Von-Thünen Model I

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QUESTION
 
A company is located at M (where x=0). The company is forced to charge marginal cost
prices for its product (i.e., c=p). Customers walk to the company to buy its products and,
aside from the purchase price, incur travel cost of (tx2
). The firm’s market reach is 9.1km
and marginal travel cost equals t=1.5.
If customers’ WTP for the product were $124.22 what is the product’s price in
equilibrium?
(2) Von-Thünen Model II
Farmers that grow wheat need to sell their product at the market M (where x=0).
Marginal cost of growing equals c=$10, marginal travel cost is t=2 (per mile and ton) and
the market’s WTP for wheat is $100 per ton.
For each distance point x=10, x=30, and x=50, say whether anybody would grow wheat
or not and what the resulting land rent would be. Note, rents cannot be negative.
(3) Von-Thünen Model III
Refer to the von-Thünen model where a city’s market reaches to the left and to the right.
The firm is competitive and its price equals marginal cost. The firm’s marginal cost is
c=1; the market reach is x=14 to the left and x=14 to the right. Marginal transportation
cost is 6.
(a) What is customers’ WTP per unit of the good in question?
(b) What is the rent for a person located at x=10?
(4) Von-Thünen Model with multiple cities
There are two cities, A and B, where B is located 14 miles east of A. Both produce an
identical good and consumers’ WTP=18 is the same everywhere. Marginal transportation
cost is identical tB=tA=2. Each city prices its good at marginal cost. We know that cB=2;
cA is unknown.
(a) What is the maximum market reach of city B to the west?
(b) Now assume there are unserved customers to the west of B (i.e., between A and B).
Calculate the minimum cA for this to be true.
(5) Locations of Transfer-Oriented Firms
The distance between the resource (R ) and the market (M) is 10 miles. A firm’s
procurement cost is given by PC=2.2×2
, where x is the distance from the resource
measured in miles. The firm’s distribution cost is given by DC=5*(10-x). Again, x is the
distance from the resource.
(a) What is the Total Freight Cost (TC) at each milestone. This question assumes there
are only 11 possible discrete locations, i.e., x=0, 1, 2, 3, …10. Locations between the
milestones are not possible. For TC, one decimal is sufficient. Where should the firm
locate?
(b) Now assume, the firm can locate between the milestones, for instance at x=5.145.
Provide a mathematically exact solution for the optimal location (with 3 decimals). What
is the optimal x? At this location, what is PC, DC and TC (three decimals).
(c) How does your answer to all parts under (b) change if the marginal procurement cost
PC were equal to zero everywhere?
(6) Median Location
Assume customers are distributed on a one-directional path (e.g., to the east) as follows.
Location A B C D E F
Distance from A 0 1 8 2 6 18
Number of customers 3 2 1 2 3 6
A company has to deliver a unit of its good to each customer and incurs a cost of $1 per
unit and mile. Each unit needs to delivered separately.
Part 1:
(a) Where is the median customer located?
(b) Where is the delivery cost minimizing location and what is its delivery cost?
Part 2:
C.Where is the median location if the number of customers at A increased from 3 to 7?

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