Question

In the city of Urbanville, land is divided between supermarkets and houses (for residents). There is a train station at x = 0.

- Urbanville residents all work from home. Their utility of living in the city is U = 20 − R, where R is the rent they pay. These residents also have the option of moving out of the city and living far away from Urbanville. Doing so gives them a utility of U₀ = 18.

- Supermarkets receive goods from the central train station. Their profits are π = P_g − 2d − R. In the profit equation, d stands for the distance to the train station.

- The price of groceries is P_g= 10.

(a) Draw the bid rent curves for supermarkets and houses (b_s and b_h).

(b) Find x* , such that land from 0 to x* is used for supermarkets and land farther away than x* is used for houses.
(c) Draw the new bid-rent curve for supermarkets (b'_s ) and find x₁, the new boundary between supermarkets and houses.

Transportation costs fall, such that supermarket profits are now given by π' = P_g − d − R.

(d) On the same diagram as before, draw the new bid-rent curve for supermarkets (b'_s ) and find x1, the new boundary between supermarkets and houses.

Answer 1

(a)

Bid-rent curve for houses: The maximum rent households would be willing to pay to live in Urbanville is R=2. This is because if they were charged a higher rent, suppose, R=3, their utility would be :

U = 20-R

U = 20-3

= 17.

This is less than the utility they wouldobtain by living outside the city, so they would prefer to move out. So the y-intercept of household bid-rent curve is 2. The x-intercept will be given by the household distance from the train station when R=0, i.e., when household is situated outside Urbanville and thus pays zero rent. This distance will be given by the distance of the train station from the boundary of Urbanville. Let's suppose this radius is r. The bid-rent curve will extend further from r on the x-axis since no rent is to be paid for living anyhere outiside Urbanville. The following figure shows household bid-rent curve.

Bid-rent curve for supermarkets: The profit function is given by (since P_g=10)

The maximum rent will be paid when d=0, i.e., supermarket is situated closent to the train station. Supermarket will pay rent to the extent of its profits. So, maximum rent can be obtained by solving:

So, maximum R paid by supermarket is 10. This is the y-intercept.

To find the maximum distance at which the supermarket will be willing to locate, we can solve the profit equation assuming R=0: .

This gives d = 5. Thus the supermarket bid-rent curve is given by the following figure:

(b)

The distance x^* will be given by the intersection of the household and superarket bid rent curves.

Equation of household bid-rent curve bh is given by: d + (r/2)*R = r. This can be simplified as: 2d + rR = 2r.

Equation for supermarket bid-rent curve b_s is given by: 2d + R = 10.

Solving the two equations:

Thus, the distance x^* = d^* = 4r / (r - 1)

(c)

New profit function is  π' = P_g − d − R. Assuming P_g=10. Using the same logic as in part (a), maximum rent paid by supermarket when d=0 is R=10. The maximum distance at which the supermarket will be willing to locate is obtained by solving: π' = P_g − d − 0 = 0. So, at R = 0, d = 10. This gives the new bid rent curve for the supermarket. The curve for households remains unchanged. Curves are shown in the figure:

x1 will be given by solving the new bid rent curve equations:

b_h : 2d + rR = 2r, and

b_s: d + R = 10

Solving the two equations, we get x1 = 8r / (r - 2).

(d) Following figure shows the two bid rent curves after change in transportation costs: