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Urban Economics FA2023 Problem Set #1 Due, Tuesday, Sep 26, 11pm EST, ONLINE ONLY! 1) No late problem sets 2) You can only upload one single file. No multiple file submissions. 3) We accept only WORD, EXCEL and PDF formats, no JPEG etc. Maximum Points: 50 Question1: 7pts Question 2: 7pts Question 3: 8pts Question 4: 8pts Question 5: 10 pts Question 6: 10pts (1) (1) Von-Thünen Model I A coffee shop is located at M (where x=0) and charges marginal cost prices for its coffee (i.e., c=p). Customers walk to the coffee shop to buy coffee and, aside from the purchase price, incur travel cost of (tx 2 ), measured from the shop (note, this is a squared function). The shop’s market reach is 2 km and t=4. If customers’ WTP for the shop’s coffee is $23.99 what is the product’s price in equilibrium? WTP = p+ tx 2 (note, c=p) 23.99 = p + 4*2 2 à p= 23.99-16=7.99 (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=$12, marginal travel cost is t=2 (per mile and ton) and the market’s WTP for wheat is $100 per ton. a) what is the maximum distance x* at which farmers would grow wheat? b) For each distance point x=10, x=20, and x=50, say whether anybody would grow wheat or not and what the resulting land rent would be. Note, rents cannot be negative.

a) First calculate x* WTP = c + tx 100 = 12 + 2*x è x*=44 b) The rent equals rent = WTP-c-tx and cannot be negative. for x=10, yes, people would grow wheat, rent = 100-12-2*10 = 68 for x=20, yes, people would grow wheat, rent = 100-12-2*20 = 48 for x=50, people would NOT grow wheat, rent = 100-12-2*50 = -12 à the rent equals ZERO (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=0; the market reach is x=14 to the left and x=14 to the right. Marginal transportation cost t equal t=5. (a) What is customers’ WTP per unit of the good in question? (5 pts) (b) What is the rent for a person located at x=12? (3 pts) Answers (needs to be adjusted) (a) WTP=c+tx à WTP=0+14*5 à WTP=70 (b) at x=10 total cost incurred is c+tx=0+5*10= 50 the rent is WTP-p-tx= 70-0-5*12 =10 (4) Von-Thünen Model with multiple cities There are two cities, A and B, where B is located 24 miles east of A. Both produce an identical good and consumers’ WTP=26 is the same everywhere. Marginal transportation cost is identical t B =t A =2. Each city prices its good at marginal cost. We know that c B =4; c A is unknown. (a) What is the maximum market reach of city B to the west? (2 pts) (b) Now assume there are unserved customers to the west of B (i.e., between A and B). Calculate the minimum c A for this to be true. (6 pts) Answer

a) calculate the market reach of city B to the west based on WTP=c+tx. 26=4+2x à x=11 (4 pts) b) since A is 24 miles away, and B reaches 11 miles to its west, A must have a market reach of LESS than 13 miles to its east to leave some unserved customers in between. again, use WTP=c+tx and set x=13 Now plug in 26=c A +2*13 à c=0. Therefore c A must be MORE than $0 (NOTE: c A =0 is incorrect, it must be c A >0) (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=3x 2 , where x is the distance from the resource measured in miles. The firm’s distribution cost is given by DC=(10-x) 2 . Again, x measures 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. Where should the firm locate? (You can do this “by hand” or in a spreadsheet)

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Locate either at x=2 or x=3 x PC DC TC 0 0 100 100 1 3 81 84 2 12 64 76 3 27 49 76 4 48 36 84 5 75 25 100 6 108 16 124 7 147 9 156 8 192 4 196 9 243 1 244 10 300 0 300 (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 2 decimals). What is the optimal x? At this location, what is PC, DC and TC (two decimals). Show your work! Best use calculus since TC = PC + DC ࠵?࠵? = 3 ࠵? ! + (10 − ࠵? ) ! = 3 ࠵? ! + ( ࠵? ! + 100 − 20࠵?) = 4 ࠵? ! − 20࠵? + 100 Build the derivative and set and set equal to zero ࠵?࠵?࠵? ࠵?࠵? = 8࠵? − 20 = 0 solve for x ∗ ࠵? ∗ = 2 .50 The best location is x*=2.50 Here, PC=18.75, DC=56.25, and TC=75 (c) How does your answer to all parts under (b) change if the procurement cost PC were equal to zero everywhere?

The company should locate at x=10 (i.e., at the market) where PC, DC, and TC all equal zero. (6) A pizzeria delivers pizzas to each customer and incurs a cost of $1 per unit and mile. Each unit needs to be delivered separately. The locations and the number of customers are provided in the Table below. Part 1: (a) Where is the median customer located? (b) Where is the delivery cost minimizing location and what is its delivery cost? Location A B C D E F Distance from A (miles) 0 10 8 2 6 18 Number of customers 3 4 1 2 3 4 (a) note: to find the median location, you first need to sort the locations according to their distance from A Location A D E C B F Distance from A 0 2 6 8 10 18 Number of customers 3 2 3 1 4 4 median location There are 17 customers. The median customer is located in C (at D there are 8 customers to the right and 8 to the left). (b) Cost is minimized at C and amounts to to A (8*3=24), D (6*2=12), E(2*3=6), C(0), B(2*4=8),

F(10*4=40) SUM= 90 Part 2: Where is the median location if the number of customers at A decreased from 3 to 1? What is the delivery cost from this location? Location A D E C B F Distance from A 0 2 6 8 10 18 Number of customers 1 2 3 1 4 4 median location à We now have 15 customers and the median customer has 7 customers to his/her left and 7 to his/her right. That is, the median location will be at B. The cost will fall to 72 A (10*1=10), D (8*2=16), E(4*3=12), C(2*1=2), B(0) F(8*4=32) SUM=72 ,

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