Question

We consider a monocentric city represented by a segment x 2 [0,xf ] where 0 stands for the CBD where everybody works and earns a uniform (exogenous) urban wage w >0 and xf is the city fringe. The rural wage a is normalized to zero. Agents located at a distance x from the CBD pay a rent R(x). The utility of an agent is her disposable income.

1 Fixed commuting cost
1.1 Private transportation
Agents drive to the CBD and incur a fixed commuting cost cc >0 per unit of distance.
1. Find the general formula for R(x) and xf .
2. What is the impact of each parameter of the model on these two outcomes?
3. Graph the rent function if cc =4 and w =100.
1.2 A metro station
Public authorities build one metro station located at point m such that 0 <m <xf . To go to the CBD, agents now either still drive to the CBD or combine private driving to the metro station and metro. Using the metro, the commuting cost per unit of distance is cm <cc .
1. Find the general formula for R(x) and xf . What is the impact of each parameter of the model on these two outcomes?
2. Graph the rent function if cc =4, cm =1, w =100 and m =20.
3. Let cc =4, cm =1, w =100 and m =50. Graph the rent function. What is the particularity of the
city in this case?
4. Can you give examples of the previous case in real life?
5. Let cc =4, cm =1, w =100. What is the size of the largest continuous city that can be achieved with only one metro station? Graph the rent function.
6. In real life, why might urban planners want to create continuous cities?

Answer 1

We need a spatial equilibrium condition s.t. a stable number of people choose to become wage workers and farmers: y(x) = w, for x ≤ x¯ Then the von Thünen rent is the maximum rent a farmer could pay at x before making a loss: P(x) = p − w − τx, for x ≤ x¯ Rent decreases with distance to the city. If we assume that beyond x¯ the rent is zero, i.e. P(¯x) = 0, we get the radius of arrable land as x¯ = p − w τ Higher price p or lower transport τpushes the maximal distance x¯further out. Florian Oswald (Sciences Po) Introduction to the monocentric Urban Model 5 / 65 Land Use - von Thünen Which Crops are planted where? Suppose we have multiple crops i with pi > pi+1 and τi > τi+1 Farmers will put land to it’s most productive use. Higher yield crops that are more expensive to transport are produced closer to the market. Dairy Farming This produces a rent function that is convex over distance. Could have setup with different labor intensity for producing different products. most labor intensive product is closest to city Florian Oswald (Sciences Po) Introduction to the monocentric Urban ModeLand Use - von Thünen Von Thünen Rings Figure: http://postoilgeography.blogspot.fr/2012/10/ remembering-von-thunen.html Florian Oswald (Sciences Po) Introduction to the monocentric Urban Model 7 / 65 Urban Land Use The Monocentric City The Monocentric Model of the City 1 We assume a city has one unique center, the central business district, CBD, where all firms are. 2 The shape of the city could be circular, or a line. We will work with a line. (It’s a line segment on R) 3 The CDB is represent by a point x = 0. 4 All workers have to commute to the CDB to work, and they face commuting costs. 5 They have to acquire housing services. 6 This model allows us to study how house prices vary with distance from the CDB, along with housing consumption, land prices, construction density and population density.