem facing the industry as a whole. So, we
solve the problem for the industry here. Set up the maximization
problem and solve for the first order conditions for E, L, and

b) Demonstrate that the bid-rent function has the normal,
negative slope.

c) Combine the first order conditions for E and L and derive an
expression showing that land density, E/L, as a function of v(x)
and w. Interpret this expression. (Divide one FOC by the other, get
expression for E/L and then take natural logs)

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