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:
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: pQ(x)=wE+v(x)L
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 x.
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)